Centre for Hospitality and Culinary Arts
MATH 1102 (full-time studies)
and
MATH 1103 (Continuing Education)
MATHEMATICS
FOR
HOSPITALITY
STUDENT WORKBOOK
revised and edited by
Bryan Bessner MBA
May 2020
This edition is dedicated to the memory of my colleague
Jim Cameron
2
Mathematics for Hospitality
Table of Contents
CHAPTER 1 REVIEW OF BASIC ALGEBRA .................................................................................................................................. 5
I. LINEAR EQUATIONS ............................................................................................................................................................................ 5
II. RATIO............................................................................................................................................................................................... 9
III. PROPORTION................................................................................................................................................................................... 10
IV. Word Problems………………………………………………………………………………………………………………………………. 13
V. SIMPLE FORMULAS .......................................................................................................................................................................... .16
CHAPTER 2 PERCENT ..........................................................................................................................................................................26
I. FINDING THE AMOUNT ..........................................................................................................................................................................30
II. FINDING THE RATE AND THE BASE .......................................................................................................................................................31
III. COMPUTING THE NET AMOUNT. .........................................................................................................................................................36
IV. TAX CALCULATIONS………………………………………………………………………………………………………………………………………………………………40
CHAPTER 3 UNITS OF WEIGHT CONVERSION .................................................................................................................... ...... 46
I. CONVERSION INTO OUNCES .................................................................................................................................................................46
II. CONVERSION FROM OUNCES INTO POUNDS AND OUNCES ......................................................................................................................46
III. METRIC CONVERSION .......................................................................................................................................................................47
IV. CONVERSION BETWEEN METRIC AND IMPERIAL SYSTEMS .......................................................................................................................47
CHAPTER 4 UNITS OF VOLUME CONVERSION ...........................................................................................................................54
I. METRIC CONVERSION ...........................................................................................................................................................................54
II. MEASURING THE VOLUME OF FLUIDS ..................................................................................................................................................54
III. IMPERIAL SYSTEM CONVERSION ........................................................................................................................................................ 55
IV. CONVERSION BETWEEN METRIC AND IMPERIAL SYSTEMS .................................................................................................................... .56
V. TEMPERATURE CONVERSIONS…………………………………………………………………………………………………….…61
CHAPTER 5 SIMPLE INTEREST......................................................................................................................... ............................... 65
..
I. DETERMINING RATE AND TIME ............................................................................................................................................................. 66
II. CALCULATING THE AMOUNT OF INTEREST .......................................................................................................................................... .67
III. FINDING THE PRINCIPAL, RATE OR TIME ........................................................................................................................................... .69
IV. ACCUMULATED VALUE..........................................................................................................................................................…………………………………. 74
CHAPTER 6 COMPOUND INTEREST ..............................................................................................………..................................... . 80
CHAPTER 7 HOSPITALITY STATISTICS ...........................................................................................................................................87
I. CENTRAL TENDENCY .......................................................................................................................................................................... 87
II. YIELD MANAGEMENT ....................................................................................................................................................................... ..97
CHAPTER 8 YIELD AND PRICE FACTORS ....................................................................................................................................102
I. %YIELD AND YIELD FACTOR ............................................................................................................................. .................................10-2
II. EDIBLE PORTION QUANTITY (EPQ) ..................................................................................................................................................104
III. AS PURCHASED QUANTITY (APQ)..............................................................................................................................................…...104
IV. EDIBLE PORTION COST (EPC) .........................................................................................................................................................106
V. PRICE FACTOR............................................................................................................................................................................…..107
VI. CONNECTION BETWEEN YIELD AND PRICE FACTORS………………………………………………………………………………….……..108
CHAPTER 9 MENU PRICING.................................................................................................................. .......................................... 115
I FOOD COSTING ......................................................................................................................................................................………………………………….....115
II. MENU PRICE......................................................................................................................................................................................119
III. MARKUP ..........................................................................................................................................................................................122
CHAPTER 10 PROFIT OR LOSS STATEMENT ...............................................................................................................................130
RULE 1 ............................................................................................................................. ................................................................... 131
RULE2............................................................................................................................. .................................................................... 131
3
Mathematics for Hospitality
Table of Contents
APPENDICES
EXERCISESOLUTIONS………………………………………………………………………………………...143
CHAPTER-BASED PROBLEMS USING EXCEL………………………………………………………….…154
4
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Chapter 1 REVIEW OF BASIC ALGEBRA
I. Linear equations
An equation is a mathematical statement indicating that two algebraic expressions are equal to each
other.
Linear equations have one literal symbol (i.e. a letter of the alphabet), representing an unknown
quantity. An equation with only one unknown quantity [and an understood exponent of 1], when
graphed, takes the shape of a straight line, which is why such an equation is called a linear equation.
Another name for them is “equations of the first degree.”
Rule 1:
5x - 1 =
8 + 2x
_____
↑
left
side
______
↑
right
side
Unknown of first
degree
To solve an equation, you must perform the same operation to both sides of the
equation.
That is:
Rule 2:
•
If the same number is added to, or subtracted from both sides of the equation,
the two sides remain equal.
•
If every term on both sides is multiplied or divided by the same number (other
than zero) the two sides remain equal.
To check the solution substitute your answer into the original equation.
Tips: Since equations can appear in many different forms, there is no one procedure for solving all
types of equations. However, the following tips will help you in solving and checking equations:
1.
Multiply both sides by a common denominator to eliminate fractions.
2.
Open all brackets.
3.
Isolate the unknown on one side of the equation. Choose the side where the unknown
will have a positive sign.
4.
Combine all like terms on each side of the equation.
5.
If the unknown has a coefficient, remove it by division.
6.
Check the answer by substituting it into the original equation.
5
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Example 1: Solve the equation
2x = 6
3
6
Mathematics for Hospitality
Solution:
Chapter 1 Review of Basic Algebra
Step1: Multiply both sides by 3, to eliminate the denominator.
(read Tip 1).
3(2x)
3
2x
= 3x6
= 18
Step 2: Divide both sides by 2 (read Tip 5).
2x
2
18
2
or
= 9
x
Step 3: Check the answer by substituting x = 9 into the original equation.
(read Tip 6)
2(9)
3
= 6
18
3
6
= 6
= 6 Equation Checks √
Example 2: Solve the equation 2(3 - x) = 5x - 8
Solution:
Step 1: Open brackets (read Tip 2).
6 - 2x = 5x - 8
Step 2: Isolate unknown on a positive side (read Tip 3).
6 + 8 = 5x + 2x
Step 3: Combine all like terms on both sides (read Tip 4).
14 = 7x
or
7x = 14
Step 4: Divide both sides by 7. (read Tip 5)
7x
14
=
or x = 2
7
7
Step 5: Check the answer in the original equation (read Tip 6).
2(3 - 2) = 5(2) - 8
2(1) = 10 - 8
2
= 2 Equation Checks √
7
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Exercise 1
Solve the following equations:
1.
4x = x+3
2.
2x – 3 = 7 - 3x
3.
3(x - 1) = 2x + 1
4.
3x + 7 = 5x - 3
5.
x + 2(x - 1) = 3(5 - x) - 5
6.
8x - 3(x+2) = 2(x+3)
7.
4x + 2(x-3) = 3(x+3) -2x
8.
3(2x + 1) = 2(x - 3) + 1
8
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
II. Ratio
A ratio shows the relationship between two quantities.
The ratio of a to b has the form
a
b
Another way of writing a ratio is with a colon (:) or with a slash (/). Therefore, the ratio a to b can be
shown as
a : b, or a / b.
Example 3: Find the ratio of 12 to 8.
Solution:
The ratio of 12 to 8 is
12
8
Reducing the fraction by 4, we obtain
3
2
Usually, the quantities in the numerator and the denominator of a ratio are expressed in the same
units, so the units cancel and the ratio is expressed without units. The original units of the problem
no longer matter; we are concerned only with the relationship between the quantities.
Example 4: A man is 45 years old, and his son is 20 years old. Find the ratio of the father’s
age to the son’s age.
Solution:
45 years
9
=
20 years
4
or the ratio of the father’s age to the son’s age is 9 : 4.
(Clearly, the ratio of the son’s age to the father’s age is 4 : 9).
Example 5: The weight of a cantaloupe is 1.5 lb. The weight of an apple is 9 oz. Determine the
ratio of the weight of the cantaloupe to the weight of the apple.
Solution:
Step1: Express both weights in the same unit (ounces).
1.5 lb. = 16 + 8 oz
= 24 oz
Step 2: Form the ratio, and simplify it.
24oz
8
=
9oz
3
or the ratio between the two weights is 8 : 3.
9
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
III. Proportion
A proportion is an equation formed of two ratios that are set equal to each other.
For example, we can easily see that
6
=
3
8
4
If the ratio a : b is equal to the ratio of c : d, we have a proportion
a:b=c:d
This proportion can be presented in a fractional form:
a/b
Rule 3:
=
c/d
To find a missing term in a proportion, cross-multiply the terms of the proportion
and form an equation.
Example 6: Find x, if x : 3 = 8 : 6
Solution:
Step 1: Convert the proportion into fractional form.
x
=
8
3
6
S t e p 2: Cross-multiply the terms.
(6) (x) = (3) (8)
or 6x=24
Step 3: To find X, divide both sides of the equation by 6.
6x / 6 = 24 / 6
or x = 4
Step 4: To check the solution, substitute x = 4 into the given proportion, and
reduce the fraction(s) to the lowest terms.
4/3
=
8/6
4/3
=
4/3
The proportion checks √
10
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Exercise 2
Solve the following proportions:
1.
x:2 = 6:4
2.
3:5 = x:10
3.
x:3 = 8:4
4.
2:x = 5:15
5.
3:4 = 6:x
6.
12:5 = 24:x
7.
3x:8 = 6:2
11
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
8.
2x:8 = 6:2
9.
5:2x = 3:12
10.
3:9 = 2:3x
11.
4:3 = 4x:9
12.
8:x = 4 (bear in mind that 4 can be expressed as 4/1)
13.
5x:8 = 10
14.
x:16 = 1:8
15.
5:9 = 20:3x
12
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
IV. Word problems
We often encounter algebra problems in real-life situations. For example, if you are driving a car
recklessly, and one of your passengers says “You’d better cut your speed in half!” you need to (very
quickly) check your speed and then find an unknown quantity-----the desired new speed.
As a hospitality professional, you will be faced with these situations all the time, whether it is the
calculation of an important hotel statistic, measurements in a recipe, and many other business
situations. Unfortunately, the algebra will not be presented to you as a formal equation, but in the
form of ordinary words which you will have to “translate” into a math format in order to solve.
For example, you might be told that as written, a recipe requires 4 cups of milk, and that the normal
yield of the recipe is 12 portions. However, your supervisor may wish you to alter the recipe so that it
will serve 18 people. How much milk should you include in the changed recipe? You would have to
recognize that this is really a proportion problem, and the equation to solve it is simply
12 / 4 = 18 / x (The answer is 6 cups of milk.)
Here is another word problem:
Example 7:
There is a number such that when seven is added to it, we obtain a quantity one greater than four
times the original number. Find the number.
In this case, two quantities are set equal to each other: the original number plus 7, and 4 times the
original number plus 1. This makes for a nice equation:
x + 7 = 4x + 1
Following the usual steps, we obtain
6 = 3x
or 2 = x
Substituting this value for x into the equation, we end up with
2 + 7 = (4 * 2) + 1
9
=
9
=
8
+1
9
Our translation has been a success!
13
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Exercise 3
Translate the following statements into algebraic equations and solve.
1. There is a number of customers such that when you add 5 to it, the result is 17 customers. Find the
number of customers.
2. If you double a certain number of steaks and then subtract 6, the result is 12 steaks. Find the
original number of steaks.
3. Adding 5 to 3/2 of a certain number of cucumbers gives a result of 11 cucumbers. Find the original
number of cucumbers.
4. Seven less than three times a certain number of avocados is equal to that certain number plus nine.
Find the original number of avocados.
14
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
5. If the sum of an unknown number of forks plus 5 is multiplied by 3, and then that product is divided
by 11, the result is exactly one-half of the original number of forks. Find it (the original number of forks)!
6. Divide me by four, then multiply what you get by two, then subtract four, and you will have ten. Who
am I?
7. Divide a certain number of kilograms by 2, and then add four more kilograms. The result will be 9
kilograms. How many kilograms did you start with?
8. Six ounces of olive oil plus an unknown quantity of ounces is equal to three times the unknown
quantity of olive oil (in ounces). Discover what it is.
15
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
V. Simple formulas
A formula is an equation that contains more than one literal symbol (letter). It may also contain
numbers, but not always.
For instance, a formula used to calculate a simple interest has four literal symbols:
l = PrT
A formula can be solved for any of its symbols. The simple interest formula above is solved
for I, but it also can be solved for P or T, or r.
Since a formula is a Literal equation, when solving a formula for a certain Literal symbol, one should
follow the same rules as for solving a regular equation and treat that symbol as unknown.
Example 8: Solve the formula A = 2B + C for B.
Solution:
Step 1: Isolate the unknown on the positive side (read Tip 4).
A-C = 2B
or
2B = A-C
Step 2: Divide both sides by a coefficient of 2 (read Tip 5).
A-C
B =
2
Example 9: Solve the formula M = R(L-t) for R.
Solution:
There is no need to open brackets in this formula (read Tip 2), since we do not
solve it for L or t. In this case (L - t) is a Literal coefficient for R.
Therefore, divide both sides by (L - t).
M
R(L-t)
=
(L-t)
(L-t)
or
R =
16
M
(L-t)
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
So the secret to solving literal formulas is to treat the letters exactly as if they were numbers.
Just as
2(x + 5) would become
2x + 10
So we see that 2(x + y) becomes
2x + 2y
Similarly, a(b + c) becomes
ab + ac
This and all the other rules must be applied to these formulas, in order to solve them.
17
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Example 10: Solve the formula M = R(L-t) for L.
Solution:
Step 1: Open brackets, since the unknown is inside them.
M = RL- Rt
Step 2: Isolate unknown on the positive side (read Tip 3).
M+Rt = RL
or
RL = M+Rt
Step 3: Divide both sides by the coefficient R.
M+Rt
L =
18
R
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Exercise 4
Solve for indicated literal symbol.
1.
N=(1-D)L
for L
2.
S=P(1+K)
for P
3.
R=
for M
M
C
4.
I = PrT
for P
5.
VR + CR = 1
for VR
6.
D = 2R(C – P)
for R
7.
Y=
EP
AP
for EP
EP
AP
for AP
8.
Y=
19
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Harder problems:
9.
A = 180-(B+C)
for B
10.
C= N+2
R
for R
11.
C= N+2
R
for N
12.
D = 2R(C - P)
for C
13.
D = 2R(C - P)
for P
20
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Review problems
Part 1
1.
2
Solve the given proportions for x:
4:3 = 8:x
15:x = 5 (similar to p. 11, #12)
3.
3:4 = x:2
4.
2:3 = 12:2x
5.
a = x:2 (again, similar to p. 11, #12)
6.
5:3x = 2:12
21
Mathematics for Hospitality
Part II
Chapter 1 Review of Basic Algebra
Solve the following equations:
7.
X+ X = 8
3
8.
2(x-3) = x
9.
6-(x-4) = 4x
10.
x-3(x-1) = 1
11.
4(x + 5) = 5-x
12.
14 - (x+2) = 5x
13.
5x - 2(2x - 3) = 3x
22
Mathematics for Hospitality
Part III
Chapter 1 Review of Basic Algebra
Solve for the indicated letter.
14.
S=PT
for T
15.
T= P(3-V)
for P
16.
PV = NrT
for T
17.
PV = NrT
for r
18.
A=KL
2
for K
19.
M=Pa-L
for L
20.
t=V
a
for a
21.
E=K
S
for K
23
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
Part IV
Translate and solve:
1. There is a number such that when ten is subtracted from it, it loses one fifth of its original value.
Find it.
2. If a recipe that serves 8 people requires 2 cups of sugar, how much sugar is required if we prepare
the recipe for a group of 19 people?
3. One-sixth of a certain number plus ten is equal to one-half of the number. What is it?
24
Mathematics for Hospitality
Chapter 1 Review of Basic Algebra
4. One-sixth of the sum of a certain number plus ten is equal to one-quarter of the number. What is it?
5. Find a number such that if it is multiplied by seven and eight is then added to the result, the new sum
will be equal to eight times the original number.
6. If a certain number is divided by two, the result will be forty-two less than twice the number. Find it.
25
Mathematics for Hospitality
Chapter 2
Chapter 2 Percent
PERCENT
The word “percent” means “per hundred” or hundredths, so percent represents a fraction with the
denominator of 100.
A clear understanding of percentage is very important in the hospitality industry, as it is used in so
many situations.
One percent = 1
100
The “%“ sign is used to replace
Therefore 3% means 3 x 1
100
RULE 1:
_1_
100
or 3 x (0.01) = 0.03 or 3% = 0.03
To convert a percent to a decimal, drop the % sign and divide the remaining number
by 100 (move the decimal point two places to the left).
Example 1: Convert 5.5% to a decimal.
Solution:
5.5 / 100
=
0.055
Therefore, 5.5% = 0.055
RULE 2:
To change a decimal to a percent, multiply it by 100 (move the decimal point two places to
the right) and add the % sign.
Example 2: Convert 0.1325 to a percent.
Solution:
0.1325 x 100% = 13.25%
RULE 3:
Example 3
To change a common fraction to a percent, first divide the numerator of the fraction by
its denominator. Then using rule 2, convert the result to a percent.
Convert the fraction
7 to a percent.
8
26
Mathematics for Hospitality
Solution:
RULE 4:
Chapter 2 Percent
7÷ 8 = 0.875
= 87.5%
To convert a percent to a fraction, first convert it to a decimal form. Then present the
decimal number as a common fraction, and reduce this fraction to its lowest terms.
Example 4: Convert 85% to a fraction.
Solution:
Step 1: Convert 85% to a decimal.
85% = 0.85
Step 2: Present 0.85 as a common fraction.
0.85 =
27
85
100
Mathematics for Hospitality
Chapter 2 Percent
Step 3: Reduce the fraction to the lowest terms.
5 is the common number by which both 85 and 100, are evenly divisible.
85 ÷ 5
100 ÷ 5
17
Therefore, 85% = 20
28
=
17
20
Mathematics for Hospitality
Chapter 2 Percent
Exercise 1
I.
Convert each decimal to a percent:
1.
0.5
2.
0.75
3.
0.25
4.
0.025
5.
0.05
II.
Convert each fraction or mixed number to a percent:
6.
½
7.
¾
8.
3/8
9.
1 5/8
10.
3 3/16
III.
Convert each percent to a decimal:
11.
12%
12.
4%
13.
40%
14.
125%
15.
0.5%
IV.
Convert each percent to a fraction or mixed number:
16.
20%
19.
225%
17.
40%
20
8%
18.
5%
29
Mathematics for Hospitality
Chapter 2 Percent
PERCENTAGE FORMULA
Equation 1:
amount = rate x base
Equation 1 is known as a percentage formula.
In this equation:
•
Rate is the percentage in a decimal form.
(e.g. interest rate, mortgage rate, tax rate)
•
Base is the whole quantity (100%).
•
Amount is a part of a whole (part of the base).
Example 5: $10 is 50% of $20.
In this statement:
$20 is the base (the whole quantity)
50% is the rate
$10 is the amount equal to 50% of the base.
I. Finding the amount
Rule 5: Rate must be expressed as a decimal in the equation.
Example 6: A provincial sale tax is 7%. What is the amount of tax on a $2,200 computer?
Solution:
Step1: Identify the given data:
7% is the rate
$2,200 is the base (whole quantity).
The amount needs to be determined.
Step 2: Calculate the amount of tax:
amount = rate x base
⇓
⇓
⇓
Tax
= 0.07 X 2,200
= $154
30
Mathematics for Hospitality
Chapter 2 Percent
II. Finding the rate and the base
Finding the rate
Example 7: What percent of 68 is 17?
Solution:
Identifying the given data:
17 is the amount
68 is the base
The rate is the value to be found.
From formula 1:
amount
base
Rate =
Rate =
17
68
= 0.25
= 25%
Finding the base
Example 8: $20 is 5% of what quantity?
Solution:
Identify the given data:
$20 is the amount (a part of the whole quantity).
5% is the rate.
The base (the whole quantity) needs to be found.
From formula 1:
amount
rate
Base =
Therefore,
Quantity =
= $400
31
$20
0.05
Mathematics for Hospitality
Chapter 2 Percent
Exercise 2.
Find the amount:
1.
20% of 100
2.
5% of 400
3.
0.5% of 600
4.
3.35% of 2000
5.
4¾ % of 800
6.
120% of 80
7.
What percent of 24 is 12?
8.
What percent of 40 is 8?
9.
10 is 50% of which number?
10.
10 is 25% of which number?
11.
Find 200% of 45.
12.
What percent of 120 is 72?
32
Mathematics for Hospitality
Chapter 2 Percent
13.
6 is 30% of which number?
14.
If 5% of a number is 5, what is the number?
15.
If 25 is 0.5% of a number, what is the number?
16.
12 is what percent of 60?
17.
If 25% of a number is 3.6, what is the number?
18.
5 is 0.5% of which number?
19.
0.14 is what percent of 3.5?
20.
What number is 12.5% of 84?
33
Mathematics for Hospitality
Chapter 2 Percent
Exercise 3.
1.
Find the occupancy rate of a 100-room hotel, when 65 rooms are occupied.
2.
How many rooms are occupied in a 40-room motel, if the occupancy rate is 15%?
3.
How many rooms should be occupied in a 250-room hotel to sustain the occupancy level at
74%?
4.
A 2,000-calorie diet includes 200 g of carbohydrates.
1 g of carbohydrates yields 4 calories.
Calculate:
a)
b)
5.
The amount of calories yielded by carbohydrates.
The percent of calories in the diet coming from carbohydrates.
30% of all calories in a 3,000 calorie diet should come from fat.
1 g of fat yields 9 calories.
Calculate:
a)
b)
the number of calories in the diet coming from fat.
the amount of fat in grams.
34
Mathematics for Hospitality
Chapter 2 Percent
6.
A 2,000 calorie diet should include 150 g of protein. 1 g of protein yields 4 calories. Calculate
the percentage of calories coming from protein.
7.
A restaurant purchased eighty 750 ml bottles and ninety 1-L bottles of wine. What
percent of all wine (by number of bottles) was purchased in 750-ml bottles?
8.
A bar sold 820 drinks in one evening. 35% of which were non-alcoholic. How many nonalcoholic drinks were sold?
9.
A bar sold 612 non-alcoholic drinks during a weekend. Find the total number of drinks sold
through the weekend, if sales of non-alcoholic drinks amounted to 34% of all sales.
10.
A restaurant sells 950 drinks on a weeknight. The nightly weekend sales are 28% better than
on weeknights. How many drinks does this restaurant sell on a weekend night?
35
Mathematics for Hospitality
Chapter 2 Percent
III. Computing the net amount.
The net amount or net price or discount price is the part of an original amount (cost, salary, income,
price) remaining after deductions, discounts, etc. have been removed. It can be calculated by
finding the amount of discount or reduction and then deducting that from the original amount.
Example 9: A set of kitchen knives retails for $60, but was sold at a 20% discount.
Calculate:
a)
b)
Solution:
the amount of discount
the discount (net) price
a) amount of discount
= original price x rate of discount
= $60 x 0.20
= $12
b) discount (net) price
= original price - amount of discount
= $60 - $12
= $48
36
Mathematics for Hospitality
Chapter 2 Percent
Computing a reduced amount or discounted price can be simplified by finding the net amount
percentage.
Referring to Example 9, the solution can be presented as follows:
Original price
$60 =
Amount of discount $12 =
Net price
$48 =
100%
20% of selling price
80% of selling price
The net price percentage, 80%, was obtained by deducting the 20% discount from 100%.
Example 10: The food cost in a snack bar is $2,600. The owner wants to reduce cost by 15%.
Determine the reduced cost.
Solution:
The net cost percentage:
100% - 15% = 85%
Reduced cost
Exercise 4.
1
Calculate the net price:
= original cost x net cost percentage
= (2600)(0.85)
= $2210
Original Price
a)
b)
c)
d)
$72.00
$122.36
$96.00
$49.98
Discount Rate Net Price
15%
25%
12.5%
10.25%
_______
_______
_______
_______
2.
Calculate how many rooms are still available in a 45-room motel, if 20% of the rooms are sold.
3,
Last month, the food cost in a bar was $23,456.00 and the beverage cost was $4,680.00. The
manager wants to reduce the food and beverage cost by 25% and 30% respectively. Calculate:
a) the new food cost
b) the new beverage cost
37
Mathematics for Hospitality
Chapter 2 Percent
4.
The net price of a freezer after a 12% discount is $572.00. Determine the original price.
5.
A hotel offers a 15% discount on all double-occupancy rooms. Determine the original price of a
room, if the discounted rooms sell for $81.60.
38
Mathematics for Hospitality
6.
Chapter 2 Percent
A 3,000 calorie diet includes 15% protein, 55% carbohydrates and 30% fat.
A person wants to reduce their fat intake by 20%. Calculate:
a)
the amount of calories from protein in the diet
b)
the amount of calories from carbohydrates in the diet
c)
the amount of calories from fat in the diet before the change
d)
the reduced amount of calories from fat after the change.
e)
total calories in the new diet
f)
the new percentage of calories from protein
g)
the new percentage of calories from carbohydrates
h)
the new percentage of fat
39
Mathematics for Hospitality
Chapter 2 Percent
IV. Tax Calculations
As of July 1, 2010, Ontario mandated the use of HST (Harmonized Sales Tax), by which the current
rates of 5% federal sales tax and 8% provincial sales tax have been combined into one sales tax rate of
13%. Thus for most products and services purchased, the final selling price includes an add-on of 13%,
which the seller must send to the federal government. Adding 13% to an item’s cost can be done in two
steps, namely calculating the amount of tax (cost x 0.13) and then adding the result to the original cost.
Example 11:
A certain new HD television set costs $1,699 before tax. What is its full price, including HST?
Solution:
$1,699 x 0.13 = $220.87.
And therefore $1,699 + $220.87 = $1,919.87
Thus the full price of the HD television including HST is $1,919.87.
However, the task of adding in the tax can be accomplished more efficiently using only one step.
Since we can treat the original cost of the item as 100%, we are adding an extra 13% to this when we
add the tax. 100% is the same as 1, so with the tax added in, we have a new total of 113%, or (in
decimal form) 1.13. Therefore, multiplying the original cost by 1.13 will provide us with the new price,
HST included, in only one step.
Example 11a:
A certain new HD television set costs $1,699 before tax. What is its full price, including HST?
Solution:
$1,699 x 1.13 = $1,919.87.
Thus the full price of the HD television including HST is $1,919.87. It is the same result as before, but
we arrived at it using only one step.
This process can be reversed as well. If we know the price of an item with the HST already included,
and we want to find out what its original before-tax price must have been, we must simply divide the
tax-included price by 1.13, and we will have what we are looking for.
Example 12:
With HST included, a certain laptop computer costs $1,412.50. What was its price before the HST
was added in?
Solution:
$1,412.50 / 1.13 = $1,250.00.
Thus the computer had a pre-tax cost of $1,250.
40
Mathematics for Hospitality
Chapter 2 Percent
Also, if we are told only the amount of HST, we can easily calculate the original pre-tax cost of the item.
Since we know that the HST represents 13% (0.13) of the amount we are seeking, we need merely to
divide the HST amount by 0.13, a calculation which simply “asks” this question: “If 13% of the original
cost is THIS much, how much was the original value on which the 13% was based?”
Example 13:
The HST charged on a bedroom furniture set is $260. What was the cost of this furniture before the
HST was added in?
Solution:
$260 / 0.13 = $2,000.
Thus the original pre-HST cost of the furniture was $2,000.
And if we wanted to find out the full price charged to the customer, we would simply add the $2,000 and
the $260 together, arriving at an HST-included price of $2,260.
Review Problems
1.
What number is 25% of 84?
2.
42 is 60% of what number?
3.
How much is 16½ % of $200?
4.
$160 is 250% of what sum?
5.
What is 0.2% of $1500?
6.
The original price of a product was $70. The price was reduced by 40%. What was the dollar
amount of the reduction?
41
Mathematics for Hospitality
7.
Chapter 2 Percent
The sales receipts for the lunch shift in a Montreal restaurant last week were:
Items
Number of Sales
A
186
B
62
C
55
D
217
Sales Mix
Calculate the sales mix (percent of total number sold for each item).
8.
The sales tax is 6%. Find the amount of tax on a set of tools with a price of $375.
9.
A frying pan retails for $48.00, but it is usually sold at 15% discount. What is the discounted
price?
10.
A person earning $45,600 pays $11,400 income tax. Calculate the percentage of earnings paid
in taxes.
11.
A waiter makes $11.80 per hour. If the waiter receives a 5% raise, what is the new hourly
rate?
12.
A set of kitchen tools retails for $225. The discounted price is $180. Calculate the rate of
discount.
42
Mathematics for Hospitality
13.
Chapter 2 Percent
A 500-room hotel had some of the rooms closed for renovations last week.
16% of the available rooms were sold on a particular night.
a) How many rooms were available each night last week, if 72 rooms were sold?
b) What percentage of its usual total number of rooms did the hotel have available last
week?
14.
Find the occupancy rate of a 60-room motel, if 9 rooms are occupied.
15.
The winter occupancy rate in a 500-room inn was 52%. Calculate how many rooms are
occupied in the summer, if the summer occupancy rate is 13% higher than in winter.
16.
A 2,500 calorie diet includes 55 g of fat and 200 g of protein. 1 g of fat yields
9 calories, and 1 g of protein yields 4 calories.
Calculate:
a)
the total number of calories from the fat and protein combined
b)
the percentage of calories from the fat and protein in the diet
43
Mathematics for Hospitality
17.
18.
Chapter 2 Percent
A chef purchased 5 lb. of pork loin for dinner. 15% of that weight was inedible fat. After
trimming another 25% (of the new weight) was lost to cooking. Calculate:
a)
the weight of the trimmed meat in pounds and ounces.
b)
the weight of the cooked meat in pounds and ounces.
c)
the number of 3-oz portions that could be served
d)
the overall rate of waste
A restaurant received an 18% discount on a purchase of fifteen 1.14 L bottles of wine that
retailed at $12.50 per bottle.
Calculate :the total cost of the purchase. Include HST on the discounted price.
19.
The yearly average occupancy rate in the Traveler’s Motel was 65%. The motel manager
wants to have at least 68 out of total 80 rooms occupied through the year. By how many
percentage points should the occupancy rate be increased in order to achieve this goal?
44
Mathematics for Hospitality
Chapter 2 Percent
20.
The Blue Star Motel sells a single room for $75, less 12%. The nearby Cozy Corner motel sells
a single room for $70, less 10%. What additional discount must the Blue Star give to meet the
competitor’s price?
21.
A restaurant supply shop is selling a case of 300 stainless steel teaspoons for $450. Find the
amount of HST that will be added to this item, and the final price to the buyer, HST included.
22. a. If the HST charged on a given item is $72.80, what was its original cost, before HST was added
in?
b. Find the full price of the item, HST included.
23. The full price of an elaborate lighting system setup for a special event, including HST, is $4,746.00.
Find the original pre-HST cost, as well as the amount of HST added in.
45
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
Chapter 3 UNITS OF WEIGHT CONVERSION
1 lb = 16 oz
1 kg = 1000 g
1 lb = 454 g
I. Conversion into ounces
Conversion factor: 16 oz/lb.
Example 1: Convert 3 lb. 9 oz into ounces.
Solution:
To convert pounds into ounces, multiply by a conversion factor of
16 oz/lb.
3 lb 9 oz = 3 x 16 oz/lb. + 9 oz
= 48 oz + 9 oz
= 57oz
II. Conversion from ounces into pounds and ounces
Conversion factor: 16 oz/lb.
Example 2: Convert 95 oz into pounds and ounces.
Solution:
Step 1: To find the number of pounds, divide the total number of ounces by a
conversion factor of 16 oz/lb.
95 oz
16 oz/lb.
= 5.9375 lb
There are 5 whole pounds in 95 oz.
Step 2: To find the remaining number of ounces, multiply the decimal part by a
conversion factor of 16 oz/lb.
0.9375 lb x 16 oz/lb.
Therefore,
= 15 oz
95 oz = 5 lb 15 oz
46
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
III. Metric conversion
Conversion factor: 1000 g/kg
-
Example 3: Convert 2 kg 560 g (2.56 kg) into grams.
Solution:
To convert kilograms to grams, multiply the number of kilograms by a conversion
factor of 1000 g/kg.
2 kg 560 g = 2 kg x 1000 g/kg + 560 g
= 2000 g + 560 g
= 2560 g
Example 4: Convert 220 g into kilograms.
Solution:
To convert grams to kilograms, divide the total number of grams by a conversion
factor of 1000 g/kg.
220 g
1000 g/kg
= 0.22 kg
IV. Conversion between metric and imperial systems
Conversion factor: 454 g/lb.
Example 5
Convert 3 lb. into kilograms.
Solution:
Step 1: Find the total number of grams by using a conversion factor of
454 g/lb.
3 lb x 454 g/lb
= 1362 g
Step 2: Convert weight in grams to kilograms.
1362 g
1000 g/kg
= 1.362 kg
47
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
Example 6: Convert 2 kg 497 g (2.497 kg) into pounds and ounces.
Solution:
Step 1: Convert the total weight to grams. Refer to Example 3.
2 kg 497 g = 2497 g
Step 2: Convert weight in kilograms to pounds by using a conversion factor of 454 g/lb.
2497 g
= 5.5 lb
454 g /lb
Step 3: Convert 5.5 lb to pounds and ounces.
5.5 lb = 5 lb 8oz
Example 7: Convert 15 oz into grams.
Solution:
Step 1: To find the number of grams in one ounce, divide 454 g by 16 (by the
number of ounces in a pound).
454 g
= 28.375 g
16
= 28.38 g rounded out to two decimal digits
= 28.4 g rounded out to one decimal digit.1
Step 2: Convert 15 oz into grams. 2
28.349 g x 15 = 425.235 g
1
2
If you need to convert 1oz into grams, it is practical to round out 28.4 g to 28 g.
If you need to convert 5 or more ounces into grams, do not truncate the decimal part. In this case the decimal part
could accumulate to several grams
48
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
Exercise 1
I. Convert into ounces
1.
2 lb
2.
7.5 lb
3.
2 lb 5 oz
4.
15 lb
5.
4 lb 7 oz
II. Perform the following operations and express your answer in pounds and ounces.
6.
2 lb 5 oz +1 lb 12 oz
7.
3 lb 6 oz
8.
3 lb 6 oz - 1 lb 12 oz
9.
4 lb 5 oz - (2 lb 9 oz
+
15 oz
+
1 lb 3 oz
+
1 lb 6 oz)
49
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
10.
2 lb 6 oz x 3
11.
1 lb 7 oz x 4
12.
3 lb ÷ 4
13.
5 lb 4 oz ÷ 3
14.
4 lb 6 oz ÷ 5 + 1 lb 7 oz x 2
15.
(3 lb 13 oz - 1 lb 15 oz) ÷ 2 + 2 lb 6 oz x 3
50
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
Exercise 2
I.
Convert the following into grams.
HINT First find the number of grams in one ounce.
16.
1 oz
17.
10 oz
18.
16 oz
19.
1 lb 6 oz
20.
2 lb
II,
Perform the following operations and express your answer in grams and kilograms.
21.
2 lb + 650 g
22.
3 kg ÷ 8 + 135 g x 5
III.
Perform the following operations and express your answer in pounds and ounces.
23.
2.27 kg + 2 lb 12 oz
24.
(8 kg 625 g – 1 kg 815 g) ÷ 3 + 1 lb 14 oz ÷ 2
51
Mathematics for Hospitality
Chapter 3 Units of Weight Conversion
Review problems
1.
A recipe calls for 1 lb 10 oz of shortening, 1 lb of sugar, 1 lb 9 oz of cake flour, 8 oz of water,
and 5 oz of eggs. Calculate the total weight of ingredients.
2.
A chef added two 75 g packages of walnuts to the recipe from question 1. Calculate the new
weight in grams and kilograms.
3.
A recipe calls for 2 lb 2 oz of paste “A”, 2 lb l4 oz of sugar, 12 oz of cake flour, 8 oz of bread
flour, and 1 lb 4 oz of egg whites. The recipe yields 24 servings, but only 18 servings are desired.
a)
Calculate the total weight of ingredients in ounces.
b)
Calculate the weight of one serving.
c)
Determine the weight of 18 servings in pounds and ounces.
52
Mathematics for Hospitality
4.
Chapter 3 Units of Weight Conversion
Convert the weight of each ingredient in question 3 into kilograms.
oz
kg
paste “A”
sugar
cake flour
bread flour
egg whites
a)
Calculate the total weight of ingredients in kilograms.
b)
Calculate the weight of one serving in grams.
c)
Calculate the weight of 18 servings in grams and kilograms.
53
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Chapter 4 UNITS OF VOLUME CONVERSION
I. Metric conversion
A system of prefixes is used in metric system to denote the conversion factor between units. The first
letter of a prefix symbolizes the conversion factor.
Common Metric Prefixes
Prefix Symbol Conversion factor
kilo
k
1000
deci
d
centi
c
milli
m
micro
µ
(thousand)
1
(one tenth)
10
1
(one hundredth)
100
1
1000
1
1000000
(one thousandth)
(one millionth)
Using metric prefixes, we can easily conclude that:
1 dl
(1 decilitre)
1
means
10
1
1cm (1 centimeter) means
1kg
of 1L (litre)
of
1m
(meter)
100
(1 kilogram) means 1000 g (gram)
II. Measuring the volume of fluids
Measuring volume requires two different units: one for solid substances and one for fluids. In the
metric system they are:
1m3 for solids.
1L for fluids.
1m3
=
1000L
1L
=
1000 ml
54
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
III. Imperial system conversion
The base imperial unit of the volume of fluid is one fluid ounce. It is widely used in the beverage
industry. For instance, one standard drink has a volume of 1 fl oz
1 gallon = 160 fl oz
1 quart
= 40 fl oz
1 pint
= 20 fl oz
It might be useful to visualize these relationships:
1 gallon
1
1 quart
1
pt.
1
Please
pin
gallon
1 quart
1
pt.
1
pt.
1 quart
1
pt.
1
pt.
1 quart
1
pt.
1
pt.
1
pt.
note that a “standard drink” contains 1 fl. oz. of the alcoholic liquid.
t
Example 1: Convert 3.4 gal into ounces.
Solution:
3.4 gal x 160 oz/gal = 544 fl oz
55
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Example 2: Convert 1030 fluid ounces into pints.
Solution:
1030 fl oz. = 51.5 pints
20 fl oz/pint
IV. Conversion between metric and imperial systems
28.41 mI/fl.oz is the primary conversion factor between the two systems.
1 fl oz = 28.41 ml
WINES
LITRE
MILLILITRES
FL OZ
1
1000
35.2
¾ litre
750
26.4
3/8 litre
375
13.2
1/10 litre
100
3.5
56
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Example 3: Determine how many standard drinks of one ounce are contained in one 1.14L
bottle.
Solution:
Step 1: Convert volume from litres to millilitres.
1.14 L x 1000 ml/L
= 1140 ml
Step 2: To convert millilitres into ounces, divide the total number of ml by a
conversion factor of 28.41 ml/oz
1140 ml
28.41 ml/oz
= 40 fl oz
Step 3: Determine the number of standard drinks.
40 oz
1 oz
= 40 drinks
If the drinks served are extra strong (1.5 fl. oz.), how many drinks will be in the
1.14 L bottle?
40 oz.
1.5 oz.
but we can only serve 26!
57
= 26.7 drinks,
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Exercise 1
Convert into fluid ounces:
1.
2L
2.
1.5 L
3.
750 ml
4.
500 ml
5.
1 L 200 ml
6.
1.5 gallons
7.
2 gallons and 3 pints
Perform the following operations. Express your answers in fluid ounces.
8.
1.14 L x 3 + 0.5 pint
9.
5 gallons - 3 pints
10.
3 gallons + 4 quarts - 1 pint
11.
(1 gallon ÷ 5) x 12
58
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Perform the following operations. Express your answer in litres and millilitres.
12.
2250 ml + 150 ml + 1050 ml
13.
1750 ml + 350 ml x 3
14.
3 L – 1650 ml ÷ 3 + 10 fl oz x 4
15.
4 L + 12 pints - 750 ml
16. Find the total volume of the following recipe:
fl oz
Sunflower oil
3 fl oz
White wine
170 ml
Wine vinegar
170 ml
Reduced stock
375 ml
Red currant jelly
3 x 20 ml
Total
59
ml
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Exercise 2
1.
A bartender serves 1,200 standard drinks per night. Determine the number of 750 ml bottles of
alcohol the bartender has to order.
2.
A bartender had three full 750 ml bottles and 400 ml of liquor left in each of two other bottles.
How many 2 oz drinks could the bartender serve from that amount of liquor?
3.
A bar sells 1,000 standard drinks per weeknight (Mon. – Thurs.) and 3,200 drinks per night
on weekends (Fri., Sat. and Sun.). Determine the number of 1.14 L bottles of liquor required
for the whole week.
4.
A bar owner ordered a 58.7 L keg of draft beer. Calculate how many pint glasses of beer could
be served from the keg.
5.
A recipe calls for two 2-oz ladles of canned tomatoes. The recipe yields eight servings.
Calculate how many 385 ml cans of tomatoes should be ordered to serve 200 portions.
6.
A recipe calls for one (15 ml) tablespoon of olive oil per portion. How many portions can be
prepared if a chef has two 16 oz cans of olive oil?
60
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
IV. Temperature Conversions
There are two systems of temperature measurement in use today, and it is important that anyone
involved in the preparation of food be able to work with either system, and to convert in either direction
between them. They are: the Fahrenheit system and the Celsius system.
The Fahrenheit system, presented to the world by the German scientist Gabriel Daniel Fahrenheit in
1714, uses degrees of heat that are quite small. In the Fahrenheit system, water freezes at 32 degrees,
while it boils at 212 degrees, a difference of 180 degrees.
The Celsius system is also named for its inventor, the Swedish scientist Anders Celsius. He introduced
his system in 1742, in which the “distance” in heat between the freezing and boiling points of water
spans only 100 degrees, from 0 to 100. For this reason, the Celsius system is often referred to as the
Centigrade system, the “cent” part of the word referring to the one hundred degrees just mentioned.
Since the Celsius system only uses 100 degrees where the Fahrenheit system uses 180, it is obvious
that Celsius degrees are larger than Fahrenheit degrees. As a result of that, the same amount of heat
will be recorded by a lower number in Celsius than in Fahrenheit. For example, a hot summer day in
Fahrenheit is approximately 90 degrees, while that same amount of heat in Celsius is only
approximately 30 degrees.
In Canada and in European countries Celsius is used, but we also purchase many American products
and cookbooks, and the cooking directions for these are often given in Fahrenheit, the system
commonly used in the U.S.
To convert from Celsius to Fahrenheit, follow these steps (remember that your new answer must be
numerically bigger):
1. Multiply the Celsius temperature by 1.8.
2. Add 32 to your new answer.
Let’s convert 20°C into °F:
1. 20 x 1.8 = 36.
2. 36 + 32 = 68°F
To convert from Fahrenheit to Celsius, follow these steps (remembering that your new answer must
be numerically smaller):
1. Subtract 32 from the Fahrenheit temperature.
2. Divide your answer by 1.8.
Let’s convert 68°F. into °C:
1. 68 – 32 = 36.
2. 36/1.8 = . 20°C, which is where we started, so we know these methods work.
61
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Exercise 3
1. Convert 10°C. into °F.
2. Convert 108 °F. into °C.
3. Which is actually colder, 0°F. or 0°C.? Prove your answer mathematically.
4. Which is actually hotter, 25°C. or 80°F.? Prove your answer mathematically.
5. There is a point (that would never be used in cooking) at which the two temperature systems meet,
i.e., the same number in both systems would express the same amount of heat. What is it?
62
Mathematics for Hospitality
Chapter 4 Units of Volume Conversion
Review Problems
1.
Divide the following sauce recipe in half:
Half-scale version
ml
meat glaze
450 ml
red wine
220 ml
tomato sauce
180 ml
sunflower oil
fl oz
1 fl oz
2.
A chef uses 2 fl oz of the sauce in question 1 for one portion. How many portions can be
served, if the chef doubles the original recipe?
3.
Determine the weight of a 12-bottle case of wine, if a bottle is 1/5 of a gallon, and a pint of
grape wine weighs 1 lb. Do your work in the following order:
4.
a)
convert the volume of one bottle into fluid ounces
b)
find the total volume of wine in a case, expressed in fl oz.
c)
convert the volume from fl oz into pints.
d)
find the total weight of the wine in the case.
Twelve 750 ml bottles of wine were used for a party.
a)
How many litres of wine were used?
b)
How many 3 oz glasses could be served from that volume of wine?
63
Mathematics for Hospitality
5.
6.
Chapter 4 Units of Volume Conversion
A chef uses one 28-ml ladle of the sauce from question 1 for one portion. 325 portions are
supposed to be served for dinner.
a)
How many ounces of sauce should be prepared?
b)
How many times should the original recipe be increased?
A 58.7 L keg of beer costs $127.86. Determine:
a)
how many 8 oz portions of beer could be sold from it.
b)
the cost of an 8 oz. portion (not the price).
7.
A bartender sold 20% of the contents of a 58.7 L keg of beer. If the bartender pours
11 oz of beer into a 12 oz glass, creating 1 oz of head-foam, how many glasses could be sold
from the remaining beer in the keg?
8.
A restaurant bought domestic Riesling wine at $6.40 per bottle. Each bottle contains
5 pints. The wine is served in 5 oz. glasses. The restaurant is to serve a party of 120
guests. Each guest will receive one glass of wine.
Determine:
a)
The restaurant’s cost for one glass of wine.
b)
How many litres of wine will be required for the party?
c)
How many bottles of wine should be ordered?
64
Mathematics for Hospitality
Chapter 5 Simple Interest
Chapter 5 SIMPLE INTEREST
Interest
=
Principal x Rate x Time
I = PrT
The amount of simple interest is calculated by the formula I = PrT,
Where
P
is the principal sum of money earning the interest
I
is the amount of interest earned
r
is the annual rate of interest (rate of money growth)
T
is the term time period - in years.
—
Although the annual rate of interest r is measured in percent, it should be used in the formula
as a decimal.
Example 1: Interest rate is 8¾ % p.a. (per annum, meaning per year). Find r.
Solution:
8¾ % = 8.75%
Therefore, r = 0.0875
The term T may be stated in years, months or days. However, since r is the annual rate, it is
essential to substitute time in years into the formula. This will require conversion of months
or days into years.
•To convert months into years, divide number of months in a term by 12.
Example 2: a) the term is 6 months:
T=
6
12
T=
11
12
b) the term is 11 months:
year
(or 0.5 year)
year
• To convert days into years, divide number of days in a term by 365.
Example 3: a) the term is 90 days
T =
90
365
year
b) the term is 122 days
T=
122
365
65
year
Mathematics for Hospitality
Chapter 5 Simple Interest
I. Determining rate and time
Exercise 1
State r and T for each of the following:
Rate
r
Time
1.
12½%
1¾ years
2.
9¾%
21 months
3.
10.25%
165 days
4.
75/8 %
1¼ years
5.
83/16%
7 months
6.
0.8%
93 days
7.
12.85%
48 months
8.
7¼%
120 days
T
66
Mathematics for Hospitality
Chapter 5 Simple Interest
II. Calculating the amount of interest: I = PrT
Example 4: Find the amount of interest earned on $1,200 in 3 years at
the rate of 5¼ % p.a.
Solution:
Step 1: Identify the elements of the formula.
P
rate
T
=
=
=
$1,200
5.25%
3 years
therefore r = 0.0525
Step 2: Calculate the amount of interest.
I = 1200 x 0.0525 x 3
I = $189
Example 5: Find the amount of interest earned on $900 in 2 years and 2 months at the
rate of 6.35% p.a.
Solution:
Step 1: Identify the elements of the formula.
P
r
T
=
$900
=
0.0635
=
2 years and 2 months
Since the term is given in mixed units, express it in months:
Term = 2 x 12 months + 2 months
= 26 months
Convert the term in months to years:
26 years
12
67
Mathematics for Hospitality
Chapter 5 Simple Interest
Step 2: Calculate the amount of interest.
I = $900 x 0.0635 x 26
12
= $123.825
= $123.83
Exercise 2
Calculate the simple interest for each of the following:
1.
$2,000 at 8% p.a. for 2 years.
2.
$350 at 10.5% p.a. for 5 years.
3.
$880 at 9.5% p.a. for 2¼ years.
4.
$1,200 at 8.75% p.a. for 7 months.
5.
$960 at 5 ¾ % p.a. for 16 months.
6.
$1,100 at 7.75% p.a. for 60 days.
7.
$1,600 at 10¼ % p.a. for 120 days.
8.
$1,000 at 4.75% p.a. for 165 days.
68
Mathematics for Hospitality
Chapter 5 Simple Interest
III. Finding the principal, rate or time
To find the principal, rate, or time, use the following derived formulas:
Principal P =
I
rT
Rate r =
I
PT
Term T =
I
rP
When the term I is given in months or days, the formulas are modified to:
For the term in months:
Principal
Rate
I x (12)
(r)(T in months)
P =
r =
I x (12)
(P)(T in months)
For the term in days:
Principal
Rate
P =
r =
69
I x (365)
(r)(T in days)
I x (365)
(P)(T in days)
Mathematics for Hospitality
Chapter 5 Simple Interest
Example 5: What principal will earn $105 simple interest in 9 months at 7% p.a.?
Solution:
For the term in months:
$105 𝑥 (12)
0.07 𝑥 9
P =
= $2,000
Example 6: Find the annual rate of interest required for $6,000 to earn $39.45 in
30 days.
Solution:
For the term in days:
$39.45 𝑥 (365)
$6000 𝑥 30
r =
= .08
or 8%
Exercise 3
Calculate the missing term for each of the following:
1.
$Principal
Rate
9,000
9.5% p.a.
1.75 years
8% p.a.
2 years
960.00
3 years
675.00
2.
3.
3,000
4.
2,500
6% p.a.
5.
4,000
7% p.a.
6.
2,000
9% p.a.
Time
$lnterest
225.00
9 months
90.00
70
Mathematics for Hospitality
7,
10% p.a.
8.
5,000
9.
7,000
10.
8,000
11.
Chapter 5 Simple Interest
6.5% p.a.
3 months
30.00
6 months
206.25
90 days
8% p.a.
71
60 days
118.36
120 days
177.53
Mathematics for Hospitality
Chapter 5 Simple Interest
Exercise 4
1.
What principal will earn $57.40 simple interest at 10.25% p.a. in 8 months?
2.
What principal will earn $71.99 simple interest at 9¾ % p.a. in 245 days?
3.
In how many months would an investment of $1,200 earn $47.25 in simple interest at 6.75%
p.a.?
4.
Find the annual rate of interest required for $1,600.00 to earn $59.18 in simple interest in 120
days.
5.
Calculate the number of years required for $745.00 to earn $178.80 in simple interest at 8%
p.a.
6.
How many days are needed for $1,500 to earn $69.04 at 10½% p.a.?
72
Mathematics for Hospitality
Chapter 5 Simple Interest
7.
Find the annual rate of interest required for $744.00 to earn $75.95 in 14 months.
8.
What is the interest on $4,000 in 2 years and 3 months at 6.5 % p.a.?
9.
Find the principal that will earn $67.80 in 219 days at 5.65% p.a.
10.
What is the interest on $4,000 invested for 2 years and 2 months at 10.5%?
11.
Find the annual rate of interest required for $1,350 to earn $35.10 in 146 days.
73
Mathematics for Hospitality
Chapter 5 Simple Interest
IV. ACCUMULATED VALUE
S = P (1 + rT)
Accumulated value is a mathematical term for what we know as the maturity value of an
investment. Since a maturity (accumulated) value is the sum of the original invested principal and
the simple interest it has earned, we will use capital letter S to denote the accumulated value.
Accumulated value S =
P+I
Since I = Prt,
S =
P + PrT
or
S =
P (1+rT)
(1 + rT) — is the accumulation factor of one dollar at simple interest.
Example 7: Calculate the accumulation factor and find the accumulated value of $5,000 invested at
8.5% p.a. for 9 months.
Solution:
1) Accumulation factor:
(1 + rT) = (1 + 0.085 x 9)
12
= 1.06375
2) Accumulated value:
S = P x (1 + rT)
= $5,000 x 1.06375
= $5,318.75
Example 8: Find the principal that will grow to $1,247.50 in 5 months at 9.5% p.a.
Solution:
1) Accumulation factor :
(1 + rT) =
(1 + 0.095 x 5)
12
= 1.039583
2) From formula 3:
Principal P
=
74
S
(1 + rT)
Mathematics for Hospitality
Chapter 5 Simple Interest
= $1,247.50
1.03953
= $1,200.00
Exercise 5
For each of the following calculate:
a) the accumulation factor (1 + rT)
b) the accumulated value S
$Principal
Rate
Time
1.
2,000
9% p.a.
2 years
2.
4,500
8% p.a.
1.5 years
3.
7,000
8.5% p.a.
9 months
4.
1,400
12% p.a.
180 days
5.
2,500
8.75% p.a.
60 days
6.
2,000
6.375% p.a.
1¼ years
7.
13,000
5.5% p.a.
7 months
8.
125,000
8¾ % p.a.
193 days
(1 + rT)
75
$S
Mathematics for Hospitality
Chapter 5 Simple Interest
Exercise 6
1.
Find the accumulated value of $4,000 invested for 16 months at 6.75%.
2.
Calculate the maturity value of $12,000 invested for 90 days, at 11¾% p.a.
3.
Calculate the accumulated amount of $480 invested for 220 days, at 12½% p.a.
4.
What is the amount to which $1,550 will grow in 11 months at 14%?
5.
Find the amount to which $900 will mature in 1.5 years, at 7¾% p.a.
6.
Find the principal that will accumulate to $6,880.90 in 204 days at 6.75% p.a.
7.
Find the principal that will accumulate to $2,627.08 in 3 years and 7 months at the rate of
8¾% p.a.
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Mathematics for Hospitality
Chapter 5 Simple Interest
Review problems
1.
Find the exact interest earned by $2,580 at 10.25% p.a. in 153 days.
2.
Find the annual rate of interest at which $2,500 could earn $156.25 in 6 months.
3.
Determine the principal that will mature to $20,000 at 8.75% p.a. in 3 months.
4.
Find the accumulated value of $4,400 invested for 10 months at 6¼% p.a.
5.
What principal will earn $134.28 interest at 12.5% in 182 days?
6.
In how many months will $4,500 grow to $4,687.50 at 5% p.a.?
7.
Find the number of days needed for $1,692 to earn $67.17 at 11.5% p.a.
77
Mathematics for Hospitality
Chapter 5 Simple Interest
8. A certain credit card company charges interest at 18¾% p.a. Mark M. had an
outstanding balance of $686.25 on his credit card for 35 days, at which time he paid
everything that he owed to the company. How much interest did Mark have to pay, and how
much did he pay in total?
9. Paul A. bought a $500 regular interest Canada Savings Bond (CSB). A regular interest
CSB pays simple interest on an annual basis. Paul’s bond pays 4% p.a. He plans to keep
the bond until it matures eight years from the purchase date.
a. How much interest will Paul earn from this bond in the first year?
b. Calculate the interest for each individual year of the bond’s term, and then find the total
accumulated interest that Paul will receive by the end of the term.
10. Tammy V. cashed in a $1000 regular interest CSB that was earning 5.8% p.a., and
received $1,348 in total. For how long had Tammy been holding this bond?
78
Mathematics for Hospitality
Chapter 5 Simple Interest
11. A term deposit owned by Andy H.. was locked in for 6 months at 3.5% p.a., and paid
$91.00 interest at maturity. Calculate the original principal that Andy must have invested.
12. Donnalu M. had $254.73 in her savings account from June 1 to August 1. She earned
$1.30 in interest during that time. The interest is calculated on a daily basis. What annual
rate of interest is receiving from investments in this account?
13. Joanne G. had $3,750.87 in her savings account on January 1. During the month of
January, she made no deposits or withdrawals.
Use these tiered rates:
Interest rate
0.20%
1.15%
3.29%
4.50%
Balance
0 - $4,999.99
$5,000 - $24,999.99
$25,000 - $49,999.99
$50,000 or more
a. How much interest did Joanne earn on this account for the month?
b. What was the new balance in Joanne’s account as of February 1?
79
Mathematics for Hospitality
Chapter 6
Chapter 6 Compound Interest
COMPOUND INTEREST
Consider $100.00 invested at 8% p.a. for 10 years.
If simple interest is paid at the end of each of the ten years, the $100.00 will grow by $100 x .08 =
$8.00 per year. That is a total of $8.00 X 10 years = $80.00 interest. Thus, the $100 will have grown
to an accumulated value (principal + interest) of $100 + $80 = $180.00.
If the interest is paid annually, and the interest is added onto the initial $100.00, so that interest can
earn interest also, year by year, then each year the amount of new interest will be greater than
that of the previous year.
In fact, we know that the $100.00 would grow during the ten years to precisely $215.89 This is a
difference of $35.89. We say that the interest in this example has been “compounded” (mixed
together with principal and reinvested). Since this has happened on an annual basis, we say that
the interest has been compounded annually.
How do we know the precise future value of the investment ahead of time?
The formula for the calculation of a compounded amount is given by: A = P (1 + i) n
Where:
A
is the accumulated amount (future value), when compound interest is used
P
is the principal
i
is the interest rate per period
n
is the number of interest periods
Also:
r
t
f
is the annual interest rate
is the length of the investment in years
is the frequency of compounding per year
Before applying the formula A = P(1 + i)n, we must properly evaluate both i and n.
Obviously, if the compounding is happening more frequently than annually, the interest being added in will
be correspondingly less. For example, if an annual interest rate of 10% is being compounded semiannually
(twice per year), the interest being given will be only 5% per half-year period. So i = r/f.
Also, the value for n is dependent on the frequency of compounding as well. If the time (t) of an investment
were 3 years, but the compounding is happening quarterly (f = 4), then n = 3 x 4 = 12.
Thus, in our simpler example, P = $100, I = .08, and n = 10, and so
A = $100(1 + .08) 10
When we work this through, A = $215.89.
80
Mathematics for Hospitality
Chapter 6 Compound Interest
Please note that before annual compounding, the future value of the investment was only
$180.00, which is less than double the original $100 value. But with compounding, the future
value grows to more than double the original value, or $215.89. The extra $35.89 may not sound
like much, but remember that all of this is based on an original investment of only $100. The
difference between simple interest and compound interest becomes more dramatic the larger your
initial investment. Also, greater compounding frequency will increase the amount, even though the
overall length of the investment remains the same.
Please note that:
If interest is compounded annually, f =1
If interest is compounded semi-annually, f = 2
If interest is compounded quarterly, f = 4
If interest is compounded monthly, f = 12
If interest is compounded daily, f = 365
However, an increase in compounding frequency causes a balanced reduction in the percentage
of interest given each period. In other words, while n (f x t) grows as f increases, I (r/f) will
decrease as f increases. Whatever factor you are multiplying by to obtain n will also be the factor
by which you are dividing to find i. With daily compounding, for example, interest is being added to
the principal every single day, and the new amount will attract the next day’s interest. But exactly
how much interest IS that?
Consider:
0.08/365 = 0.000219178 per day, or 0.0219178%
0.12/365 = 0.000328767 per day, or 0.0328767%
Nevertheless, more frequent compounding does indeed help the investor earn more interest. In
our example, if the compounding were done on a daily basis (f = 365), then
I = 0.08/365 = .000219178, and n = 10 x 365 = 3,650.
Applying these to our compound interest formula, we obtain:
A = $100 ( 1 + .000219178) 3,650
So A = $222.53
The daily compounding factor (as opposed to annual) has allowed the investor to gain an extra
$6.64 ($222.53 – 215.89). Again, this may not sound like much, but we began with only $100.
81
Mathematics for Hospitality
Chapter 6 Compound Interest
EXAMPLE 1:
Investigate how $1,000 will grow over 10 years if it is compounded (i) annually (ii) semi-annually and
(iii) quarterly at interest rate of 8% p.a.
Solution:
(i) Annually
Now
A
P
f
r
t
i
n
=
=
=
=
=
=
=
?
$1,000
1
0.08
10 years
r/f = 0.08/1 = 0.08
t x f = 10 x 1 = 10
Thus A = $1,000 x (1 + 0.08)10
= $1,000 x 1.08x 1.08x1.08x1.08x1.08x1.08x1.08x1.08x1.08x1.08
= $2,158.90
(ii) Semi-annually. Here ‘f’ is different, thus ‘i’ and ‘n’ will be different.
(iii) Quarterly
f
=
2
i
=
r/f = 0.08/2 = 0.04
n
=
t x f = 10 x 2 = 20
Thus A
=
$1,000 x (1 + 0.04) ²º
=
$1,000 x 1.04 ²º
=
$1,000 x 2.191123143
=
$2,191.12
Here f
i
= 4
= 0.08/4=0.02
n = 10 x 4=40
Thus A = $1,000 x (1 + 0.02)40
= $1,000 x 1.0240
= $1,000 x 2.208039664
= $2,208.04
82
Mathematics for Hospitality
Chapter 6 Compound Interest
EXAMPLE 2:
The bank rate seldom remains stationary. Currently, it may change weekly according to the rate
quoted by the Bank of Canada.
Consider a deposit of $2,000 that is earning interest at 10.5% p.a. compounded quarterly. After two
and a half years, the interest rate is changed to 12% compounded monthly. Find the amount
accumulated after six years.
Solution:
Initially
p
f
t
i
n
=
=
=
=
=
$2,000
4
2.5 year
0.105/4 = 0.02625
2.5 x 4 = 10
Then
A
=
=
=
=
2,000 x (1 + 0.02625)10
2,000 x 1.0262510
2,000 x 1.295781279
$2,591.56
After 2.5 yrs
p
f
t
i
n
=
=
=
=
=
$2, 591.56
12
3.5 year
0.12/12 = 0.01
3.5 x 12 = 42
Then
A
=
=
=
=
2,591.56 x (1 +0.01)42
2,591.56 x 1.0142
2,591.56 x 1.518789895
$3,936.04
Notice that the first accumulated amount became the principal of the second accumulated amount.
83
Mathematics for Hospitality
Chapter 6 Compound Interest
Table for Calculating Compound Interest
The following table will help you to organize the calculation of the amount accumulated for
compound interest growth:
Principal
Rate
Time
Frequency
Number of
Periods
(n)
Amount
(f)
Interest
Period
(i)
(P)
(r)
(t)
$
express as a
decimal
Year
times/year
i = r/f
n=txf
A = P (1 + i)ⁿ
(A)
EXERCISE 1
1. Find the compounded amount for each of the following;
Principal
Nominal
Rate
Time
Frequency
10%
8 yr
Annually
$1,000.00
8%
6 yr
Semi-annually
$1,250.00
6%
4 yr
Quarterly
$1,600.00 7.5%
2 yr
Monthly
$480.00
6.5 yr
Semi-annually
$400.00
6.5%
f
i
n
A
2. Dario G. owns a five-year term deposit of $5,000.00 with interest at 6.5% p.a.
compounded semi-annually. Find its maturity value and the amount of interest Dario will
earn.
3. Roy’s father made a trust deposit of $1,000.00 on October 31, 1978 to be withdrawn on Roy’s
18th birthday, July 31, 1996. What would the deposit be worth on that date if interest was
compounded quarterly throughout the length of the investment at 10% p.a.?
84
Mathematics for Hospitality
Chapter 6 Compound Interest
4. The Canadian consumer price index was approximately 200 at the beginning of 1980. If inflation
continued at an average annual rate of 10%, what would the index have been at the beginning of
1990?
5. The population of Quahog, R.I. on December 31, 2001 was 10,000. The town is growing at a rate of
2% per annum. What is Quahog’s projected population on December 31, 2010?
6. Tim F. had an investment account of $2,500.00 earning interest at 12% p.a. compounded monthly
for 3 years. After the 3 years, the interest rate was changed to 9.6% compounded quarterly. How much
would Tim have in the account one and a half years after the change of interest rate?
85
Mathematics for Hospitality
Chapter 6 Compound Interest
7. Jayne D. opened an RHOSP deposit account on December 1, 1990 with a deposit of
$1,000.00. She added $1,000.00 on June 1, 1991 and $1,000.00 on September 1, 1992.
How much was in Jayne’s account on March 1, 1994 if interest was compounded
monthly at12% p.a.?
8. Accumulate $1,500.00 at 8.4% p.a. compounded monthly from March 1, 2002 to
July 1, 2004, and thereafter at 9.2% p.a. compounded quarterly. What was the accumulated value
on April 1, 2007?
9. a) Marco Z. has just invested $10,000. Find the accumulated value that this will grow to, if Marco
invested it for 12 years at an interest rate of 8% p.a., compounded annually.
b) If the interest in the above investment were compounded quarterly, find the accumulated value.
c) How much MORE interest would Marco earn in part b than in part a?
86
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
Chapter 7 HOSPITALITY STATISTICS
Statistics is the science of gathering, organizing and interpreting information. Statistical
analysis of the collected information helps us to make decisions.
Definitions:
Population - a large group being researched: e.g. all people in Canada earning an
annual income of $35,000 or less
-
Sample - a smaller group randomly selected from a population for the purpose of
statistical research: e.g. women earning an annual income of $35,000 or less, who live
in Ontario.
I. Central tendency
We often want to find one value to represent all the data collected in a sample. For
example, we may have interviewed people of different ages to get their opinions about
something, and we want to show the general level of their responses. This number will
be some central value around which other numbers seem to cluster. This central value is
known as a measure of central tendency.
It is important to find this number in order to have a standard against which to measure
the amount of deviation from this most representative value in a sample group.
The arithmetic mean, the median, the mode, and the midrange are widely used
measures of central tendency.
1. The arithmetic mean or average is calculated by finding the sum of all values, and
then dividing it by the number of those values. In this approach to a central value, each
item of data is given an equal level of importance.
Example 1: The Cozy Corner inn had the following numbers of room sales during 2005:
January
February
March
April
May
June
104
94
112
110
104
130
July
August
September
October
November
December
125
136
128
115
104
94
Find the average (mean) number of room sales per month.
87
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
Solution:
Calculate the total number of rooms sold throughout the year, and
divide by 12 (months).
mean =
104+94+112 +110 +104 +130+125+136+128+115 +104 +94
12
=
1356
12
=
113 rooms
Therefore the average room occupancy (mean) is 113 rooms.
88
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
2. The weighted mean is used when different parts of a list of data have different levels of
importance. A weighted mean is calculated by multiplying (weighting) each number according to
its importance. The sum of all products is then divided by the total number of weights.
Example 2: Several computer stores carry the same printer. Using the prices and number
of sales shown below, find:
a) the average price per store
b) the average price per printer.
Store
A
B
C
D
E
No. sold
40
60
160
90
50
Price
$600
$550
$425
$500
$600
Solution:
a) To find the average price per store, we have to divide the sum of all prices at all stores
by the total number of stores.
Average price per store
= $600 + $550 + $425 + $500 +$600
5
= $535
The average price per store was calculated as a simple arithmetic mean, and is equal to
$535.
b) To determine the average price per printer, all prices must be weighted according to
the number of printers sold at each of the different prices.
Step 1: Weighting prices:
Number
40
60
160
90
50
Total
x
x
x
x
x
$ Price
600
550
425
500
600
$ Weighted Product
24,000
33,000
68,000
45,000
30,000
400
$200,000
Step 2:
Average price per printer =
$200,000
400
=
$500.00
89
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
3. The median is a midpoint (or middle number) of a group of numbers. There are
as many numbers above the median as below it in a distribution.
Given the numbers 7, 4, 14, 8, 10, 3, 5, 1, 11, 4, 8, we must first arrange them in
ascending order, and then identify the middle number.
1,3,4,4,5,7,8,8,10,11,14
⇑
middle number
There is a total of 11 numbers in the given distribution, and the middle number is the
sixth one (with five numbers on each side of it).
Therefore the median is 7.
The formula for the position (not the value) of the median of a set of n values is n + 1
2
So if a data list contains 37 values, once we order the values, we know that the median
will be found in position 19, whatever that value may be. This is because 37 + 1 = 38,
And 38/2 = 19.
Returning to our opening data set from above, if we exclude 14 from the set of
numbers we worked with, and leave only ten numbers in the set, which is an even
number of values, the median then would be the number halfway between the fifth
figure (which is 5) and the sixth figure (which is 7).
1,3,4,4,5,7,8,8,10,11
⇑⇑
middle numbers
The median would therefore be 6, the arithmetic mean of the two central values.
4.
The midrange is the arithmetic mean between the highest and the lowest
numbers in a set of numbers. For the set given above------ 1, 3, 4, 4, 5, 7, 8, 8, 10, 11
midrange
=
1 + 11
2
The midrange is 6, and in this particular case it is equal to the median.
90
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
5. The mode is the value that appears most frequently in a set. The mode of the
numbers
2,0,2,5,8,12,5,4,3,5,11
is 5, since it appears three times in the given set, more than any other number.
If there are no repeating numbers in a set, the set then does not have a mode. However,
if there is more than one value that appears an equal number of times in a set, such a
set will have more than one mode.
If the above set had another number 2 at the end
2, 0, 2, 5, 8, 12, 5, 4, 3, 5, 11, 2
the modes would be 2 and 5.
Example 3: Find the mean, median, midrange and mode for the given set:
10, 15, 23, 13, 35, 27, 25, 31, 18, 23
Solution: There are 10 numbers in the set.
1)
To find the mean divide the sum of all numbers by 10.
mean:
=
10+15+23+13+35+27+25+31+18+23
10
= 22
2)
To find the median, first arrange the numbers in order
10, 13, 15, 18, 23, 23, 25, 27, 31, 35
There are two numbers in the middle of ten numbers set the fifth - 23, and the sixth - 23.
The midway value is 23.
The median is 23
3)
To find the midrange calculate the arithmetic mean of the lowest and the
highest values.
Midrange
=
4)
The mode is 23, since this
value appears twice, more than any
other value.
91
10 + 35
2
=22.5
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
Exercise I
1. Calculate mean occupancy rates for each hotel using the following data:
Hotel A
Hotel B
Hotel C
Saturday
50%
40%
60%
Sunday
50%
40%
60%
Monday
60%
70%
70%
Tuesday
80%
69%
70%
Wednesday
80%
90%
65%
Thursday
80%
90%
80%
Friday
62%
70%
50%
Totals:
Mean Rates:
2. For the following sample of monthly sales, find the indicated measures of
central tendency.
$15000,$22000,$27000,$37000, $19000, $22000,
$36000, $35000,$19000,$18000,$16000,$22000.
a)
the mean
b)
the median
c)
the midrange
d)
the mode
92
Mathematics for Hospitality
3.
Chapter 7 Hospitality Statistics
An employer wishes to determine which of his three salespersons had the highest
mean monthly sales. The following information was available:
Month
January
Blake
$22000
Tanaka
$24000
Rodriguez
$29000
February
20400
24600
26000
March
26000
29800
24200
April
22000
18600
23000
May
18600
22000
24600
June
32000
28800
29000
July
30000
30000
32000
August
28400
32000
34000
September
24800
30600
28800
October
22000
24000
28600
November
29000
28600
31600
December
32000
31600
36000
Mean Sales:
4.
For the following sets of data, determine the measures of central tendency as
indicated below:
Set A: 6, 2, 4, 9, 6, 1, 8, 3, 6
Set B: 21, 19, 23, 20, 27, 32, 33, 25, 37, 27, 33
Set C: 9, 12, 13, 11, 19, 13, 15, 21, 10, 17
For set A determine:
a.
the mean
c.
the midrange
b.
the median
d.
the mode
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Mathematics for Hospitality
Chapter 7 Hospitality Statistics
For set B determine:
e.
the mean
f.
the median
g.
the midrange
h.
the mode
For set C determine:
i.
the mean
j.
the median
k.
the midrange
l.
the mode
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Mathematics for Hospitality
5.
Chapter 7 Hospitality Statistics
Given that “quality points” are numerical equivalents of letter grades (A = 4, B = 3,
etc.), calculate the grade point (weighted) average for a student whose credit hours
and quality points are listed below. Refer to Example 2.
Credit
hours
Final
grade
Quality
points
Weighted
product
4
B
3
______
5
C
2
______
5
A
4
______
3
B
3
______
2
C
2
______
3
B
3
______
Weighted Average ___________________
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Mathematics for Hospitality
6.
Chapter 7 Hospitality Statistics
The Gorgonzola Motel employs 12 people in the positions and at the wages shown below.
Position
No. of people
Weekly wages
Manager
1
$300
Clerks
3
120
Cleaners
6
80
Bookkeeper
1
140
Plumber
1
100
Weighted product
a)
What is the average (mean) wage of each individual position?
b)
What is the mean wage, considering all employees as one group?
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Mathematics for Hospitality
Chapter 7 Hospitality Statistics
II. Yield management
Yield management, also called revenue management, is a method used to maximize
room revenues. Yield statistics is an important part of yield management, since we use it
as a measure of performance. Statistical yield is calculated as:
Yield = Occupancy rate x Achievement factor
where the Achievement factor itself is the rooms revenue yield:
Achievement factor
Actual Average room rate
Standard Average room rate
(rack rate)
=
Example 4: The Yellow Moon Inn has 120 rooms. On average, the inn sells 85 rooms at an average
room rate of $68 per night. The rack (standard) rate in the inn is $80 per night. Calculate
the statistical yield.
Solution: Step 1:
Determine the occupancy rate.
= 85
120
Occupancy rate = 0.71 (or 71%)
Step 2:
Determine the achievement factor.
= $68
$80
0.85
0.85
(or 85%)
Achievement factor = 0.85 (or 85%)
Step 3:
Calculate the yield.
Yield = 0.71 x 0.85 = 0.603
= 60.3%
Therefore the inn operates at only 60.3% of its
potential.
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Chapter 7 Hospitality Statistics
Exercise 2
1
A manager of a 150-room motel wishes to determine the motel yield in a high and low
season. The following information is available:
a)
b)
2.
Season
Number of
Rooms Sold
Room
rate
Rack
rate
High
Low
125
80
$86
$68
$90
$90
Calculate the achievement factor for a 230-room hotel that on average sells 186 rooms,
and maintains a yield of 0.61.
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Mathematics for Hospitality
3.
Chapter 7 Hospitality Statistics
Calculate the statistical yield for each of the following:
Total Rooms Occupancy Rack Room
rooms sold
rate
rate rate
Achievement
factor
Yield
factor
Hotel A
160
125
_______
$100
$78
_______
______
Hotel B
145
95
_______
$ 95
$76
_______
______
Hotel C
150
78
_______
$110
$92
_______
_______
Hotel D
140
108
_______
$100
$86
_______
_______
Fill in the blanks:
Total Rooms Occupancy Rack Room
rooms sold
rate
rate rate
Achievement
factor
Yield
factor
4.
200
145
______
$110 ______
______
0.522
5.
160
______
______
$ 95
$76
______
0.54
6.
150
______
72%
$125
$92
______
______
7.
140
105
______
______ $76
______
0.57
8.
______
126
______
______
0.54
$ 92
99
$69
Mathematics for Hospitality
Chapter 7 Hospitality Statistics
Review problems
A 120-room hotel had the following record of room sales in the first week of March, 2006:
Day
Rooms Sold
Sunday
114
Monday
102
Tuesday
102
Wednesday
108
Thursday
84
Friday
90
Saturday
% Occupancy
114
a)
From the above data, calculate the daily occupancy rate for each day of the week.
b)
Determine the mean (average) occupancy rate.
2. Determine the indicated measures of central tendency for the following data set:
75, 74, 70, 68, 79, 82, 66, 68, 70, 68
a.
the mean
b.
the median
c.
the midrange
d.
the mode
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Mathematics for Hospitality
3.
Chapter 7 Hospitality Statistics
The Dew Drop Inn has the following record of room occupancy for 2015:
Category of
Guest
Length of
Stay
Number of
Guests
Weighted
Product
A
1 day
160
______
B
2 days
420
______
C
3 days
100
______
D
4 days
30
______
For the average length of stay calculate:
4.
a)
the simple average (based on category of guests)
b)
the weighted average (based on numbers AND classes of guests)
Determine the statistical yield for a 180-room motel that sells 156 rooms at $88 per night
in the summertime, if the rack rate is $110.
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Chapter 8 YIELD AND PRICE FACTORS
I. %Yield and Yield Factor
Most foods (especially meat) lose weight as a result of peeling, trimming, and cooking.
The weight of cooked foods is called the Edible Portion Quantity (EPQ) or yield.
The yield can be expressed as a percentage of weight of the raw food, e.g., the
As Purchased Quantity (APQ). In this case it is called % yield.
The decimal value of the %yield is known as yield factor.
For instance, if the %yield = 91%, then the yield factor = 0.91
%yield =
Edible Portion Quantity (EPQ)
As Purchased Quantity (APQ)
X 100%
and
Edible Portion Quantity (EPQ)
Yield factor = As Purchased Quantity (APQ)
Since for this calculation we are always placing the smaller number on the top of the fraction, yield
factors will always be less than 1.00.
Example 1: 6 lb of oranges were purchased to make freshly squeezed orange juice. 5 lb of orange
juice was produced. Determine the %yield and yield factor.
Solution:
1)
Edible Portion Quantity (EPQ)
As Purchased Quantity (APQ)
Yield factor =
= 5/6
Yiel2)d
f
= 0.833
6
2) % Yield = 0.833 x 100 = 83.3%
l
83.3%
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Chapter 8 – Yield and Price Factors
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Exercise 1
1. A chef ordered a 50 lb. bag of potatoes. After peeling and cooking, there were
37.5 lb of potatoes available to serve. Calculate the percentage yield.
2. Four 16-oz cans of peas were ordered by a restaurant. The yield, after cooking, was 50 oz
Calculate the %yield.
3. 12.4 kg of veal was ordered for a party. The Edible Portion Quantity (EPQ) of the veal is 10.8 kg.
a) Calculate the %yield.
b) Find the yield factor.
4. A 7.4 lb turkey lost 0.6 lb of its As Purchased Quantity (APQ) in the process of roasting.
a) Calculate the %yield
b) Find the yield factor.
5. A chef ordered nine 12-oz cans of mushrooms. After cooking there are 5 lb of mushrooms
available to serve. Calculate the yield factor.
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Chapter 8 – Yield and Price Factors
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II. Edible Portion Quantity (EPQ)
We can determine the Edible Portion Quantity (EPQ) as:
Edible Portion Quantity (EPQ) = As Purchased Quantity (APQ) x yield factor
Example 2: The yield factor of veal is 0.80. Calculate the Edible Portion Quantity (EPQ) of 3 lb.
7oz of veal.
Solution:
Step 1: Convert the weight of the raw veal into ounces.
3 lb 7oz = (3 x 16) + 7
= 55 oz
Step 2: Calculate the Edible Portion Quantity (EPQ) in ounces.
Edible Portion Quantity (EPQ)
= 55 oz x 0.8
= 44 oz
Step 3: Express the Edible Portion Quantity (EPQ) in pounds and ounces.
44 oz
16 oz/lb
= 2.75 lb
Edible Portion Quantity (EPQ) =
= 2 lb 12 oz
III. As Purchased Quantity (APQ)
The expression for As Purchased Quantity (APQ) can also be found easily:
As Purchased Quantity (APQ)
=
Edible Portion Quantity (EPQ)
yield factor
Example 3: How many kilograms of apples should be purchased to make 8.64 kg of apple juice,
if apples have a 0.72 yield factor?
Solution:
As Purchased Quantity (APQ)
=
8.64 kg
0.72
= 12 kg
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Chapter 8 – Yield and Price Factors
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Exercise 2
1. Find the Edible Portion Quantity (EPQ) of 32 lb. of meat, if the meat has a 0.85 yield factor.
2. The As Purchased Quantity (APQ) of raw cauliflower is 16 lb. Find the Edible Portion Quantity
(EPQ), if the %yield is 80%.
3. A restaurant owner needs 12 lb. of cooked meat. How many pounds of raw meat should be
purchased, if the meat has a 0.8 yield factor?
4. What is the As Purchased Quantity (APQ) of 4.2 kg of broiled turkey with a %yield of 84%?
5. What is the Edible Portion Quantity (EPQ) of 5 kg of meat with a %yield of 88%?
6. Find the As Purchased Quantity (APQ) of 5.6 kg of processed fruit, with a yield factor of 0.70.
7. 32 servings of cooked beans are to be prepared. Each serving is 3oz
How many 12-oz cans of beans should be ordered, if the yield factor is 0.8?
8. A chef planned to serve 4.5 oz steaks to 180 people. How much raw flank steak should the chef
order, if the trimming, cooking and portioning waste was expected to be 25%?
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Chapter 8 – Yield and Price Factors
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IV. Edible Portion Cost (EPC)
To calculate the food cost per portion, we have to take into consideration the difference between
the As Purchased Cost (APC) and the Edible Portion Cost (EPC).
The As Purchased Cost (APC) is the cost of a unit of weight of the raw product. For instance, the As
Purchased Cost (APC) of veal is $5.00. per lb or $11.00 per kg.
Since the weight of the cooked product differs from the weight of the raw product, the value of one
unit of cooked weight also differs from the As Purchased Cost (APC). The total cost of what was
purchased will not change after the food is processed, but cooked food will always be more
expensive per unit of weight than the food in its raw form.
Therefore the Edible Portion Cost (EPC) is the value of one unit of cooked weight.
As Purchased Cost (APC) x APQ
Edible Portion Quantity (EPQ)
Edible Portion Cost (EPC) =
Example 4: (imperial units)
10 lb. of meat were purchased at $3.59 per lb. The Edible Portion Quantity (EPQ)
after cooking was 8 lb. Find the processed price of meat per lb.
Solution:
1) The total or As Purchased Cost (APC) of 10 lb. of meat:
$3.59 x 10 = $35.90
2) The Edible Portion Cost (EPC) of cooked meat:
Edible Portion Cost (EPC)
=
=
APC X APQ
EPQ
$3.59 x 10lbs.
8 lb
= $4.49 per lb.
Example 5: (metric units)
Five kg. of carrots were purchased, the total cost (APC) was $20.85.
The Edible Portion Quantity (EPQ) was 4.2 kg. Calculate the Edible Portion Cost
per kg. (EPC) for these carrots.
Solution:
EPC =
APC x APQ
EPQ
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Chapter 8 – Yield and Price Factors
Mathematics For Hospitality
=
$20.85
4.2 kg
= $4.96 per kg
V. Price Factor
The price factor is the ratio of the Edible Portion Cost (EPC) to the As Purchased Cost (APC). This
ratio is used to indicate the change in value of one unit of weight of the purchased product, as a result
of cooking. Because for this calculation we are always placing the larger number (EPC) on the top of
the fraction, a price factor will always be greater than 1.00.
Price Factor =
Edible Portion Cost (EPC)
As Purchased Cost (APC)
Example 6: The As Purchased Cost was $4.30 per kg. The Edible Portion Cost (EPC) is $5.00
per kg. Calculate the price factor.
Solution:
Price factor =
Edible Portion Cost (EPC)
As Purchased Cost (APC)
Price factor
$5.00
$4.30
=
= 1.1627
= 1.163 (let’s agree to round to 3 decimal places)
Example 7: A 2.5 kg chicken was purchased for $10.60. Find the Edible Portion Cost (EPC) of
one kg of cooked chicken, if the price factor was 1.125.
Solution:
1) Determine the As Purchased Cost (APC) per kg.
As Purchased Cost (APC)
$10.60
2.5 kg
$4.24 per k
=
= $4.24 per kg
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Chapter 8 – Yield and Price Factors
Mathematics For Hospitality
Edible Portion Cost (EPC) = As Purchased Cost (APC) x Price Factor
Edible Portion Cost (EPC)
= = $4.24 x 1.125
= $4.77 per kg
VI. Connection between Yield and Price Factors
Note an interesting relationship: price factors and yield factors are inversely related! This is to say that a
given price factor is the yield factor for the same food, turned upside down.
Proof: Let’s use the data from Example 4 on page 103. In that example, 10 lbs. of meat cost $35.90.
The yield was 8 lb., which makes the yield factor 0.8. We calculated an EPC of $4.49 per lb, and so the
price factor here must be $4.49/ $3.59, which works out to 1.25.
1.25 is simply 125/100.
If we turn this fraction upside down, we have 100/125, which works out to 0.8. So we are right back to
the same yield factor!
Proof:
Yield factor = 8lb./10 lb. = 0.8, or 0.8/1
Price factor = $4.49/$3.59 = 1.25, or 1.25/1
And 0.8/1 can be reduced to 4/5 (divide by 0.2, both numerator and denominator)
But 1.25/1 can be reduced to 5/4 (divide by 0.25, both numerator and denominator)
Thus the two fractions are upside-down versions of each other.
Therefore yield factor/1 = 1/price factor. This will work for any price and yield factor combination!
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Chapter 8 – Yield and Price Factors
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Exercise 3
1. A chef ordered a 50 lb bag of potatoes. The potatoes cost 12 cents per pound. After peeling and
cooking, there were 37.5 lb. of potatoes available to serve. Calculate the Edible Portion Cost
(EPC).
2. 12.5 kg of veal was ordered for a party. The total cost of veal was $68.20. The Edible Portion
Quantity (EPQ) of veal was 10 kg.
a) Calculate the As Purchased Cost (APC) per kg.
b) Calculate the Edible Portion Cost (EPC).
C) Find the price factor.
3. A 7.4 lb. turkey lost 0.9 lb of its As Purchased Quantity (APQ) in the process of broiling. The total
Cost (APC) was $48.84.
a) Calculate the As Purchased Cost (APC) per lb.
b) Calculate the Edible Portion Cost (EPC) per lb.
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Chapter 8 – Yield and Price Factors
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c) Find the price factor.
4. A restaurant paid $179.45 for 18.5 kg of fish. The price factor was 1.22. Calculate the Edible
Portion Cost (EPC) per kg.
5. A chef ordered nine 12-oz cans of mushrooms. The total cost was $16.20. After cooking there were
5 lb. of mushrooms ready-to-serve.
a) Calculate the As Purchased Cost (APC) of 1 lb. of mushrooms
b) Calculate the Edible Portion Cost (EPC) per lb.
c) Find the price factor.
6. A chef ordered 15 kg of turkey at a total cost of $173.85. The price factor was 1.145. Calculate
the Edible Portion Cost (EPC) of one kg of turkey.
7. A restaurant owner needed 14.5 lb. of cooked meat to serve for dinner. He figured that the
trimming and cooking waste was 27.5%. The cost of raw meat is $8.60 per lb.
a) Find the As Purchased Quantity (APQ) of meat.
b) Calculate the total As Purchased Cost (APC).
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Chapter 8 – Yield and Price Factors
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c) Calculate the Edible Portion Cost (EPC).
d) Find the price factor.
8. A chef needs to prepare 45 3-oz. portions of cooked broccoli. This vegetable loses 29% of its weight
when cooked. If raw broccoli costs $4.59 per lb.,
a. What is the yield factor for broccoli?
b. How many pounds of raw broccoli must be purchased (to the nearest whole ounce)?
c. What is the EPC per cooked pound of broccoli?
d. What is the price factor for broccoli?
e. Describe the relationship between the price and yield factors that is seen in this problem.
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Review problems
1. What is the %yield of a product that reduces its weight from 6.5 kg to 5.8 kg in the process of
cooking?
2. A restaurant ordered 12.7 kg of meat with a percentage yield of 80%. Determine the Edible
Portion Quantity (EPQ) of meat.
3. The Edible Portion Quantity (EPQ) of potatoes is 5 lb 12 oz Find the required As Purchased
Quantity (APQ), if potatoes have a yield factor of 0.80.
4. A chef ordered 20 kg of cabbage. After cooking, the As Purchased Quantity (APQ) of cabbage
decreased by 4 kg. Calculate the yield factor and %yield.
5. A 24.5 lb piece of beef lost 12% of its As Purchased Quantity (APQ) in the process of
cooking. Calculate the Edible Portion Quantity (EPQ) for this piece of beef.
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Chapter 8 – Yield and Price Factors
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6. Fifty 3 oz steaks are supposed to be served for dinner. Beef loses 25% of its weight in the
process of cooking. Calculate how many pounds of raw beef should be ordered.
7. How many 2.5 kg broilers should be purchased, if a party requires 18 kg of cooked boneless
meat, and a broiler has a 0.60 yield factor (which includes deboning)?
8. If potatoes lose 12% to peeling and trimming, how many pounds of diced potatoes could be
prepared from a 50 lb. bag of raw, unpeeled potatoes?
9. A raw turkey loses 35% of its weight in the process of cooking.
Calculate:
a) how much usable meat would a 15 lb. turkey yield?
b) how many 3-oz portions could be served?
10. A chef ordered 15 kg of turkey at a total cost of $173.85. The price factor was 1.154. Calculate
the processed price of one kg of turkey.
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Chapter 8 – Yield and Price Factors
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11. Given what you know about the relationship between yield factors and price factors, complete the
following table:
Yield Factor
a.
b.
Price Factor
.80
___________
________
c.
.69
d.
_________
1.666
____________
1.333
e.
.85
___________
f.
_________
1.087
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Mathematics for Hospitality
Chapter 9 Menu Pricing
Chapter 9 MENU PRICING
I. FOOD COSTING
For costing, or comparative shopping, it is necessary to calculate the cost of a single unit, as well
as a single portion as served.
Unit Cost
The cost of one item when a large quantity (case, sack, box, and so on) is purchased is known as
unit cost (or cost per unit). It is calculated by dividing the total cost by the quantity purchased.
Cost per Unit
As Purchased Cost (Total)
Number of Units
=
Example 1: The price of a case of oranges containing 150 oranges was $30. Calculate the unit
cost per orange.
Solution:
As Purchased Cost
Cost per Unit =
Number of Units
$30
150
=
=
$0.20 per orange
Portion cost as purchased
Each purchased unit could be divided into smaller portions equivalent to one serving. A
calculation of portion cost often requires conversion into a different unit.
Portion cost = Unit cost x Portion size
Example 2: (metric units)
Raw top sirloin roasts cost $11 per kg. What is the cost of a 175 g portion?
Solution:
Convert the price from dollars per kg to dollars per g, and multiply by the size
of a portion in grams. Use a conversion factor of 1000 g/kg.
Portion cost =
Purchased cost
per gram
⇓
$11.00
1000g/kg
Portion size
in grams
⇓
x 175g
= $1.93 (rounded to the nearest cent)
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Mathematics for Hospitality
Chapter 9 Menu Pricing
Example 3: (imperial units)
Medium ground beef costs $2.99 per lb. Calculate the cost of a 4 oz portion.
Solution:
Convert the cost from dollars per lb to dollars per oz and multiply by the
size of a portion in ounces.
Portion cost =
=
Purchased
price per oz
⇓
$2.99
16 oz/lb
$0.75
116
Portion size
in oz
⇓
x 4 oz
Mathematics for Hospitality
Chapter 9 Menu Pricing
Exercise 1
Solve the following problems, rounding out your answer to the nearest cent.
1. A jar contains 15 pickles. The jar costs $2.99. Calculate the unit cost per pickle.
2. A case of cherries contains 24 jars. A case costs $25.68. What is the unit cost per jar?
3. One pound of bananas will make 4 portions when sliced. The cost of bananas is
$0.24 per lb. How much does each portion cost?
4. A menu item requires fresh mushrooms. They cost $4.39 per kg. Calculate the cost of a
50 g portion.
5. Fresh asparagus costs $4.39 per kg. A chef can figure on 24 portions per kg. What is the cost per
portion?
6. A chef purchased a case of 150 pears for $15.00. Each serving uses one and a half pears.
Calculate the cost per serving.
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Mathematics for Hospitality
Chapter 9 Menu Pricing
7. A portion size of ready-to-serve pork side ribs is 9 oz The cost of raw side ribs is $3.99 per
lb.. Assuming that the after-cooking yield for side ribs is 75% (i.e., wastage is 25%), find the
cost per portion.
8.
The cost of lamb is $6.59 per kg. A portion size of raw lamb is 225 g. What does each portion of
raw lamb cost?
9.
Pork loin costs $6.89 per kg. Calculate the cost of a 180 g portion of raw meat.
10, The purchasing manager for a restaurant has two choices when buying canned tomatoes .
.Option A is four 12-oz cans of tomatoes for $2.88. Option B is five 16-oz cans of tomatoes for $4.00.
a) What is the unit cost per ounce for Option A?
118
b) What is the unit cost per ounce for Option B?
c) The purchasing manager has to order 75 lb of canned tomatoes. How much will he save by
choosing the better option?
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Mathematics for Hospitality
Chapter 9 Menu Pricing
II. Menu price
One of the most important decisions in the food business is the decision on a menu price. Overpricing
menu items might result in the loss of patrons to competition; on the other hand, a menu priced too low
could leave the business without a profit.
The strategy of every business is to keep costs low. After figuring %yield and price factors, one can
have a clear idea of what the cost of ingredients is per menu item. To cover the cost of labour and
other expenses (hydro, water, etc.), the menu price (selling price) must be higher than the cost of
the ingredients.
In order to coordinate the food cost and selling price, the food cost is expressed as a percentage of
selling price.
%Food cost =
$Food cost
Menu price
Example 4:
Sandwich ingredients cost $0.78. Calculate the selling price, if the desired
%Food cost is 28%.
Solution:
Find the Menu price from the %Food cost expression above:
Menu Price =
$Food cost
% Food cost
= $0.78
0.28
= $2.79
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Mathematics for Hospitality
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Example 5: After some marketing research, a bistro owner wanted to price a chicken dinner
the same as the competition, at $9.95. What should be the $Food cost, if a 30%
food cost level is desired?
Solution:
The $Food cost can also be found from the %Food cost expression:
$Food cost = $Menu price x %Food cost
= $9.95x 0.30
= $2.99
Exercise 2
Calculate the missing term for each of the following:
$Food cost
1.
$1.68
2.
$2.88
3.
$1.68
5.
$2.72
6.
8.
$9.00
$4.50
$6.00
32%
28.5%
$3.34
Menu price
30%
27%
4.
7,
%Food cost
$8.75
31.5%
A baker figured that the cost of ingredients per blueberry muffin was $0.27. The baker decided
that the food cost% should be 30%. Calculate:
a) the selling price of one muffin.
b) the selling price of 6 muffins, if a 10% price discount is allowed on that quantity.
c) the selling price of a dozen muffins, if a 15% reduction in price is allowed on that quantity.
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Mathematics for Hospitality
Chapter 9 Menu Pricing
9.
A broccoli soup recipe costs $0.97 per serving. Determine the menu price, if a 29% food cost
percentage is desired.
10.
The market price for a hamburger with fries was $6.95. Calculate the food cost, if the Food cost%
was 28%.
11.
Determine the Food cost % of a fruit platter, priced at $8.99, if the cost of the fruit was $2.79.
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Mathematics for Hospitality
Chapter 9 Menu Pricing
III. Markup
In the food business the menu price must cover:
•Food cost
•Labour cost
•Other Expenses
•Some profit (we hope!!)
Markup is the difference between the menu price and the food cost, which is used to cover
labour cost, expenses, and generate some profit. The dollar amount of markup (the dollar markup)
is calculated as:
$Markup = $Menu price - $Food cost
Markup could also be expressed as a percentage of the menu price, in which case it is referred to as
margin.
$Markup
Margin
=
Menu price
Example 3:
The menu price is $8.20, and the food cost is $2.46. Determine the markup
in dollars.
Solution:
$Markup = $8.20 - $2.46
= $5.74
Example 4: The menu price is $8.20, and the markup is $5.74.
What is the margin?
Solution:
Margin
=
$5.74
$8.20
= 70%
Example 5: The menu price is $8.20 and the margin is 68%. Determine the
markup in dollars.
Solution:
$Markup = Margin x Menu price
= 0.68 x $8.20
= $5.58
123
Mathematics for Hospitality
Chapter 9 Menu Pricing
Exercise 3
1. Given: Food cost% = 32% and Menu price = $9.70
Calculate: a)$Food cost
______
b)$Markup
______
c) Margin
______
2.Given: Food cost% = 29.6% and Menu price = $7.95
Calculate: a) $Food cost
______
b) $Markup
______
c) Margin
______
3. Given: Food cost% = 30.4% and $Food cost = $1.98
Calculate: a) Menu price
b) $Markup
c) Margin
______
_______
______
124
4.Given: Food cost%=28.2% and $Food cost =$1.63
Calculate: a) Menu price
_______
b) $Markup
______
c) Margin
______
5. Given: Food cost = $2.08 and Menu price = $6.75
Calculate: a) Food cost%
______
b) $Markup
_______
c) Margin
______
125
Mathematics for Hospitality
Chapter 9 Menu Pricing
6. Given: $Food cost = $4.62 and Menu price = $14.20
Calculate: a)Food cost%
______
b)$Markup
______
c)Margin
______
7. Given: $Food cost = $2.84 and Food cost% = 31.7%
Calculate: a)Menu price
______
b)$Markup
______
c)Margin
______
8. Given: Food cost% = 32.6% and $Food cost = $5.95
Calculate: a) Menu price
______
b) $Markup
______
c) Margin
______
126
9. Given: $Food cost = $3.03 and Menu price = $9.95
Calculate: a) Food cost%
______
b) $Markup
______
c) Margin
______
10. Given: Food cost% = 29.5% and Food cost = $3.10
Calculate: a)Menu price
______
b)$Markup
______
c)Margin
______
11. Given: Food cost% = 25% and Menu price = $6.60
Calculate: a)$Food cost
______
b)$Markup
_______
c)Margin
______
127
Mathematics for Hospitality
Chapter 9 Menu Pricing
Review problems
1. The menu price for a salad bar is $5.99 per person. Your manager has
decided that the cost of food should not exceed 31%. Calculate:
a) $Food cost
b) $Markup
c) Margin
2. A chef purchased 10 lb. of ground veal at $3.68 per lb. The chef figured that cooking
and portioning waste was 26%. The portion size of cooked veal is 6 oz
Calculate:
a) the Edible Portion Quantity (EPQ)
b) the cost of cooked veal per lb.
c) the portion cost
3. Determine the %markup for a serving of Greek Salad, if the menu price was $6.49 and the
cost of raw ingredients was $2.08.
128
Mathematics for Hospitality
Chapter 9 Menu Pricing
4. 8-oz steaks (after cooking) were served to 56 guests in a restaurant. The trimming, cooking
and portioning waste was 30%. The restaurant paid $3.49 per pound of raw meat.
Calculate:
a) the total Edible Portion Quantity (EPQ)
b) the As Purchased Quantity (APQ) of meat
c) the total As Purchased Cost (APC) of purchased meat
d) the cost of one ready-to-serve steak
5. A restaurant purchased 15 kg of pork loin roast at 8.60 per kg. 150-g portions of cooked
meat were served. The meat had a waste factor of 32%.
Calculate:
a) how many portions of cooked meat were served.
b) the cost of the Edible Portion Quantity (EPQ) per kg.
c) the cost of a portion.
6. Calculate the menu price per portion in question 5, if the restaurant maintained a %food cost of
29%.
129
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Chapter 10 PROFIT OR LOSS STATEMENT
Sooner or later, every business owner or manager wants or needs to know exactly how well (or
how poorly) the business is doing. The Income Statement is one way to measure that level of
success. In essence, this document is very simple: it starts with how much money has been
taken in by the business, a.k.a. the Sales. For foodservice organizations, this is usually broken
down into how much money was taken in through food sales, and how much through beverage
sales.
After the sales have been shown, which is the pleasant part of the business’ story, we must then
show how much money has been taken away from those hard-earned sales, to pay for the
expenses of running the business. For a foodservice organization, the first expenses shown are
the Food Costs and Beverage Costs, which together are referred to as the Cost of Goods Sold.
Whatever is left over is called the Gross Profit, but it is not the end of the story, since there are
other costs to pay, such as Labour, Rent, Supplies, Insurance, etc. (To keep things simple here,
we will just refer to all these other costs aside from Labour as “Expenses”.) Only when all these
are subtracted as well do we reach the end of the story, generally called Net Profit or Net
Income.
So here is a typical Income Statement:
Sam ‘n’ Ella’s Diner
Income Statement
September 30, 2009.
Sales
Food
Beverage
Cost of Goods Sold
Food
Beverage
$60,000
40,000
100,000
20,000
12,000
32,000
Gross Profit
68,000
Expenses
Labour
Total Costs
33,000
25,000
58,000
Net Profit
$10,000
=====
130
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
When we review the performance of the business we either speak in dollar terms or in percentage
terms. Most of the percentages are based on the total sales figure, but food and beverage cost
percentages are based on food and beverage sales, respectively. Let’s return to Sam ‘n’ Ella’s
income statement, to see what it looks like with the percentages included (rounded to 1 decimal
place only----rounding to 2 decimal places will produce slightly different answers in some cases):
Sam ‘n’ Ella’s Diner
Income Statement
September 30, 2009.
Sales
Food
Beverage
Cost of Goods Sold
Food
Beverage
$60,000
40,000
100,000
60%
40%
100%
20,000
12,000
32,000
33.3%
30%
32%
Gross Profit
68,000
68%
Expenses
Labour
Total Costs
33,000
25,000
58,000
33%
25%
58%
$10,000
=====
10%
==
Net Profit
Now, let’s review all this:
The purpose of a Profit or Loss statement is to establish whether a business operates at a
profit or loss.
There are two types of calculations in the Profit or Loss statements: one is in dollars, the
other in percentages.
In general, all entries are expressed as a percentage of a sales figure. Since there will be
three sales entries (food sales, beverage sales, and total sales) you have to follow the rules
below when calculating cost percentages.
RULE 1:
Beverage cost% is based on Beverage sales
Food cost% is based on Food sales
Total cost% is based on Total sales
RULE2:
Percentages for all other than cost entries: food and beverage sales,
profit, expenses, etc., are based on Total sales.
131
Mathematics for Hospitality
Example 1:
Chapter 10 Profit or Loss Statement
Fill in the blanks for the incomplete statement below:
$
SALES
COST
%
Food
60
Beverage
40
Total sales 150.000.00
100
Food
32
Beverage
28
Total cost
_____
GROSS PROFIT
Payroll
38
Expenses
14
NET PROFIT
Solution
SALES
Food
Beverage
Total sales
$
90,000
60,000
150,000
COST
Food
Beverage
Total cost
28,800
16,800
45,600
GROSS PROFIT
Payroll
Expenses
NET PROFIT
Notes
150,000 x 0.60
150,000 x 0.40
32
90,000 x 0.32
28
60,000 x 0.28
30.4% of Total Sales
104,400
69.6%
57,000
21,000
38
14
26,400
132
%
60
40
100
17.6%
of Total Sales
150,000 x 0.38
150,000 x 0.14
of Total Sales
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Notes:
1.
Gross profit is calculated as:
Gross Profit = Total sales - Cost of Goods Sold
2.
Net profit is calculated as:
Net Profit = Gross Profit - (Payroll + Expenses)
3.
The value of Net Profit can be greater, equal to, or smaller than zero.
That value shows us the financial status of the business:
Net Profit < 0 (the business operates at loss)
Net Profit = 0 (the business breaks even)
Net Profit > 0 (the business makes some profit)
133
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Exercise 1
Complete the following statement on your own. Follow the steps outlined below:
1.
Calculate the $Food sales and $Beverage sales as a percentage of the Total sales.
2.
Calculate the Food cost as a percentage of the Food sales.
3.
Calculate the Beverage cost as a percentage of the Beverage sales.
4.
Calculate the dollar figure of the Total costs.
5.
Determine the Total cost% based on the Total sales.
6.
Calculate the Gross Profit (see the formula above).
7.
Determine the Gross profit% based on the Total sales.
8.
Calculate the Payroll $ and Expenses $, based on the Total sales
9.
Calculate the Net Profit (see the formula above).
10.
Determine the Net profit % based on Total sales.
$
SALES
COST
%
Food
65
Beverage
35
Total sales 200,000
100
Food
35
Beverage
30
Total cost
________________
_____
GROSS PROFIT
Payroll
28
Expenses
15
NET PROFIT
134
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Exercise 2
Fill in the blanks in the following incomplete statements:
1
SALES:
COST
Food
Beverage
Total sales
COST
Food
Beverage
Total sales
COST
a)
b)
100
31
28
$
56,000
34,500
90,500
%
a)
b)
100
Food
c)
Beverage d)
3
SALES:
%
Food
c)
Beverage d)
2
SALES:
$
80,000
60,000
140,000
Food
Beverage
Total sales
Food
c)
Beverage d)
135
32
30
$
72,400
43,500
115,900
%
a)
b)
100
32.7
31.4
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
4
SALES:
COST
Food
Beverage
Total sales
$
76,800
43,200
120,000
Food
Beverage
%
100
32
25
Total Cost
Gross Profit
Payroll
Expenses
40
12
Net Profit
18.5
5
$
SALES:
COST
Food
Beverage
Total sales
180,000
%
60
40
100
Food
Beverage
Total Cost
31.5
27.5
Payroll
Expenses
41
14
Gross Profit
Net Profit
136
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
When an incomplete Profit or Loss statement does not contain the sales figures, they can still
be determined, if at least one other entry has both a dollar figure and a percentage based on
sales.
Example 2: Payroll costs are $58,650. Payroll cost is 42.5% of the total sales. Calculate the
total sales.
Solution:
Solve the formula
Payroll cost% =
$Payroll cost
$Total sales
for the $Total sales:
$Payroll cost
$Total sales = Payroll cost%
=
$58,650
0.425
= $138,000.00
Example 3: Food cost is $22,000. The food cost is 32.4% of the food sales. Calculate the
food sales.
Solution:
$Food sales =
=
$Food cost
Food cost%
$22,000
0.324
= $67,901.235
Rounded
to nearest cent
= $67,901.24
137
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Exercise 3
Calculate the missing sales figures:
1
$
SALES
Food
Beverage
Total sales
COST
Food
Beverage
2
16,800.00
8,490.00
$
SALES
Food
Beverage
Total sales
COST
Food
Beverage
3
24,759.00
15,035.00
$
SALES
Food
Beverage ___________
Total sales
Gross Profit
82,440.
138
%
32
30
%
31.5
31
%
65.0
68.7
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Exercise 4
Complete the following statements by filling in the missing entries:
1
SALES
Food
$
76,800
%
Beverage
Total sales
COST
100
Food
32
Beverage
25
Total cost
GROSS PROFIT
Payroll
50,400
42
Expenses
NET PROFIT
14
139
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
$
2
SALES
Food
Beverage
56,000
Total sales
COST
%
60
40
100
Food
28,560
Beverage
16,800
Total cost
GROSS PROFIT
Payroll
39
Expenses
12
NET PROFIT
140
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
3
SALES
COST
Food
$
84,500
%
Beverage
35
Total sales
100
Food
34
Beverage
Total cost
32.60
GROSS PROFIT
Payroll
52,000
Expenses
23,400
NET PROFIT
141
Mathematics for Hospitality
Chapter 10 Profit or Loss Statement
Review problem
Given: Beverage sales are $32,200. Beverage sales are 40% of the Total sales. %Food cost is 28%
and %Beverage cost is 32%. Expenses are 12% and the payroll cost is 34%.
1.
Calculate the Total sales.
2.
Calculate the $Food sales.
3.
Calculate the $Food cost.
4.
Calculate the $Total cost.
5.
Calculate the $Gross profit.
6.
Calculate the Gross profit%.
7.
Calculate the $Expenses.
8.
Calculate the $Payroll costs.
9.
Calculate the $Net profit.
10.
Calculate the Net profit%
142
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 1 – Review of Algebra
Exercise 1
Exercise 2
Exercise 3
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1 x=
2x=
3x=
4x=
5x=
6x=
x= 1
x= 2
x= 4
x= 5
x= 2
x= 4
x= 3
x = -2
x= 3
x= 6
x= 6
x= 6
x= 8
x = 10
x= 8
x = 12
x = 10
x= 2
x= 3
x= 2
x = 16
x= 2
x = 12
12
9
4
8
6
28
7 x = 10 kg.
8 x = 3 oz.
Exercise 4
1 L = N/(1 – d)
2 P = S/(1+K)
3 M = RC
4 P = I/rT
5 VR = 1 – CR
6 R = D/2(C-P)
7 EP = Y(AP)
8 AP = EP/Y
9 B = 180 – C – A
10 R = (N + 2)/C
11 N = CR – 2
12 C = (D + 2RP)/2R or C = (D/2R )+ P
13 P = (2RC – D)/2R or P = C – (D/2R)
143
Mathematics for Hospitality
Exercise Answers
Review Problems, Chapter 1
Part I
1
2
3
4
5
6
x=
x=
x=
x=
x=
x=
Part III
6
3
1.5
9
2a
10
14
15
16
17
18
19
20
21
T = S/P
P = T/(3 – v)
T = PV/NR
r = PV /NT
K = 2A/L
L = Pa - M
a = V/T
K = ES
Part II
Part IV
7 x= 6
8 x= 6
9 x= 2
10 x = 1
11 x = -3
12 x = 2
13 x = 3
1 x = 50
2 x = 4 ¾ cups
3 x = 30
4 x = 20
5x= 8
6 x = 28
144
Appendix
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 2 - Percent
Exercise 1
Exercise 2
1
50%
2
75%
3
25%
4 2.5%
5
5%
6
50%
7
75%
8 37.5%
9 162.5%
10 318.75%
11 0.12
12 0.04
13 0.40
14 1.25
15 0.005
16 1/5
17 2/5
18 1/20
19 9/4 or 2 ¼
20 2/25
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Exercise 3
20
20
3
67
38
96
50%
20%
20
40
90
60%
20
100
5000
20%
14.4
1000
4%
10.5
1
2
3
4a
4b
5a
5b
6
7
8
9
10
Exercise 4
Review Problems
1a
1b
1c
1d
2
3a
3b
4
5
6a
6b
6c
6d
6e
6f
6g
6h
1
2
3
4
5
6
7a
7b
7c
7d
8
9
10
11
12
13a
13b
$61.20
$91.77
$84.00
$44.86
36 rooms
$17,592
$3,276
$650
$96.00
450 cal.
1650 cal.
900 cal.
720 cal.
2820 cal.
16%
58.5%
25.5%
21
70
$33
$64
$ 3
$28
35.8%
11.9%
10.6%
41.7%
$22.50
$40.80
25%
$12.39
20%
450 rooms
90%
145
14
15
16a
16b
17a
17b
17c
17d
18
19
20
21
22a
22b
23
65%
6 rooms
185 rooms
800 cal.
40%
900 cal.
100 g.
30%
47%
287 drinks
1800 drinks
1216 drinks
15%
325 rooms
1,295 cal.
52%
4 lbs. 4 oz.
3 lbs. 3 oz.
17 portions
36%
$173.74
20%
4%
$58.50; $508.50
$560
$632.80
$4,200; $546
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 3 – Units of Weight Conversion
Exercise 1
Exercise 2
Review Problems
1 32 oz.
2 120 oz.
3 37 oz.
4 240 oz.
5 71 oz.
6 4 lbs. 1 oz.
7 5 lbs. 8 oz.
8 1 lb. 10 oz.
9 6 oz.
10 7 lb. 2 oz.
11 5 lb.12 oz.
12 12 oz.
13 1 lb. 12 oz.
14 3 lbs. 12 oz.
15 8 lbs. 1 oz.
16 28 g.
17 284 g.
18 454 g.
19 625 g.
20 908 g.
21 1 kg. 558 g.
22 1 kg. 50 g.
23 7 lbs. 12 oz.
24 5 lbs. 15 oz.
1
2
3a
3b
3c
4a
4b
4c
5 lbs.
2 kg. 420 g.
120 oz
5 oz.
5 lbs. 10 oz.
3 kg. 405 g.
142 g.
2 kg. 556 g.
Chapter 4 – Units of Volume Conversion
Exercise 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
70.2 fl. oz.
52.7 fl. oz.
26.4 fl. oz.
17.6 fl. oz.
42.1 fl. oz.
240 fl. oz.
380 fl. oz.
130 fl. oz.
740 fl. oz.
620 fl. oz.
384 fl. oz.
3 L 450 ml
2 L 800 ml
3 L 586 ml
10L 66 ml
30 fl. oz.;
860 ml.
Exercise 2
1
2
3
4
5
6
Exercise 3
46 bottles
53 drinks
339 bottles
103 pints
8 cans
60 portions
146
1 50 F
2 42.2 C
3 0F
4 80 F
5 -40 degrees C and F
Mathematics for Hospitality
Exercise Answers
Appendix
Review Problems, Chapter 4
1 225 ml; 7.9 fl. oz.
110 ml; 3.9 fl. oz.
90 ml; 3.2 fl. oz.
14 ml; 0.5 fl. oz.
2 30 portions
3a 32 fl. oz.
3b 384 fl. oz.
3c 19.2 pints
3d 19.2 lbs.
4a
9L
4b 105 glasses
5a 320.4 oz.
5b 10.4 times
6a 258 portions
6b $0.50
7 150 glasses
8a $0.32 per glass
8b
just over 17 L
8c
6 bottles
Chapter 5 – Simple Interest
Exercise 1
Exercise 2
r
1
2
3
4
5
6
7
8
Exercise 4
1
2
3
4
5
6
7
8
9
10
11
1
2
3
4
5
6
7
8
9
10
11
T
0.125
0.0975
0.1025
0.0763
0.0819
0.008
0.1285
0.0725
1.75
21/12
165/365
1.25
7/12
93/365
48/12
120/365
1
2
3
4
5
6
7
8
$320.00
$183.75
$188.10
$ 61.25
$ 73.60
$ 14.01
$ 53.92
$ 21.47
Exercise 5
1 + rT
1
2
3
4
5
6
7
8
Exercise 3
1.18
1.12
1.06375
1. 0591781
1. 0143836
1. 0796875
1. 0320833
1.0462671
$1,496.25
$ 6,000.00
7.50%
1.5 years
$ 210.00
6 months
$ 1,200.00
8.25%
$ 112.19
9%
$ 6,749.84
Exercise 6
S
$ 2,360.00
$ 5,040.00
$ 7,446.25
$ 1,482.85
$ 2,535.96
$ 2,159.38
$ 13,417.08
$130,783.39
1
2
3
4
5
6
7
147
$ 4,360.00
$12,347.67
$ 516.16
$ 1,748.92
$ 1,004.63
$ 6,630.75
$ 2,000.00
$ 840.00
$1,100.00
7 months
11.25%
3 years
160 days
8.75%
$ 585.00
$ 2,000.00
$ 910.00
6.50%
Mathematics for Hospitality
Exercise Answers
Appendix
Review Problems, Chapter 5
1
2
3
4
5
6
7
8
$ 110.85
12.50%
$19,571.87
$ 4,629.17
$ 2,154.38
10 months
126 days
$ 12.34; $698.59
9a
9b
$ 20.00
$160.00
10
11
12
13a
13b
6 years
$5,200.00
3% p.a.
$ 0.63
$ 3,751.50
Chapter 6 – Compound Interest
Exercise 1
f
1a
1b
1c
1d
1e
2
3
4
5
6
7
8
9a
9b
9c
1
2
4
12
2
i
n
A
0.1
0.04
0.015
0.00625
0.0325
8
12
16
24
13
$ 857.44
1601.03
1586.23
1858.07
727.46
$6,884.47; $1884.47
$5,772.91
518.75
11,951
$4,123.91
$4,058.96
$2,341.80
$25,181.70
$25,870.70
$ 689.00
148
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 7 – Hospitality Statistics
Exercise 1
1a
1b
1c
2a
2b
2c
2d
3
4a
4b
4c
4d
66%
67%
65%
$24,000
$22,000
$26,000
$22,000
Rodriguez, $28,900
5
6
5
6
4e
4f
4g
4h
4i
4j
4k
4l
5
6a
6b
27
27
28
27 & 33
14
13
15
13
2.91
$148
$115
Exercise 2
1a
1b
2
3a
3b
3c
3d
0.796
0.403
0.754
0.609
0.524
0.435
0.663
Total
Rooms
Rooms
Sold
4
Rack
Rate
72.5%
5
108
6
108
7
8
Occ.
Rate
67.5%
Achiev.
Factor
$79.20
0.72
Yield
0.80
0.736
75.0%
175
Room
Rate
$100
72.0%
0.76
0.75
149
0.53
Mathematics for Hospitality
Exercise Answers
Appendix
Review Problems, Chapter 7
1a
1b
2a
2b
2c
2d
3a
3b
4
Sun.
95%
85%
72
70
74
68
2.5 days
2.0 days
0.694
Mon.
85%
Tues.
85%
Wed.
90%
Thurs.
70%
Fri.
Sat.
75% 95%
Chapter 8 – Yield and Price Factors
Exercise 1
Exercise 2
Exercise 3
1
2
3a
3b
4a
4b
5
1 27.2 lb.
2 12.8 lb.
3 15.0 lb.
4 5 kg.
5 4.4 kg.
6 8.0 kg.
6 12.5 lb.
7 10 cans
8 67.5 lbs.
1 $0.16/lb.
2a $5.46/kg.
2b $6.82/kg.
2c 1.25
3a $6.60/lb.
3b $7.513846/lb
75%
78.13%
87.1%
0.871
91.89%
0.919
0.741
3c 1.138462
4 $11.83/kg.
5a $2.40/lb.
5b
5c
6
7a
7b
7c
7d
8a
8b
8c
8d
8e
$3.24/lb.
1.35
$13.27/kg.
20 lb.
$172.00
$11.86/lb.
1.38
0.71
11 lbs., 14 oz.
$6.46
1.41
reciprocal;
1/1.41 = 0.71
150
Review Problems
1
2
3
4
5
7
8
9a
9b
10
11a
11b
11c
11d
11e
11f
89%
10.16 kg.
7 lb. 3 oz.
80%; 0.8
21.56 lb.
12 broilers
44 lb.
9.75 lb.
52 portions
$13.37
1.25
0.60
1.45
0.75
1.1765
0.92
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 9 – Menu Pricing
Exercise 1
Exercise 2
1
2
3
4
5
6
7
8
9
10a
10b
10c
1 $ 5.60
2 32%
3 $ 1.22
4 28%
5 $ 8.50
6 $ 2.49
7 $10.60
8a $0.90
8b $ 4.86
8c $ 9.18
9 $ 3.34
10 $ 1.95
11 31%
$ 0.20
$ 1.07
$ 0.06
$ 0.22
$ 0.18
$ 0.15
$ 2.99
$ 1.48
$ 1.24
$ 0.06
$ 0.05
$12.00
Exercise 3
1a
1b
1c
2a
2b
2c
3a
3b
3c
4a
4b
4c
5a
5b
5c
6a
6b
6c
$3.10
$6.60
68%
$2.35
$5.60
70.4%
$6.51
$4.53
69.6%
$5.78
$4.15
71.8%
30.8%
$4.67
69.2%
32.5%
$9.58
67.5%
7a
7b
7c
8a
8b
8c
9a
9b
9c
10a
10b
10c
11a
11b
11c
Review Problems
1a
1b
1c
2a
2b
2c
3
$ 1.86
$ 4.13
69%
7.4 lbs.
$ 4.97
$ 1.86
68%
4a
4b
4c
4d
5a
5b
5c
6
151
28 lbs.
40 lbs.
$139.60
$ 2.49
68 portions
$ 12.65
$ 1.90
$ 6.55
$ 8.96
$ 6.12
68.3%
$18.25
$12.30
67.4%
30.5%
$ 6.92
69.5%
$10.51
$ 7.41
70.5%
$ 1.65
$ 4.95
75%
Mathematics for Hospitality
Exercise Answers
Appendix
Chapter 10 – Profit or Loss Statement
Exercise 1
Food Sales
Beverage Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Labour Cost
Expenses
Net Profit
$130,000
70,000
45,500
21,000
66,500
133,500
56,000
30,000
47,500
33.25%
66.75%
23.75%
Exercise 2
1a
1b
1c
1d
2a
2b
2c
2d
3a
3b
3c
3d
57.1%
42.9%
$24,800
$16,800
61.9%
38.1%
$17,920
$10,350
62.5%
37.5%
$23,675
$13,659
4 Food Sales
Beverage Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Payroll
Expenses
Net Profit
$76,800
64.0%
$43,200
36.0%
$24,576
$10,800
$35,376
29.5%
$84,624
70.5%
$48,000
$14,400
22,220 (may vary due to rounding)
5 Food Sales
Beverage Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Payroll
Expenses
Net Profit
$108,000
$ 72,000
$ 34,020
$ 19,800
$ 53,820
$126,180
$ 73,800
$ 25,200
$ 27,180
152
29.9%
70.1%
15.1%
Mathematics for Hospitality
Exercise Answers
Appendix
Exercise 3
1 Food Sales
$52,500
Beverage Sales $28,300
Total Sales
$80,800
2 Food Sales
$78,600
Beverage Sales $48,500
Total Sales
$ 127,100
61.8%
38.2%
3 Food Sales
$78,000
Beverage Sales $42,000
Total sales
$120,000
Exercise 4, Chapter 10
1 Food Sales
Beverage Sales
Total Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Payroll
Expenses
Net Profit
2 Food Sales
Beverage Sales
Total Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Payroll
Expenses
Net Profit
3 Food Sales
Beverage Sales
Total Sales
Food Cost
Beverage Cost
Total Cost
Gross Profit
Payroll
Expenses
Net Profit
65.0%
35.0%
35.0%
Review Problem, Chapter 10
$ 43,200
$120,000
$ 24,576
$ 10,800
$ 35,376
$ 84,624
$ 17,424
$ 16,800
64.0%
36.0%
1
2
3
4
5
6
7
8
9
10
29.5%
70.5%
14.5%
$ 84,000
$140,000
$
$
$
$
$
45,360
94,640
54,600
16,800
23,240
34.0%
30.0%
32.4%
67.6%
16.6%
65.0%
$ 45,500
$130,000
$ 28,730
$ 13,650
$ 42,380
$ 87,620
$ 12,220
30.0%
67.4%
40.0%
18.0%
9.4%
153
$80,500
$48,300
$13,524
$23,828
$56.672
70.4%
$ 9,660
$27,370
$19,642
24.4%
CHAPTER-BASED PROBLEMS USING EXCEL
CHAPTER 2
You own an eclectic restaurant that sells different items from various cuisines. You want to create a
worksheet containing a database of 22 basic items listed below, showing the price at which the item is
sold, the dollar amount of HST (13%) that would be added to the price, and the resulting total price to
customers. (All amounts must be rounded to the nearest cent.) Thus your worksheet will end up having
3 columns in total.
Create the worksheet, giving the following titles to your three columns in boldface italic: Pretax Price,
HST, Aftertax Price. Once your worksheet is complete, reorder (sort) the food items by total price, from
highest to lowest. That worksheet is Deliverable 1. Also, present the same worksheet, with the food
items sorted alphabetically---- that is Deliverable 2.
Deliverable: 2 3-column worksheets
Item
Pretax Price
Hamburger
Coconut Cream Pie
Lemon Pepper Wings
Garden Salad
T-bone steak
Creamed Spinach
Crème Brulée
Artichoke Purée
Napoleon
Fried Squid
Vegan Hamburger
General Tao Chicken
Hot and Sour Soup
Goulash paprikash
Fruit Salad
Eggplant Parmesan
Club Sandwich
Rhubarb-Strawberry Pie
Caprese Salad
Tiramisu
Beef Barley Soup
Bulgarian Kavarma
154
$ 13.50
5.95
7.85
4.00
18.99
5.00
3.75
4.95
4.30
8.50
14.50
9.75
6.99
9.50
3.85
10.45
7.90
5.95
4.25
6.35
6.99
12.95
CHAPTER 3
Canada uses the Metric weight system, while the United States uses the Imperial weight system.
Create a worksheet that converts the given list of Metric weights to pounds and ounces, and the given
list of Imperial weights to kilograms and grams. These lists could help maintain good relations between
the two neighbouring countries!
List A
6
4.2
762
85
25
100
2,783
625
42
464
kg
kg
kg
kg
kg
g
kg
g
kg
g
List B
4 pounds and 3 ounces
900 pounds
62.5 pounds
13 ounces
2 pounds
1
134 pounds
2,200 pounds
450 pounds
8 ounces
0.5 ounce
Deliverable: one worksheet. 1st section with two columns (titles): Metric and Imperial
2nd section with two columns (titles): Imperial and Metric
As before, column titles must be in Boldface Italic
155
CHAPTER 4
a. Using formulas created in Excel, convert the following metric quantities to Imperial quantities
(fluid ounces).
1.
2L
2.
1.5
3.
750 ml
4.
500 ml
5.
1 L 200 ml
6.
1.5 gallons
7.
2 gallons and 3 pints
b. Using Excel, convert your answers from part a above back to metric quantities, using ml. for all.
c. Convert the following temperatures from Celsius to Fahrenheit, using an Excel formula:
1. -20° C
2. 45° C
3.
8° C
4.
0° C
5. 100° C
6. -40° C
7. 180° C
8. -80°C
d. Now, by creating the reverse formula, convert your answers from c above back into Fahrenheit.
Note: the keyboard shortcut for the degree symbol (°) is CTRL-SHIFT-@ pressed together,
followed by a space.
Deliverable: 2 worksheets, one containing your answers to parts a and b, the other containing
your answers to parts c and d.
156
CHAPTER 5
You did some simple interest calculations in this chapter. Now you will do the same calculations using
Excel.
For each of the following, create a formula in Excel to calculate:
a) the accumulation factor (1 + rT)
b) the accumulated value S
Add a row at the bottom of all this in your $S column, to show its total.
$Principal
Rate
Time
1.
2,000
9% p.a.
2 years
2.
4,500
8% p.a.
1.5 years
3.
7,000
8.5% p.a.
9 months
4.
1,400
12% p.a.
180 days
5.
2,500
8.75% p.a.
60 days
6.
2,000
6.375% p.a.
1¼ years
7.
13,000
5.5% p.a.
7 months
8.
125,000
8¾ % p.a.
193 days
(1 + rT)
$S
The answers to this problem are already in this workbook, since it is an exact copy of Ex. 5 on page 75.
Solutions are provided on page 147, so you can immediately see if your formulas were effective.
Were you able to simply create one formula and use it over and over, or were modifications necessary?
Explain why you answered this question as you did.
Deliverable: 1 worksheet containing 5 columns.
157
CHAPTER 6
You did some compound interest calculations in this chapter. Now you will do similar calculations using
Excel.
For each of the following, create a formula in Excel to calculate:
a) the amount of interest
b) the accumulated value A
$Principal
Rate
Time
Compounding
Frequency
$ Interest
$ A
1.
2,000
9% p.a.
2 years
semi-annual
_________
__________
2.
4,500
8% p.a.
1.5 years
quarterly
_________
__________
3.
7,000
8.5% p.a. 9 months
monthly
_________
__________
4.
1,400
12% p.a.
180 days
daily
_________
__________
5.
2,500
8.75% p.a. 60 days
daily
_________
__________
6.
2,000
6.375% p.a.1¼ years
quarterly
_________
__________
7.
13,000
5.5% p.a.
monthly
_________
__________
8.
125,000
8¾ % p.a. 193 days
daily
_________
__________
7 months
You may or may not have noticed that the principal, rate and time choices were precisely the same as
in the Excel exercise from Ch. 5, but with a compounding frequency added.
After completing this exercise, please create two new columns on this worksheet, titled Simple Interest
Result (S) and Difference. In the Simple Interest Result column, you will copy the solutions you found
for each problem in the Ch. 5 exercise, and in the Difference column, you will have Excel measure how
much larger the same investment grew, once compounding was involved. Add the row for Total at the
bottom, and have Excel calculate a total for each of your four right-hand columns.
Deliverable: one worksheet containing 8 columns.
158
CHAPTER 7
You manage a restaurant that serves a variety of 5 different main dishes, from various national
cuisines. In order to have enough inventory on hand, you have decided to create an inventory system.
The first part of that will be statistical calculations about how many servings of each main dish are sold,
each day of the week. Data are as follows (for the most recent two weeks):
Day
Roast Beef
Chili
Lasagna
Bun Cha
Escargots
Mon 1
23
14
32
28
10
Tues 1
20
18
29
35
8
Wed 1
18
12
25
27
14
Thurs 1
25
15
30
22
18
Fri 1
34
25
33
30
15
Sat 1
42
33
40
37
20
Sun 1
34
30
35
32
17
Mon 2
25
17
26
27
6
Tues 2
21
20
30
25
12
Wed 2
16
14
26
19
15
Thurs 2
23
21
36
28
13
Fri 2
33
29
42
31
15
Sat 2
36
33
39
30
19
Sun 2
29
24
31
22
13
Enter these data into an Excel spreadsheet, with the same rows and columns as presented above.
Provide a column for the total of each row, and a row for the total of each column, and a total of totals
in the far right corner. Also provide a row of average amount sold for each main dish. Properly program
Excel to calculate these row and column totals, as well as the row of averages.
Then, answer the following questions:
a. Which dish was the most popular one served, during these two weeks?
b. Which dish was the least popular one served, during these two weeks?
c. On which day of the week was the restaurant most busy, during the two-week period? (Assume
each customer has ordered a main dish.) Is the answer the same for both weeks?
159
d. On which day of the week was the restaurant least busy over the two-week period? (Assume
each customer has ordered a main dish.) Is the answer the same for both weeks?
e. If you calculate the average number of each dish sold within each week separately, are these
averages similar to each other per dish, or do they vary widely?
Deliverable:
One worksheet, and the answers to the five questions above, supported by your numeric data. For any
averages you calculate, be sure to use the special Excel AVG function…… do not create the formulas
yourself!
CHAPTER 8
Your task in this chapter’s Excel problem is to create formulas that will calculate the missing data.
Please note that each row has its own individual set of relationships; you will need to figure out the
correct formula and cell references for each missing figure. Whatever food this is, doesn’t matter!
As Purchased
Quantity
Price per
Unit
Yield
Factor
Edible Portion
Size
Edible Portion
Cost
Price
Factor
1.
20 kg
$3.00
/kg
85%
200 g
__________
________
2.
10 kg
$15.89
/kg
_____
500 g
__________
1.3333
3.
15 lbs
$ 35
/lb
80%
4 oz
__________
________
4.
_______
$7
/lb
70%
8 oz
5.
2 kg
$15
/kg
______
_____
6. 1 litre
$27
90%
100 ml
$5.00
$3.75
_________
________
1.25
________
Deliverable:
One worksheet in which this table is reproduced, with all 12 missing figures shown in bolded Bernard MT
Condensed font.
160
CHAPTER 9
The mathematics involved in this chapter is not too complicated, and it should be possible to easily
redo a few of the exercises already seen, using Excel. Try this:
A. (Ex. 2, p. 121)
Calculate the missing term for each of the following:
1.
2.
$Food cost
%Food cost
$1.68
30%
$2.88
3.
$1.68
5.
$2.72
6.
7,
$9.00
27%
4.
$4.50
$6.00
32%
28.5%
$3.34
Menu price
$8.75
31.5%
And now these: (Review problems 4, 5 and 6, p. 129)
B. 8-oz steaks (after cooking) were served to 56 guests in a restaurant. The trimming, cooking and
portioning waste was 30%. The restaurant paid $3.49 per pound of raw meat.
Calculate:
a) the total Edible Portion Quantity (EPQ)
b) the As Purchased Quantity (APQ) of meat
c) the total As Purchased Cost (APC) of purchased meat
d) the cost of one ready-to-serve steak
161
C. A restaurant purchased 15 kg of pork loin roast at 8.60 per kg. 150-g portions of cooked meat
were served. The meat had a waste factor of 32%.
Calculate:
a) how many portions of cooked meat were served.
b) the cost of the Edible Portion Quantity (EPQ) per kg.
c) the cost of a single portion.
D. Calculate the menu price per portion in question C, if the restaurant has a %food cost of
29%.
NOTE: Problem A should be set up as a small worksheet with three columns.
Problems B, C and D should be set up with as many columns as you will need (on the same
worksheet), including your known data as well as the values you are trying to calculate, so that
you can enter the data you have into formulas that will calculate the unknown values.
Deliverable:
One worksheet, on which all of problems A, B, C and D are presented. Be sure to copy in all the
given data, coloured black, while the values you calculate should be coloured red and made
bold.
162
CHAPTER 10
As you now understand from going through the chapter, this material is really all about the relationships
in the Profit and Loss Statement. Once you have a firm grasp of them, the math is easy. And therefore,
creating correct Excel formulas that will provide the results you need depends on that same set of
relationships.
PLEASE remember that Food and Beverage cost percentages are connected to their respective
revenues, while everything else is presented as a percentage of Total Revenue.
Find Excel ways to fill the empty spots:
A.
$
SALES
Food
Beverage
56,000
Total sales
COST
%
60
40
100
Food
28,560
Beverage
16,800
Total cost
GROSS PROFIT
Payroll
39
Expenses
12
NET PROFIT
163
C.
SALES
Food
COST
$
84,500
%
Beverage
35
Total sales
100
Food
34
Beverage
Total cost
32.60
GROSS PROFIT
Payroll
52,000
Expenses
23,400
NET PROFIT
Deliverable: One worksheet showing these two problems. Use columns A, B, C, and D only, placing
problem A above problem B.
And you can check your work, as you previously solved these problems (didn’t they seem vaguely
familiar?) as #2 and #3, pages 140 and 141!
The End
164