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Calculations using percentages

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Calculations using fractions
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Solving problems

13.4 Calculations using percentages

Remember that per cent means per hundred. For example, \(\frac{2}{10} = \frac{20}{100}\), which is \(20\) per hundred, is equal to \(20\%\). Because \(2\) tenths is \(20\) hundredths, the fraction \(\frac{2}{10}\) can also be expressed as \(20\%\).

But what about something like \(7\) twentieths or \(\frac{7}{20}\)?

This fraction can also be expressed in percentage notation, but first we need to find an equivalent fraction with a denominator of \(100\). \(20 \times 5 = 100\), so multiply the numerator and multiply the denominator by \(5\) to change the fraction:

\[\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100}\]

\(\frac{7}{20}\) is called the simplest form of \(\frac{35}{100}\) because \(\frac{35}{100}\) cannot be expressed with a smaller numerator than \(7\).

So, now that the fraction has the denominator \(100\), it is \(35\) per \(100\) or \(35\%\). Because \(7\) twentieths is \(35\) hundredths, the fraction \(\frac{7}{20}\) can be expressed as \(35\%\).

To convert a fraction to a percentage, find the equivalent fraction with a denominator of \(100\). The numerator of the new fraction shows you the amount per hundred, or per cent.

Worked example 13.31: Converting a fraction to a percentage

Convert the following fraction to a percentage:

\[\frac{12}{25}\]

Find the equivalent fraction that has a denominator of \(100\).

Multiply the numerator and multiply the denominator by \(4\) to change the fraction:

\[\frac{12}{25} = \frac{12 \times 4}{25 \times 4} = \frac{48}{100}\]

Convert to a percentage.

\(48\) out of \(100\) is \(48\)%, so \(\frac{12}{25} = \frac{48}{100} = 48\%\).

To convert percentages to fractions, follow the process in reverse:

  1. Write the percentage as a fraction out of \(100\).
  2. Then simplify the fraction to its simplest form.

Worked example 13.32: Converting a percentage to a fraction

Convert the following percentage to a fraction: \(62\%\).

Write the percentage as a fraction over \(100\).

\[62\% = \frac{62}{100}\]

Simplify the fraction to its simplest form.

\[\frac{62}{100} = \frac{62 \div 2}{100 \div 2} = \frac{31}{50}\]

So, \(62\%\) is \(\frac{31}{50}\) in fraction form.

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Exercise 13.1

Convert the following fractions to percentages:

  1. \[\frac{6}{10}\]
  2. \[\frac{11}{25}\]
  3. \[\frac{13}{20}\]
  4. \[\frac{16}{50}\]
  5. \[\frac{7}{25}\]
  6. \[\frac{3}{5}\]
  7. \[\frac{1}{4}\]
  1. \[\frac{6}{10} = \frac{6\ \times \ 10}{10\ \times \ 10} = \frac{60}{100} = 60\%\]
  2. \[\frac{11}{25} = \frac{11\ \times \ 4}{25\ \times \ 4} = \frac{44}{100} = 44\%\]
  3. \[\frac{13}{20} = \frac{13\ \times \ 5}{20\ \times \ 5} = \frac{65}{100} = 65\%\]
  4. \[\frac{16}{50} = \frac{16\ \times \ 2}{50\ \times \ 2} = \frac{32}{100} = 32\%\]
  5. \[\frac{7}{25} = \frac{7\ \times \ 4}{25\ \times \ 4} = \frac{28}{100} = 28\%\]
  6. \[\frac{3}{5} = \frac{3\ \times \ 20}{5\ \times \ 20} = \frac{60}{100} = 60\%\]
  7. \[\frac{1}{4} = \frac{1\ \times \ 25}{4\ \times \ 25} = \frac{25}{100} = 25\%\]

Convert the following percentages to fractions.

  1. \[27\%\]
  2. \[18\%\]
  3. \[33\%\]
  4. \[15\%\]
  5. \[125\%\]
  6. \[100\%\]
  7. \[77\%\]
  1. \[27\% = \frac{27}{100}\]
  2. \[18\% = \frac{18}{100} = \frac{18 \div 2}{100 \div 2} = \frac{9}{50}\]
  3. \[33\% = \frac{33}{100}\]
  4. \[15\% = \frac{15}{100} = \frac{15 \div 5}{100 \div 5} = \frac{3}{20}\]
  5. \[125\% = \frac{125}{100} = \frac{125 \div 25}{100 \div 25} = \frac{5}{4} = 1\frac{1}{4}\]
  6. \[100\% = \frac{100}{100} = \frac{1}{1} = 1\]
  7. \[77\% = \frac{77}{100}\]

To calculate percentages of whole numbers, you use your knowledge of fractions.

For example, to calculate \(60\)% of \(105\), first convert \(60\)% to a fraction out of \(100\), simplify it, and then multiply this fraction by the given whole number.

You do not need to write the fraction in its simplest form before multiplying by the whole number, but this makes calculations much easier when you do not have a calculator.

Amount \(= \frac{60}{100}\ \times \ 105\ = \frac{60\ \div\ 20}{100\ \div\ 20} \times \frac{105}{1} = \frac{3}{5} \times \frac{105}{1} = \frac{3\ \times\ 105}{5\ \times\ 1} = \frac{315}{5} = 63\)

So, \(60\%\) of \(105\) is \(63\).

Worked example 13.33: Finding percentages of whole numbers

Find \(40\)% of \(315\).

Convert the percentage to a fraction.

\[40\% = \frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}\]

Multiply the fraction by the given number.

\[\frac{2}{5} \times 315 = \frac{2}{5} \times \frac{315}{1} = \frac{2 \times 315}{5} = \frac{630}{5} = 126\]

So, \(40\)% of \(315\) is \(126\).

It was possible to cross cancel \(5\) in calculations:

\[\frac{2}{5} \times 315 = \frac{2}{\mathbf{1}} \times \frac{\mathbf{63}}{1} = \frac{2 \times 63}{1} = 126\]

The answer is the same.

To calculate percentages of fractions is no different from calculating percentages of whole numbers.

For example, to calculate \(60\)% of \(\frac{1}{5}\), we first convert \(60\)% to a fraction out of \(100\), simplify it, and then multiply this fraction by the given fraction. So, \(60\%\) of \(\frac{1}{5} = \frac{60}{100} \times \frac{1}{5} = \frac{3}{5} \times \frac{1}{5} = \frac{3}{25}\).

Worked example 13.34: Calculating the percentage of a fraction

Find \(30\)% of \(\frac{3}{11}\).

Convert percentage to a fraction.

\[30\% = \frac{30}{100} = \frac{30 \div 10}{100 \div 10} = \frac{3}{10}\]

Multiply the fraction by the given number.

\[\frac{3}{10} \times \frac{3}{11} = \frac{3 \times 3}{10 \times 11} = \frac{9}{110}\]

So, \(30\)% of \(\frac{3}{11}\) is \(\frac{9}{110}\).

If the final and the original amounts are known, how do you work out the percentage of the final amount in relation to the original amount?

For example, what percentage of \(\text{R}\,\text{3,50}\) is \(70\)c? In this case we can think of \(70\)c as the final amount and \(\text{R}\,\text{3,50}\) as the original amount. To calculate the percentage, divide the two amounts and multiply by \(100\).

Remember to use the same units when dividing different amounts. Here, you are given cents and rands, so you need to make sure that both amounts are in the same unit.

\(\text{R}\,\text{3,50} = 350\) cents

Percentage \(= \frac{70}{350} \times \frac{100}{1} = \frac{1}{5} \times \frac{100}{1} = \frac{100}{5} = 20\%\)

So, \(70\)c is \(20\)% of \(\text{R}\,\text{3,50}\).

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Exercise 13.2

Find the percentage of the given amounts.

  1. Find \(40\)% of R\(300\).
  2. Find \(20\)% of R\(150\).
  3. Find \(60\)% of \(250 \text{ ml}\).
  4. Find \(45\)% of \(80 \text{ cm}\).
  5. Find \(27\)% of \(75 \text{ kg}\).
  1. \(40\)% of R\(300\) \(= \frac{40}{100} \times \frac{300}{1} = \frac{2}{5} \times \frac{300}{1} = \frac{600}{5} =\) R\(120\)
  2. \(20\)% of R\(150\) \(= \frac{20}{100} \times \frac{150}{1} = \frac{1}{5} \times \frac{150}{1} = \frac{150}{5} =\) R\(30\)
  3. \(60\)% of \(250\text{ ml}\) \(= \frac{60}{100} \times \frac{250}{1} = \frac{3}{5} \times \frac{250}{1} = \frac{3 \times 50}{1} = 150 \text{ ml}\)
  4. \(45\)% of \(80\text{ cm}\) \(= \frac{45}{100} \times \frac{80}{1} = \frac{9}{20} \times \frac{80}{1} = \frac{9 \times 4}{1 \times 1} = \frac{36}{1} = 36 \text{ cm}\)
  5. \(27\)% of \(75\text{ kg}\) \(= \frac{27}{100} \times \frac{75}{1} = \frac{27}{4} \times \frac{3}{1} = \frac{81}{4} = 20\frac{1}{4} = \text{20,25} \text{ kg}\)

Find the percentage of the given fractions.

  1. Find \(20\)% of \(\frac{2}{9}\).
  2. Find \(50\)% of \(\frac{13}{22}\).
  3. Find \(70\)% of \(\frac{16}{49}\).
  4. Find \(35\)% of \(\frac{30}{105}\).
  5. Find \(66\)% of \(\frac{44}{125}\).
  1. \(20\)% of \(\frac{2}{9}\) \(= \frac{20}{100} \times \frac{2}{9} = \frac{1}{5} \times \frac{2}{9} = \frac{2}{45}\)
  2. \(50\)% of \(\frac{13}{22}\) \(= \frac{50}{100} \times \frac{13}{22} = \frac{1}{2} \times \frac{13}{22} = \frac{13}{44}\)
  3. \(70\)% of \(\frac{16}{49}\) \(= \frac{70}{100} \times \frac{16}{49} = \frac{7}{10} \times \frac{16}{49} = \frac{1}{5} \times \frac{8}{7} = \frac{1 \times 8}{5 \times 7} = \frac{8}{35}\)
  4. \(35\)% of \(\frac{30}{105}\) \(= \frac{35}{100} \times \frac{30}{105} = \frac{35 \div 5}{100 \div 5} \times \frac{30 \div 15}{105 \div 15} = \frac{7}{20} \times \frac{2}{7} = \frac{14}{140} = \frac{1}{10}\)
  5. \(66\)% of \(\frac{44}{125}\) \(= \frac{66}{100} \times \frac{44}{125} = \frac{33}{50} \times \frac{44}{125} = \frac{33}{25} \times \frac{22}{125} = \frac{33\ \times \ 22}{25\ \times \ 125} = \frac{726}{3\ 125}\)

What percentage of \(\text{R}\,\text{7,20}\) is \(9\)c?

\(\frac{9}{720} \times \frac{100}{1} = \frac{1}{80} \times \frac{100}{1} = \frac{10}{8} = 1\frac{2}{8} = 1\frac{1}{4} = \text{1,25}\%\)
\(9\)c is \(\text{1,25}\%\) of \text{R}\,\text{7,20}\).

To calculate a percentage increase or decrease:

Calculate the difference between the two amounts.

  • If it is a percentage increase, then subtract the original amount from the new amount.
  • If it is a percentage decrease, then subtract the new amount from the initial amount.

Divide the difference by the original amount and multiply the answer by \(100\).

For example, we need to calculate the percentage increase if the R\(40\) price of a bus ticket is increased to R\(64\).

Amount increased = R\(64\) − R\(40\) = R\(24\).

Divide the difference, \(24\), by the original amount, \(40\), and multiply the answer by \(100\).

The percentage increase \(= \frac{24}{40} \times \frac{100}{1} = \frac{24 \div 8}{40 \div 8} \times \frac{100}{1} = \frac{3}{5} \times \frac{100}{1} = \frac{300}{5} = 60\%\)

Worked example 13.35: Calculating percentage increase or decrease

Calculate the percentage decrease if the price of petrol goes down from \(\text{R}\,\text{22,00}\) a litre to \(\text{R}\,\text{21,72}\) a litre.

Find the amount of the decrease.

Amount decreased = \(\text{R}\,\text{22,00} - \text{R}\,\text{21,72} = \text{R}\,\text{0,28} = \text{28 cents}\).

Find the percentage of the decrease.

Original price = \(\text{R}\,\text{22,00} = \text{2 200 cents}\).

Percentage decrease \(= \frac{28}{2\ 200} \times \frac{100}{1} = \frac{28}{2\ 200 \div 100} \times \frac{100 \div 100}{1} = \frac{28}{22} = \frac{14}{11} = \text{1,27}\%\)

Hint: Use a calculator to work out the answer to \(14 \div 11\) and round off to \(2\) decimal places.

So, a \(28\) cent decrease is \(\text{1,27}\)% of the original price \(\text{R}\,\text{22,00}\).

Worked example 13.36: Calculating an amount given a percentage increase or decrease

Calculate how much a bicycle will cost if its original price of R\(900\) is reduced by \(15\)%.

The calculation involves finding \(15\)% of R\(900\) and then subtracting that amount from the original price.

Find 15% of R900.

\(15\)% of \(900\) \(= \frac{15}{100} \times \text{R}900 = \frac{3}{20} \times \frac{900}{1} = \frac{3}{20 \div 20} \times \frac{900 \div 20}{1} = \frac{3}{1} \times \frac{45}{1} = 135\)

So, \(15\)% of \(900\) is R\(135\).

Find the new price.

The new price of bicycle = R\(900\) – R\(135\) = R\(765\).

Another way of calculating the new price is to calculate \(100\% - 15\% = 85\%\) of the original price. This is because the price decrease of \(15\)% essentially means that the new price is \(85\)% of the old price.

\(85\)% of \(900\) \(= \frac{85}{100} \times \text{R}900 = \frac{85 \div 5}{100 \div 5} \times \frac{900}{1} = \frac{17}{20} \times \frac{900}{1} = \frac{17}{20 \div 20} \times \frac{900 \div 20}{1} = \frac{17}{1} \times \frac{45}{1} = 765\)

So, the new price is \(\text{R}765\). The answer is the same.

Exercise 13.3

Use a calculator to answer the questions below. Round your answers to two decimal places.

Calculate the percentage increase if:

  1. the price of petrol goes up from \(\text{R}\,\text{22,00}\) a litre to \(\text{R}\,\text{23,52}\) a litre
  2. the price of milk goes up from \(\text{R}\,\text{16,00}\) a litre to \(\text{R}\,\text{16,65}\) a litre
  3. the price of apple juice goes up from \(\text{R}\,\text{33,99}\) a litre to \(\text{R}\,\text{34,15}\) a litre
  4. the price of a house goes up from \(\text{R}\,\text{750 000,00}\) to \(\text{R}\,\text{820 000,00}\)

a) increase from \(\text{R}\,\text{22,00}\) a litre to \(\text{R}\,\text{23,52}\) a litre:
\(\text{23,52} - \text{22,00} = \text{1,52}\)
\(\dfrac{\text{1,52}}{\text{22,00}} \times \dfrac{100}{1} = \dfrac{152}{(2\ 200)} \times \frac{100}{1} = \dfrac{152}{22} = \text{6,90}\%\) increase

b) increase from \(\text{R}\,\text{16,00}\) a litre to \(\text{R}\,\text{16,65}\) a litre:
\(\text{16,65} - \text{16,00} = \text{0,65}\)
\(\dfrac{\text{0,65}}{\text{16,65}} \times \dfrac{100}{1} = \dfrac{65}{(1\ 665)} \times \dfrac{100}{1} = \dfrac{13}{333} \times \frac{100}{1} = \frac{(1\ 300)}{333} = \text{3,90}\%\) increase

c) increase from \(\text{R}\,\text{33,99}\) a litre to \(\text{R}\,\text{34,15}\) a litre:
\(\text{34,15} - \text{33,99} = \text{0,16}\)
\(\dfrac{\text{0,16}}{\text{33,99}} \times \dfrac{100}{1} = \dfrac{16}{(3\ 399)} \times \dfrac{100}{1} = \dfrac{(1\ 600)}{(3\ 399)} = 0,47\%\) increase

d) increase from \(\text{R}\,\text{750 000,00}\) to \(\text{R}\,\text{820 000,00}\):
\(\text{820 000} - \text{750 000} = \text{70 000}\)
\(\dfrac{\text{70 000}}{\text{750 000}} \times \dfrac{100}{1} = \dfrac{7}{75} \times \dfrac{100}{1} = \dfrac{700}{75} = \text{9,33}\%\) increase

Calculate the percentage decrease if:

  1. the price of petrol goes down from \(\text{R}\,\text{22,00}\) a litre to \(\text{R}\,\text{20,50}\) a litre
  2. the price of butter goes down from \(\text{R}\,\text{140,00}\) per kg to \(\text{R}\,\text{138,00}\) per kg
  3. the price of a car goes down from \(\text{R}\,\text{722 000,00}\) to \(\text{R}\,\text{621 000,00}\)
  4. the price of meat goes down from \(\text{R}\,\text{212,00}\) per kg to \(\text{R}\,\text{205,70}\) per kg

a) decrease from \(\text{R}\,\text{22,00}\) a litre to \(\text{R}\,\text{20,50}\) a litre:
\(\text{22,00} - \text{20,50} = \text{1,50}\)
\(\frac{\text{1,50}}{\text{22,00}} \times \frac{100}{1} = \frac{150}{2\ 200} \times \frac{100}{1} = \frac{150}{22} = \text{6,81}\%\) decrease

b) decrease from \(\text{R}\,\text{140,00}\) per kg to \(\text{R}\,\text{138,00}\) per kg:
\(140 - 138 = 2\)
\(\frac{2}{140} \times \frac{100}{1} = \frac{1}{70} \times \frac{100}{1} = \frac{10}{7} = \text{1,43}\%\) decrease

c) decrease from \(\text{R}\,\text{722 000,00}\) to \(\text{R}\,\text{621 000,00}\):
\(\text{722 000} - \text{621 000} = \text{101 000}\)
\(\frac{\text{101 000}}{\text{722 000}} \times \frac{100}{1} = \frac{101}{722} \times \frac{100}{1} = \frac{10\ 100}{722} = \text{13,99}\%\) decrease

d) decrease from \(\text{R}\,\text{212,00}\) per kg to \(\text{R}\,\text{205,70}\) per kg:
\(\text{212,00} - \text{205,70} = \text{6,30}\)
\(\frac{\text{6,30}}{\text{212,00}} \times \frac{100}{1} = \frac{630}{21\ 200} \times \frac{100}{1} = \frac{630}{212} = \text{2,97}\%\) decrease

Calculate the new price.

  1. The original price of a fridge, \(\text{R}\,\text{2 400}\), is reduced by \(20\)%.
  2. The original price of a table, \(\text{R}\,\text{12 500}\), is increased by \(12\)%.
  3. The original price of a phone, \(\text{R}\,\text{3 999}\), is reduced by \(16\)%.
  4. The original price of an ice cream, \(\text{R}\,\text{14,00}\), is increased by \(7\)%.
  5. The original price of jeans, \(\text{R}\,\text{380}\), is reduced by \(11\)%.

a) The new price of a fridge = \(\text{R}\,\text{2 400}\) reduced by \(20\)%
\(= \frac{80}{100} \times \frac{\text{2 400}}{1} = 80 \times 24 = \text{R}\,\text{1 920}\).

b) The new price of a table = \(\text{R}\,\text{12 500}\) increased by \(12\)%
\(= \text{12 500} + \left( \frac{12}{100} \times \frac{12\ 500}{1} \right)\)
\(= \text{12 500} + (12 \times 125)\)
\(= \text{12 500} + \text{1 500}\)
\(= \text{R}\,\text{14 000}\)

c) The new price of a phone = \(\text{R}\,\text{3 999}\) reduced by \(16\)%
\(= \text{3 999} - \left( \frac{16}{100} \times \frac{3\ 999}{1} \right)\)
\(= \text{3 999} - (\text{639,84})\)
\(= \text{R}\,\text{3 359,16}\)

d) The new price of an ice cream = \(\text{R}\,\text{14,00}\) increased by \(7\)%
\(= \text{14,00} + \left( \frac{7}{100} \times \frac{14}{1} \right) = 14,00 + \left( \frac{7}{50} \times \frac{7}{1} \right)\)
\(= \text{14,00} + \frac{49}{50}\)
\(= \text{14,00} + \text{0,98}\)
\(= \text{R}\,\text{14,98}\)

e) The new price of jeans = \(\text{R}\,\text{380}\) reduced by \(11\)%
\(= \frac{89}{100} \times \frac{380}{1}\)
\(= \frac{89}{100 \div 20} \times \frac{380 \div 20}{1}\)
\(= \frac{89}{5} \times \frac{19}{1}\)
\(= \frac{\text{1 691}}{5}\)
\(= \text{R}\,\text{338,20}\)

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