SESSION 13 & 14

Last Updated8th March 2011

Percentages and Proportions

Lecturer: Florian BoehlandtUniversity: University of Stellenbosch Business SchoolDomain: http://www.hedge-fund-analysis.net/pages/ve

ga.php

Learning Objectives

1. Percentages and Proportions2. Ratios3. Assignment One

Percentages

To use 100 as a standard denominator:

Expressed as:

Numerator

Denominator

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

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

2. Solve for X (Divide both sides by 1.15):X * 1.15 / 1.15 = 16 / 1.15 = 13.913

Representing 15%

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%

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

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%

Representing the unknown VAT

Decrease UnknownExample: 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) = 452. Solve for X (Divide by 50):

50 * (1 - X) / 50 = 45 / 50 = 0.93. Solve for X cont. (Subtract 1):

(1 - X) - 1 = 0.9 - 1 = - 0.14. 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):

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 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) = 132.82. 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) = 382. Solve for X (Divide by 23):

23 * (1 + X) / 23 = 38 / 23 = 1.6523. 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) = X2. 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 X2. 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%

•Divide 1350 in the ratio 2, 3, 4.•Divide 850 in the ratio 25%, 30%, 45%.

Exercises (3/4)