Last Updated
8th March 2011
SESSION 13 & 14
Percentages and Proportions
Lecturer:
University:
Domain:
Florian Boehlandt
University of Stellenbosch Business School
http://www.hedge-fundanalysis.net/pages/vega.php
Learning Objectives
1. Percentages and Proportions
2. Ratios
3. Assignment One
Percentages
To use 100 as a standard denominator:
Numerator
Denominator
Expressed as:
Proportion vs Percentage
Multiplying a Proportion by 100 yields the
percentage:
Proportion
Fraction
Percentage
10/100
0.1
10
5/25
0.2
20
33/429
0.13253
13.253
4 out of 10
0.4
40
Ratio vs Percentage
Here, the sum of the ratios yields the total to
base the fraction on:
Ratio
Total
Fraction
Percentage
1:9
10
0.1 and 0.9
10 and 90
5:20
25
0.2 and 0.8
20 and 80
33:396
429
0.13 and 0.87
13 and 87
4 versus 6
10
0.4 and 0.6
40 and 60
Calculation – Rule of Three
Example: What is 30% of 40?
Step
1
2
3
Percentage
100
1
30
→
→
→
Number
40
0.4
= 40 / 100
12
= 30 * 0.4
Example: What Percentage is 12 out of 40?
Step
1
2
3
Number
40
1
12
→
→
→
Percentage
100
2.5
= 100 / 40
30
= 12 * 2.5
Base Value Unknown
Example: Price of a product including 15% VAT is
ZAR 16. What is the net price?
1. Set up formula with an unknown variable:
X * (1 + 0.15) = 16
Representing 15%
2. Solve for X (Divide both sides by 1.15):
X * 1.15 / 1.15 = 16 / 1.15 = 13.913
Alternative – Rule of Three
Consider that ZAR 16 represents 100% + 15% =
115% of the total price:
Step
1
2
3
Percentage
115
1
100
→
→
→
ZAR
16
0.13913
13.913
= 16 / 115
= 100 * 0.13913
To double-check:
Step
1
2
3
ZAR
13.913
1
16
→
→
→
Percentage
100
7.188
= 100 / 13.913
115
= 16 * 7.188
Target Value Unknown
Example: A product costs ZAR 20 net. What does
it cost including 15% VAT?
1. Set up formula with an unknown variable:
20 * (1 + 0.15) = X
Representing 15%
2. Solve for X:
20 * 1.15 = 23
Increase Unknown
Example: A product costs ZAR 30 net and ZAR 34.5 incl.
tax. What is the VAT in %?
1. Set up formula with an unknown variable:
30 * (1 + X) = 34.5
Representing the unknown VAT
2. Solve for X (Divide by 30):
30 * (1 + X) / 30 = 34.5 / 30 = 1.15
3. Solve for X cont. (Subtract 1):
(1 + X) – 1 = 1.15 – 1 = 0.15 = 15%
Decrease Unknown
Example: Product’s original price is ZAR 50. The product on sale
now costs ZAR 45. What is the discount?
1. Set up formula with an unknown variable:
50 * (1 - X) = 45
2. Solve for X (Divide by 50):
50 * (1 - X) / 50 = 45 / 50 = 0.9
3. Solve for X cont. (Subtract 1):
(1 - X) - 1 = 0.9 - 1 = - 0.1
4. Solve for X cont. (Multiply by -1):
- X * (- 1) = -0.1 * (-1)  X = 0.1 = 10%
Alternative – Rule of Three
Consider that ZAR 50 represents 100% (or the
total price):
Step
1
2
3
ZAR
50
1
45
→
→
→
Percentage
100
2
= 100 / 50
90
= 45 * 2
Subtracting 90 from 100 gives the percentage
decrease:
100% - 90% = 10%
Example 3
Example: Change the following percentage to a
fraction:
42% = 42 / 100 = 21 / 50
Simplify by using the largest common denominator (2).
Example 4
Example: The new price of an article after a price
increase of 18% is R132.80. Calculate the original
price.
1. Set up formula with an unknown variable:
X * (1 + 0.18) = 132.8
2. Solve for X (Divide by 1.18):
X * 1.18 / 1.18 = 132.8 / 1.18 = 112.54
Example 5
Example: A piece of iron is warmed up form a
temperature of 23ºC to 38ºC. Calculate the
percentage change in rise of temperature.
1. Set up formula with an unknown variable:
23 * (1 + X) = 38
2. Solve for X (Divide by 23):
23 * (1 + X) / 23 = 38 / 23 = 1.652
3. Solve for X cont. (Subtract 1):
(1 + X) – 1 = 1.652 – 1 = 0.652 = 65.2%
Example 6
Example: A farmer produces 18000 bags of mealies
one year and the next year the produce decreases
by 20%. Calculate the amount of bags produced
after production decrease.
1. Set up formula with an unknown variable:
18000 * (1 - 0.2) = X
2. Solve for X:
18000 * 0.8 = 14400
Exercises (1/4)
• Calculate 28% of 4122.
• The fuel tank of a motor vehicle holds 65 l. The manufacturer
decides to increase the capacity with 24%. What is the new
capacity of the fuel tank?
• What percentage of R4200 are 380?
• If a motorist has to travel a distance of 1800 km and he covers
32 % during the first day, calculate the distance he still needs
to travel.
• Express 2/11 as a percentage.
• Express 23/30 as a percentage.
Exercises (2/4)
The 13% sales tax on an article in a shop is totalling R 24,56. Calculate the
following.
•
The price without the tax
•
The price with the tax.
A family earns R8500 per month. They have to pay 12% of the money on
electricity and 22% on house instalments. From the money that is left, after
paying all debts previously mentioned, the family banked 48% in a savings
account. Calculate the following:
•
The amount paid for the electricity.
•
The amount paid on the house instalment.
•
The amount of money left after paying the debts.
•
The amount of money, which they do save.
•
The amount of money left after doing everything above.
•
What is the percentage of money saved from the total salary?
Ratios and Proportions
A ratio is simply another way to express
fraction. The formula to be used is:
TOTAL / Sum of Ratios = Total per part
A proportion is another word for a decimal
fraction (i.e. 0.75 = 75 / 100 = 3 / 4)
Example 7
Divide 221 in the ratio 7:6:4.
1. The Total = 221 and the Sum of Ratios is 7 + 6
+ 4 = 17. Thus, using the formula:
221 / 17 = 13
2. Calculate the shares accordingly:
Share 1
7
= 7 * 13
91
Share 2
6
= 6 * 13
78
Share 3
4
= 4 * 13
52
Example X
Example: A group of 4 friends has ZAR 1000. Bill
gets half of what Jack receives. Jack gets the
same share as John, whereas Andrew makes 3
times the amount of Bill’s.
1. Start with the smallest share (i.e. Bill). That share
represents 1X. Since Jack makes twice what Bill
gets, his share represents 2X. So does Jack’s share.
Andrew’s share is 3X since it is three times the
amount of Bill’s share.
Example X
2. Build a formula including all shares:
1X + 2X + 2X + 3X = ZAR 1000
8X = ZAR 1000
3. Solve for X:
X = ZAR 1000 / 8 = ZAR 125
4. Calculate the shares according to the ratios:
Bill
1
= 1* 125
125 ZAR
Jack
2
= 2 * 125
250 ZAR
John
2
= 2 * 125
250 ZAR
Andrew
3
= 3 * 125
375 ZAR
Example 7
The following data is given to you: A recent survey
of Saturday shoppers at a local suburban shopping
centre, found the following amounts spent by
individual shoppers:
R100 R518 R325 R80 R455 R280 R918
R122 R144 R475 R290 R177
Share 1
7
= 7 * 13
91
Share 2
6
= 6 * 13
78
Share 3
4
= 4 * 13
52
Example - Proportion
The following data is given to you: A recent survey
of Saturday shoppers at a local suburban shopping
centre, found the following amounts spent by
individual shoppers:
R100 R518 R325 R80 R455 R280 R918
R122 R144 R475 R290 R177
Example - Proportion
1. What proportion of the shoppers spends more
than R500?  2/12 = 0.167 = 16.7%
2. What percentage of shoppers bought between
R200 and R500?  5/12 = 0.417 = 41.7%
Exercises (3/4)
•
•
Divide 1350 in the ratio 2, 3, 4.
Divide 850 in the ratio 25%, 30%, 45%.