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## Percentages

Definition 1: Percentage
A number represented as a part of 100.

## How to calculate a percentage of an amount

1. Write the percentage as a fraction with the denominator 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.

### Example 1: Working out percentages of amounts

#### Question

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 43% of 50 586 757 key in:
$$\text{43} \div \text{100} \times \text{50 586 757} =$$
OR
$$\text{43}\% \times \text{50 586 757} =$$
2. $$\text{80}\% = \text{80} \div \text{100}$$
$$\frac{\text{80}}{\text{100}} \times \text{1291} = \text{1032}$$ 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.

### Example 2: Working out one amount as a percentage of another amount

#### Question

Top Teenage T-shirts printed 120 T-shirts. They sold 72 T-shirts immediately. What percentage of the T-shirts were sold?

72 of the 120 T-shirts were sold

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

### Exercise 1: Calculating the percentages of amounts

Calculate the following without a calculator:

1. 25% of R 124,16

2. 50% of 30 mm

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

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

1. 15% of R 3500

2. 12% of 25 litres

3. 37,5% of 22 kg

4. 75% of R 16,92

5. 18% of 105 m

6. 79% of 840 km

1. R 525

2. 3 litres

3. 8,25 kg

4. R 12,69

5. 18,9 m

6. 663,6 km

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

1. 120 of 480

2. 23 of 276

3. 3500 ml of 5 litres

4. 750 g of 2 kg

5. 4 out of 5 for a test

6. 2 out of 14 balls

1. 25%

2. 8,3%

3. 70%

4. 37,5%

5. 90%

6. 14,3%

## Percentage discounts and increases

Look at the following extracts from newspaper articles and adverts:

Definition 2: Cost price
The amount that the dealer / trader / merchant pays for an article.
Definition 3: Marked price
This is the price of the article.
Definition 4: Selling price
This is the price after discount.
Definition 5: Profit
Sale price $$-$$ cost price.

### Exercise 2: Discounts and increases

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

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

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

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

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

1. $$\text{R 239,96} - \text{R 59,75} = \text{R 180,21}$$

2. $$\text{R 299,50} - \text{R 44,925} = \text{R 1254,58}$$

3. $$\text{R 9875} + \text{R 790} = \text{R 10 665}$$

4. $$\text{R 15 995} + \text{R 799,75}= \text{R 16 794,75}$$

Calculate the percentage discount on each of these items:

1. $$\frac{\text{R 1360}}{\text{R 1523}} = \text{89}\%$$. So discount is $$\text{100}\% - \text{89}\% = \text{11}\%$$

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