Fractions, decimals and percentages – Year 10 GCSE maths revision notes to download as PDF

Worked example 1

Work out \(3\frac{1}{2}\div1\frac{3}{4}\).

\(\frac{7}{2}\div\frac{7}{4}=\frac{7}{2}\times\frac{4}{7}=\frac{28}{14}=2\).

Worked example 2

A coat costs £84 after a 20% increase. Find the original price.

The new price is 120% of the original. \(84\div1.2=£70\).

Worked example 3

Increase 40 to 52. Find the percentage change.

Change \(=12\). \(\frac{12}{40}\times100=30\%\) increase.

Worked example 4 (Higher tier)

Write \(0.\dot{3}\dot{6}\) as a fraction.

Let \(x=0.363636...\). Then \(100x=36.3636...\). Subtract: \(99x=36\), so \(x=\frac{36}{99}=\frac{4}{11}\).

Worked example 5 (adding mixed numbers)

Work out \(2\frac{3}{5}+1\frac{7}{10}\) [3 marks].

Add the whole numbers: \(2+1=3\). Add the fractions with a common denominator: \(\frac{6}{10}+\frac{7}{10}=\frac{13}{10}=1\frac{3}{10}\). Total \(=4\frac{3}{10}\). Show each step: the method marks are for the common denominator and for combining the whole numbers.

Worked example 6 (fraction of an amount, then the rest)

A school has 640 pupils. \(\frac{3}{8}\) travel by bus and 25% walk. The rest cycle. How many cycle? [4 marks].

Bus: \(640\div8\times3=240\). Walk: \(0.25\times640=160\). Cycle: \(640-240-160=240\).

Worked example 7 (percentage of a percentage change chain)

A house price rises by 10% in year 1 and by 5% in year 2. What is the overall percentage increase? [3 marks].

Multiplier \(1.1\times1.05=1.155\). The overall increase is 15.5%, not 15%.

Worked example 8 (comparing offers)

Shop A: \(\frac{1}{3}\) off £90.

Shop B: 30% off £88. Which is cheaper? A: \(90\times\frac{2}{3}=£60\). B: \(88\times0.7=£61.60\). Shop A is cheaper by £1.60. Always finish with a statement answering the question.

Worked example 9: mixed numbers

Work out \(2\dfrac35+1\dfrac34\) without a calculator [3 marks].

Add the whole numbers: \(2+1=3\). Add the fractions: \(\dfrac35+\dfrac34=\dfrac{12}{20}+\dfrac{15}{20}=\dfrac{27}{20}=1\dfrac7{20}\). Combine: \(3+1\dfrac7{20}=4\dfrac7{20}\). Check with decimals: \(2.6+1.75=4.35\) and \(\dfrac7{20}=0.35\). The check confirms the answer.

Worked example 10: compound percentage with a table

£1500 is invested at 3.2% compound interest per year. Find the value after 3 years to the nearest penny [3 marks].

The multiplier is 1.032. Year 1: \(1500\times1.032=1548\). Year 2: \(1548\times1.032=1597.536\). Year 3: \(1597.536\times1.032=1648.657\). The value is £1648.66. Using a power gives the same: \(1500\times1.032^3=1648.66\). Round only at the very end, or the rounding errors build up.

Worked example 11: percentage increase and decrease together

A house price rose by 12% and then fell by 5%. The final price is £213 750. Find the original price [3 marks].

The combined multiplier is \(1.12\times0.95=1.064\). Original \(=213\,750\div1.064=200\,893\) to the nearest pound. The overall change is an increase of 6.4%, not 7%. Never combine the two percentages by simply adding or subtracting them.

Worked example 12: finding a percentage of a percentage

In a school 45% of the pupils are boys. Of the boys, 60% play a sport. What percentage of the whole school are boys who play a sport? [2 marks]

Multiply the two decimals: \(0.45\times0.6=0.27\). So 27% of the whole school are boys who play a sport. If the school has 1200 pupils then \(1200\times0.27=324\) pupils.