Chapter 4
Ratios and
Proportions
Copyright © 2020 McGraw-Hill Education Limited.
Microsoft® PowerPoint® Presentation by Julie Howse, St. Lawrence College.
Previous edition updated by Rob Sorenson, Camosun College.
Learning Objectives
LO1 Set up and manipulate ratios.
LO2 Set up and solve proportions.
LO3 Use proportions to allocate or prorate an
amount on a proportional basis.
LO4 Use quoted exchange rates to convert
between currencies.
LO5 Related currency exchange rate movement to
currency appreciation or depreciation.
(exclude)
LO6 Interpret and use index numbers. (exclude)
4-2
Introduction
• Ratios and proportions are widely used in
business to compare associated quantities.
• For example, profit compared to total sales
is a meaningful relationship that a business
person may need to report or present. See
top of page 104 for what we use ratios for.
• This chapter explores ratios and proportions
and then studies currency exchange rates.
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4-3
Ratios
• A ratio is a comparison of two or more
quantities.
• Ratios may be expressed using a colon, as a
fraction, as a decimal or as a percent.
5
5: 10
0.5
50%
10
See example 4.1 A on page 105 in your textbook
10 nurses caring for 60 patience….
4-4
From page 105
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4-5
From page 105
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4-6
Ratios. Can be three terms.
• A ratio is a comparison of more than 2
terms.
Example: suppose a store sells $2,000
worth of product X; $1,500 of product Y;
and $2,500 of product Z.
That gives us:
2000: 1500: 2500
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4-7
Easy way to find a common factor
Hint: Take the smallest term in the ratio and use that to divide the other
values into. In the example above 75 is the smallest value.
So, 150/75 = 2. And 225/75 = 3.
This may not always be possible.
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4-8
Reduced to its lowest terms
• A ratio can be reduced to its lowest terms.
1. If all terms are whole numbers (not fractions),
divide every term by a common factors.
2. If one or more of the terms are decimal
numbers (such as 1.20: 1.68) make all the
terms whole numbers by moving the decimal
point and then reduce as above (120:168).
See next slide
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4-9
See example 4.1C on top of page
106
Be mindful that moving the decimal
does not give you 12 instead of 120
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10
Converting Ratios
• Sometimes, it is preferable to convert a ratio
to one where the lowest term is 1.
• This can make the relative size of the terms
more apparent.
• For example:
15: 26 → 1: 1.73
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4-11
Skill Check
1. Reduce the following ratios to lowest
terms
i. 250: 375
ii. 0.05: 0.30: 0.45
Please look at exercise 4.1 on page 107.
Do number: a, c, e, and j and k
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4-12
Skill Check
Please look at exercise 4.1 on page 107. Do number:
a) 5:75 =
c) 0.20:0.80 =
e) ½ : ¼ =
j) 3: 1.5 : 6 =
k) 4: 28: 48 =
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4-13
Proportions
• A proportion is a statement of the equality
of two ratios.
• Consider the statement “the ratio of the
sales of X to the sales of Y is 4:3”
• This can be expressed as: 𝑥: 𝑦 = 4: 3
• Graphically, the
columns x and y are
proportional to 4 and 3.
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4-14
Proportions
• We can use proportions to solve problems.
• First, convert the ratio to its equivalent
fraction.
𝑥 4
=
𝑦 3
• Then, given either x or y, you can solve for
the other variable.
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4-15
Proportions
• Let’s solve for y when x is $1800:
𝑥
𝑦
=
4
3
$1800 4
=
𝑦
3
4𝑦 = $1800 × 3
𝑦 = $1350
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4-16
Example: 4.2B
Betty and Lois have already invested $8960 and $6880,
respectively, in their partnership. If Betty invests another
$5000, what amount should Lois contribute to maintain
their investments in the original ratio?
Lois’s investment : Betty’s investment = $6880 : 8960
Let Lois’s additional investment be x. Then
𝑥
$6880
=
$5000 $8960
Therefore,
$6880 × $5000
𝑥=
= $3839.29
$8960
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4-17
Ratios: Reducing to the lowest terms.
Example: suppose a store sells $2,000
worth of product X; $1,500 of product Y;
and $2,500 of product Z.
That gives us:
2000: 1500: 2500
= 200:150:250
= 20: 15: 25
= 4: 3: 5
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4-18
Proportions with Three Variables
• When we are given a proportion with three terms,
we can separate the proportion into three
equations (one for each pair of terms).
• For example:
𝑥: 𝑦: 𝑧 = 4: 3: 5
• Gives us:
𝑥: 𝑦 = 4: 3 𝑦: 𝑧 = 3: 5 𝑥: 𝑧 = 4: 5
• Which becomes:
𝑥 4
𝑦 3
𝑥 4
=
=
=
𝑦 3
𝑧 5
𝑧 5
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4-19
Example: 4.2D
A 560-bed hospital operates with 232 registered nurses and 185
other support staff. The hospital is about to open a new 86-bed wing.
Assuming the same proportionate staffing levels, how many more
nurses and support staff will need to be hired?
Let n represent the number of additional nurses and s the number of
additional staff. Then n and s must satisfy the proportion
Beds : Nurses : Staff = 560 : 232 : 185 = 86 : n : s
Therefore,
560 86
=
232
𝑛
560𝑛 = 86 × 232
𝑛 =35.6
560 86
=
185
𝑠
560𝑠 = 86 × 185
s = 28.4
Rounding, the hospital should hire 36 nurses and 28 staff.
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4-20
See question 18 on page 114
• The West Essex School Board
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4-21
One of the homework video questions in Connect
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4-22
An alternative way to do the
question…
And get to the same answer…say you
switched the order in which you stated the
equation.
7412 : 348 = 7780 : n
Convert the ratio to a fraction
7412 = 7780
348
n
Explanation continues on the next slide…
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4-23
An alternative way to do the
question…
7412
7780
=
348
n
7412 x n = 7780 x 348
n = 7780 x 348
7412
n = 365
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4-24
Skill Check
i.
The Smiths wish to purchase a larger house to accommodate
their growing family. The current year’s property tax on their
home amounts to $3658 based on its assessed value of
$425,000. The assessed value of a property they are seriously
considering is $572,000. What property tax can the Smiths
expect to pay on this home if property taxes are in the same
ratio as assessed values?
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4-25
Skill Check
Based on past experience, a manufacturing process requires 4.5
hours of direct labour for each $3000 worth of raw materials
processed. If the company is planning to consume $150,500 worth of
raw materials, what total amount should it budget for labour at
$27.50 per hour?
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4-26
Allocation and Proration
• Often, money must be allocated among
partners, departments, cost centers, etc.
• If the allocation is not made equally, we can
use a procedure called proration.
• Proration allows us to allocate money on a
proportionate basis.
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4-27
Example: 4.3A
The partnership of Mr. X, Mr. Y, and Ms. Z has agreed to distribute
profits in the same proportion as their respective capital investments
in the partnership. How will the recent period’s profit of $28,780 be
allocated if Mr. X’s capital account shows a balance of $34,000, Mr.
Y’s shows $49,000, and Ms. Z’s shows $54,500?
The total amount invested by all three partners = $34,000 + $49,000 + $54,500 = $137,500
We can use this ratio to determine the amount each partner will receive:
𝑃𝑎𝑟𝑡𝑛𝑒𝑟′𝑠 𝑠ℎ𝑎𝑟𝑒 𝑃𝑎𝑟𝑡𝑛𝑒𝑟′𝑠 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
=
𝑇𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑓𝑖𝑡
𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
Mr. X:
𝑀𝑟.𝑋 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒
$28,780
=
$34,000
$137,500
→ 𝑀𝑟. 𝑋 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒 =
$34,000
×
$137,500
Mr. Y:
𝑀𝑟.𝑌 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒
$28,780
=
$49,000
$137,500
→ 𝑀𝑟. 𝑌 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒 =
$49,000
× $28,780
$137,500
= $10,256.15
Mr. Z:
𝑀𝑟.𝑍 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒
$28,780
=
$54,500
$137,500
→ 𝑀𝑟. 𝑍 ′ 𝑠 𝑠ℎ𝑎𝑟𝑒 =
$54,500
× $28,780
$137,500
= $11,407.35
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$28,780 = $7,116.51
4-28
Skill Check
Three business partners David, Samantha and
Gurpreet invested money into a coffee shop
in the ratio of 2:5:8 respectively. After the first
fiscal year profits of $650,700 were to be
divided between the three of the business
owners. How much of the profit should go to
each of the owners.
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4-29
Let us do: Question 4 on page 120
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4-30
Let us do: And question 5 on page 120
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4-31
Let us do: And question 5 on page 120
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4-32
Currency Conversion
• The exchange rate between two currencies
is the amount of one currency needed to
purchase one unit of the other.
• Suppose that C$1.00 = US$0.75569
• To get the equivalent C$ of US$1.00, divide
both sides by 0.75569
See the next slide….
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4-33
Currency Cross-Rate Table
How many units of the currency in the left column would be
required to buy one unit of the currency on the right?
How much of these,
for one of these?
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Currency Conversion
Calculate the cost in C$ to purchase US$600.
We have two equivalent forms:
𝑈𝑆$0.75569
𝐶$1.323294
𝑜𝑟
𝐶$1.00
𝑈𝑆$1.00
𝑈𝑆$0.75569
𝑈𝑆$600
=
𝐶$1.00
𝐶$𝑥
Cross multiply to find C$793.98
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4-35
Currency Conversion
Calculate the cost in C$ to purchase US$600.
We have two equivalent forms: What if you used
the second fraction?
𝑈𝑆$0.75569
𝐶$1.323294
𝑜𝑟
𝐶$1.00
𝑈𝑆$1.00
𝐶$1.323294
𝐶$𝑥
=
𝑈𝑆$1.00
𝑈𝑆$600
Cross multiply to find C$793.98. (See next slide)
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4-36
Currency Conversion
𝐶$1.323294
𝐶$𝑥
=
𝑈𝑆$1.00
𝑈𝑆$600
C$1.323294 x US$600 = US$1 x C$𝑥
C$𝑥 = C$1.323294 x US$600
US$1
= C$793.98
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4-37
Skill Check
i.
Using the Currency Cross Table on page
129, how much will it cost in Canadian
dollars to purchase € 700?
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4-38
Currency Cross-Rate Table
How many units of the currency in the left column would be
required to buy one unit of the currency on the right?
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4-39
Skill Check
i.
Using the Currency Cross Table, how
much will it cost in Canadian dollars to
purchase 700 Euros?
€0.65537 = €700
C$1
= C$?
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4-40
Skill Check
i.
How much will it cost in Canadian dollars
to purchase € 700, is it cost C$1.52586 to
purchase €1 ?
C$1.52586 = C$?
€1
€700
= C$?
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4-41
Let us look at page 129 one last time
• Question 1 of exercise 4.4. Using the crossmultiplication method
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4-42
Currency Cross-Rate Table
How many units of the currency in the left column would be
required to buy one unit of the currency on the right?
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4-43
Let us look at page 129 one last time
• Question 1 of exercise 4.4. Using the crossmultiplication method
• US$1856 = C$? Use table 4.2
US$1
=
C$1.32329
US$1856
CS?
• CS? = 1856 x 1.32329 =
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4-44
Let us look at page 129 one last time
• Question 2 of exercise 4.4. Using the crossmultiplication method
£123.50 = €? Use table 4.2
£1
=
€ 1.10720
£ 123.50
€?
€? = 1.10720 x 123.50 =
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4-45
Let us do question 9 on page 140
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4-46
Let us do, question 13 on page 140
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4-47
Let us do, question 13 on page 140
C$1.5947 – C$1.632 = (C$0.0373). Amount the Canadian dollar weakened by compared
to the British Pound during the eight weeks the Percival’s were in Great Britain. Which
also means the Pound strengthened by that amount.
The currency (i.e., “asset”) the Percival’s had upon returning was £ 242 – which was
now worth more in Canadian dollars than when they bought it initially. More by how
much? £ 242 x 0.0373 = C$9.0266. So even though the C$ weakened, for the Percival’s
it was a gain.
4-48
Let us look at example 4.4F
• On page 128….
• When dealing with 2 moving parts.
…see the next 2 slide
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4-49
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Limited.
50
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52
Chapter 4
End of Chapter
Copyright © 2020 McGraw-Hill Education Limited.
Microsoft® PowerPoint® Presentation by Julie Howse, St. Lawrence College.
Previous edition updated by Rob Sorenson, Camosun College.