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Percentages

1.7 Percentages (EMGZ)

Percentage
A number represented as a part of \(\text{100}\).

How to calculate a percentage of an amount (EMG32)

  1. Write the percentage as a fraction with the denominator \(\text{100}\), for example \(\text{20}\% = \frac{\text{20}}{\text{100}}\). OR write the percentage as a decimal fraction, for example \(\text{20}\% = \text{0,2}\).
  2. Multiply this fraction / decimal fraction with the amount that is given.

Let's see how this works in an example.

Worked example 16: Working out percentages of amounts

Use a calculator to answer the following questions:

  1. How many people live in rural areas?

  2. How many T.B. patients are H.I.V. positive?

  3. How many people had never voted before the 1994 election?

  1. \(\text{43}\% = \text{43} \div \text{100}\) \(\frac{\text{43}}{\text{100}} \times \text{50 586 757} = \text{21 752 305}\) people live in rural areas. With a calculator: To find \(\text{43}\%\) of \(\text{50 586 757}\) key in: \(\text{43} \div \text{100} \times \text{50 586 757} =\) OR \(\text{43}\% \times \text{50 586 757} =\)
  2. \(\text{80}\% = \text{80} \div \text{100}\) \(\frac{\text{80}}{\text{100}} \times \text{1 291} = \text{1 032}\) patients
  3. \(\text{73}\% = \text{73} \div \text{100}\) \(\frac{\text{73}}{\text{100}} \times \text{21 700 000} = \text{15 841 000}\) people had never voted before.

Worked example 17: Working out one amount as a percentage of another amount

Top Teenage T-shirts printed \(\text{120}\) T-shirts. They sold \(\text{72}\) T-shirts immediately. What percentage of the T-shirts were sold?

\(\text{72}\) of the \(\text{120}\) T-shirts were sold

\(\text{72} \div \text{120} \times \text{100} = \text{60}\%\). So \(\text{60}\%\) of the T-shirts were sold.

Calculating the percentages of amounts

Exercise 1.10

Calculate the following without a calculator:

\(\text{25}\%\) of \(\text{R}\,\text{124,16}\)

\(\text{25}\% = \frac{\text{1}}{\text{4}}\). \(\frac{\text{1}}{\text{4}} \text{ of } \text{R}\,\text{124,16} = \text{R}\,\text{124,16} \div \text{4} = \text{R}\,\text{31,04}\)

\(\text{50}\%\) of \(\text{30}\) \(\text{mm}\)

\(\text{50}\% = \frac{\text{1}}{\text{2}}\). \(\frac{\text{1}}{\text{2}} \text{ of } \text{30}\text{ mm} = \text{30}\text{ mm} \div \text{2} = \text{15}\text{ mm}\)

Using your calculator and calculate:

\(\text{15}\%\) of \(\text{R}\,\text{3 500}\)

\(\text{R}\,\text{525}\)

\(\text{12}\%\) of \(\text{25}\) litres

\(\text{3}\) litres

\(\text{37,5}\%\) of \(\text{22}\) \(\text{kg}\)

\(\text{8,25}\) \(\text{kg}\)

\(\text{75}\%\) of \(\text{R}\,\text{16,92}\)

\(\text{R}\,\text{12,69}\)

\(\text{18}\%\) of \(\text{105}\) \(\text{m}\)

\(\text{18,9}\) \(\text{m}\)

\(\text{79}\%\) of \(\text{840}\) \(\text{km}\)

\(\text{663,6}\) \(\text{km}\)

Calculate what percentage the first amount is of the second amount (you may use your calculator):

\(\text{120}\) of \(\text{480}\)

\(\text{25}\%\)

\(\text{23}\) of \(\text{276}\)

\(\text{8,3}\%\)

\(\text{3 500}\) \(\text{ml}\) of \(\text{5}\) litres

\(\text{70}\%\)

\(\text{750}\) \(\text{g}\) of \(\text{2}\) \(\text{kg}\)

\(\text{37,5}\%\)

\(\text{4}\) out of \(\text{5}\) for a test

\(\text{90}\%\)

\(\text{2}\) out of \(\text{14}\) balls

\(\text{14,3}\%\)

Percentage discounts and increases (EMG33)

Look at the following extracts from newspaper articles and adverts:

Cost price
The amount that the dealer / trader / merchant pays for an article.
Marked price
This is the price of the article.
Selling price
This is the price after discount.
Profit
Sale price \(-\) cost price.

Discounts and increases

Exercise 1.11

The price of a tub of margarine is \(\text{R}\,\text{6,99}\). If the price rises by \(\text{10}\%\), how much will it cost?

New price is \(\text{R}\,\text{6,99}\) + \(\text{10}\%\) of \(\text{R}\,\text{6,99}\)= \(\text{R}\,\text{6,99}\) + \(\text{70}\) \(\text{c}\) (rounded off) = \(\text{R}\,\text{7,69}\) OR New price is (\(\text{100}\) + \(\text{10}\))\% of \(\text{R}\,\text{6,99}\) = \(\text{110}\%\) of \(\text{R}\,\text{6,99}=\frac{\text{110}}{\text{100}} \times \frac{\text{6,99}}{\text{1}}= \text{R}\,\text{7,69}\) (rounded off)

Top Teenage T-shirts have a \(\text{20}\%\) discount on all T-shirts. If one of their T-shirts originally cost \(\text{R}\,\text{189,90}\), what will you pay for it now?

You only pay \(\text{80}\%\) (\(\text{100}\%\) \(-\) \(\text{20}\%\) discount). Thus: \(\frac{\text{80}}{\text{100}} \times \text{189,901} = \text{R}\,\text{151,92}\) OR \(\text{20}\%\) of \(\text{R}\,\text{189,90} = \frac{\text{20}}{\text{100}} \times \text{189,901}\). The discount is thus \(\text{R}\,\text{37,98}\). You pay \(\text{R}\,\text{189,90} - \text{R}\,\text{37,98} = \text{R}\,\text{151,92}\).

Look at the pictures below. What is the value of each of the following items, in rands?

\(\text{R}\,\text{239,96} - \text{R}\,\text{59,75} = \text{R}\,\text{180,21}\)

\(\text{R}\,\text{299,50} - \text{R}\,\text{44,925} = \text{R}\,\text{1 254,58}\)

\(\text{R}\,\text{9 875} + \text{R}\,\text{790} = \text{R}\,\text{10 665}\)

\(\text{R}\,\text{15 995} + \text{R}\,\text{799,75}= \text{R}\,\text{16 794,75}\)

Calculate the percentage discount on each of these items:

\(\frac{\text{R}\,\text{1 360}}{\text{R}\,\text{1 523}} = \text{89}\%\). So discount is \(\text{100}\% - \text{89}\% = \text{11}\%\)

\(\frac{\text{R}\,\text{527,40}}{\text{R}\,\text{586}} = \text{90}\%\). So discount is \(\text{100}\% - \text{90}\% = \text{10}\%\)