SESSION 13 & 14 Last Updated 8 th March 2011 Percentages and Proportions
Dec 30, 2015
Lecturer: Florian BoehlandtUniversity: University of Stellenbosch Business SchoolDomain: http://www.hedge-fund-analysis.net/pages/v
ega.php
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?
Example: What Percentage is 12 out of 40?Step Number Percentage
1 40 → 1002 1 → 2.5 = 100 / 403 12 → 30 = 12 * 2.5
Step Percentage Number1 100 → 402 1 → 0.4 = 40 / 1003 30 → 12 = 30 * 0.4
Alternative – Rule of Three
Consider that ZAR 16 represents 100% + 15% = 115% of the total price:
To double-check:
Step Percentage ZAR1 115 → 162 1 → 0.13913 = 16 / 1153 100 → 13.913 = 100 * 0.13913
Step ZAR Percentage1 13.913 → 1002 1 → 7.188 = 100 / 13.9133 16 → 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
2.Solve for X:20 * 1.15 = 23
Representing 15%
Alternative – Rule of Three
Consider that ZAR 50 represents 100% (or the total price):
Subtracting 90 from 100 gives the percentage decrease:100% - 90% = 10%
Step ZAR Percentage1 50 → 1002 1 → 2 = 100 / 503 45 → 90 = 45 * 2
Example 3
Example: Change the following percentage to a fraction:42% = 42 / 100 = 21 / 50Simplify by using the largest common denominator (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
• 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 (1/4)
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?
Exercises (2/4)
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 = 132.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 10008X = 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%