Introduction to Percentages

Worked example

1. 1

   $0.45\times 100=45\%$ — Per cent means out of 100, so multiply by 100. The digits move 2 places left: 0.45 becomes 45.

2. 2

   $0.3=0.30$ — 0.3 has only one decimal place. Add a zero so there are two places to move. The value does not change.

3. 3

   $0.30\times 100=30\%$ — Now move the digits 2 places left: 30. So 0.3 is 30%, not 3%.

4. 4

   $0.06\times 100=6\%$ — Move 2 places left: 0.06 becomes 6. The zeros in front fall away, so it is 6%, not 60%.

Worked example

1. 1

   $\frac{9}{25}=\frac{9\times 4}{25\times 4}$ — 25 × 4 = 100, so multiply the bottom by 4. Multiply the top by 4 too, so the fraction keeps its value.

2. 2

   $=\frac{36}{100}$ — 9 × 4 = 36 and 25 × 4 = 100. Now the fraction is out of 100.

3. 3

   $=36\%$ — Per cent means out of 100, so 36 out of 100 is 36%.

Worked example

1. 1

   $\frac{3}{8}=3÷8=0.375$ — No whole number times 8 makes 100, so divide the top by the bottom. This turns the fraction into a decimal.

2. 2

   $0.375\times 100=37.5\%$ — Now change the decimal to a percentage the usual way: multiply by 100.

Worked example

1. 1

   $\frac{33}{150}$ — The question. 150 does not go into 100, but the fraction can be made smaller first.

2. 2

   $=\frac{33÷3}{150÷3}=\frac{11}{50}$ — Both 33 and 150 divide by 3. Dividing top and bottom by the same number keeps the value.

3. 3

   $=\frac{11\times 2}{50\times 2}=\frac{22}{100}$ — 50 × 2 = 100, so multiply top and bottom by 2 to get a number out of 100.

4. 4

   $=22\%$ — 22 out of 100 is 22%.

Worked example

1. 1

   $\frac{4}{9}=4÷9=0.4444\dots$ — 9 never makes 100, so divide. The 4s go on for ever, so write a few and then dots.

2. 2

   $0.4444\dots \times 100=44.44\dots \%$ — Multiply by 100 to make it a percentage. The digits move 2 places left.

3. 3

   $\approx 44.4\%$ — The answer never ends, so round it. Give 3 significant figures unless the question asks for something else.

Worked example

1. 1

   $36\%=36÷100=0.36$ — Per cent means out of 100, so divide by 100. The digits move 2 places right: 36 becomes 0.36.

2. 2

   $4\%=4÷100=0.04$ — 4 has only one digit, so put a 0 in front of it to fill the second decimal place. It is 0.04, not 0.4.

3. 3

   $143\%=143÷100=1.43$ — More than 100% means more than one whole, so the decimal is bigger than 1. You still divide by 100.

4. 4

   $64\%=\frac{64}{100}$ — For a fraction, write the percentage as a number out of 100.

5. 5

   $=\frac{64÷4}{100÷4}$ — Both 64 and 100 divide by 4. Divide top and bottom by the same number, so the value stays the same.

6. 6

   $=\frac{16}{25}$ — 64 ÷ 4 = 16 and 100 ÷ 4 = 25. No number divides both 16 and 25, so this is the simplest form.

Worked example

1. 1

   $45\%=\frac{45}{100}$ — Write the percentage as a fraction out of 100.

2. 2

   $=\frac{45÷5}{100÷5}=\frac{9}{20}$ — The shape has 20 squares, so we want a bottom of 20. 100 ÷ 5 = 20, so divide the top by 5 too.

3. 3

   $\text{shade }9\text{ squares}$ — 9 out of 20 squares is 45% of the shape. Any 9 squares will do.

Worked example

1. 1

   $100\%-34\%=66\%$ — Everyone on the bus is 100%. Standing and sitting make up the whole, so take the standing part away from 100%.

Worked example

1. 1

   $\frac{1}{4}=25\%$ — The parts are in different forms. Change the fraction to a percentage first, so they can be added.

2. 2

   $40\%+25\%=65\%$ — Add the parts you know: walk and bus together.

3. 3

   $100\%-65\%=35\%$ — The whole class is 100%. Take away the parts you know to find the rest.