Fractions – Year 7 maths lesson to download as PDF

Example 1

Simplify \(\dfrac{24}{36}\).

The HCF of 24 and 36 is 12. \(\dfrac{24\div12}{36\div12}=\dfrac{2}{3}\).

Example 2

Write \(3\dfrac{2}{5}\) as an improper fraction and \(\dfrac{23}{6}\) as a mixed number.

\(3\times5+2=17\), so \(\dfrac{17}{5}\). \(23\div6=3\) remainder 5, so \(3\dfrac{5}{6}\).

Example 3

Which is larger, \(\dfrac{5}{6}\) or \(\dfrac{7}{9}\)?

The common denominator is 18. \(\dfrac{15}{18}\) and \(\dfrac{14}{18}\). So \(\dfrac{5}{6}\) is larger.

Example 4

Work out \(\dfrac{3}{4}+\dfrac{2}{3}\).

The common denominator is 12. \(\dfrac{9}{12}+\dfrac{8}{12}=\dfrac{17}{12}=1\dfrac{5}{12}\).

Example 5

Work out \(3\dfrac{1}{2}-1\dfrac{3}{4}\).

As improper fractions: \(\dfrac{7}{2}-\dfrac{7}{4}=\dfrac{14}{4}-\dfrac{7}{4}=\dfrac{7}{4}=1\dfrac{3}{4}\).

Example 6

Find \(\dfrac{3}{8}\) of 56.

\(56\div8=7\) and \(7\times3=21\).

Example 7

Work out \(\dfrac{4}{9}\times\dfrac{3}{8}\).

Cancel: 4 and 8 share 4, and 3 and 9 share 3. This gives \(\dfrac{1}{3}\times\dfrac{1}{2}=\dfrac{1}{6}\).

Example 8

Work out \(1\dfrac{1}{2}\times2\dfrac{2}{3}\).

\(\dfrac{3}{2}\times\dfrac{8}{3}=\dfrac{24}{6}=4\).

Example 9

Work out \(\dfrac{3}{4}\div\dfrac{3}{8}\).

\(\dfrac{3}{4}\times\dfrac{8}{3}=\dfrac{24}{12}=2\).

Example 10

Sam eats \(\dfrac{1}{4}\) of a pizza and Jo eats \(\dfrac{1}{3}\). What fraction is left?

\(\dfrac{1}{4}+\dfrac{1}{3}=\dfrac{7}{12}\), so \(1-\dfrac{7}{12}=\dfrac{5}{12}\) is left.

Example 11

A jug holds \(2\dfrac{1}{2}\) litres. How many \(\dfrac{1}{4}\)-litre glasses can be filled?

\(\dfrac{5}{2}\div\dfrac{1}{4}=\dfrac{5}{2}\times4=10\) glasses.

Example 12

A class has 30 pupils. \(\dfrac{2}{5}\) walk to school, \(\dfrac{1}{3}\) come by bus and the rest are driven. How many are driven?

Walk: \(\dfrac{2}{5}\times30=12\). Bus: \(\dfrac{1}{3}\times30=10\). Driven: \(30-12-10=8\) pupils.

Example 13

Work out \(\dfrac{2}{3}\) of \(\dfrac{3}{4}\) of 24.

\(\dfrac{3}{4}\) of 24 is 18, and \(\dfrac{2}{3}\) of 18 is 12.

Example 14

Order \(\dfrac{3}{4}\), \(\dfrac{5}{8}\) and \(\dfrac{7}{10}\) from smallest to largest.

Use a common denominator of 40: \(\dfrac{30}{40}\), \(\dfrac{25}{40}\), \(\dfrac{28}{40}\). The order is \(\dfrac{5}{8}\), \(\dfrac{7}{10}\), \(\dfrac{3}{4}\).

Worked example in context

A recipe needs \(\dfrac{3}{4}\) of a cup of flour. Lucy wants to make half the recipe. How much flour does she need?

Half of \(\dfrac{3}{4}\) means \(\dfrac{1}{2}\times\dfrac{3}{4}=\dfrac{3}{8}\). Lucy needs \(\dfrac{3}{8}\) of a cup.

Example 1

Work out \(3\dfrac{2}{3}-1\dfrac{3}{4}\).

One way is to change both numbers into improper fractions. \(3\dfrac{2}{3}=\dfrac{11}{3}\) and \(1\dfrac{3}{4}=\dfrac{7}{4}\). The lowest common multiple of 3 and 4 is 12, so we write \(\dfrac{44}{12}-\dfrac{21}{12}=\dfrac{23}{12}\). As a mixed number this is \(1\dfrac{11}{12}\). A sensible check: \(3\dfrac{2}{3}\) is a little less than 4 and \(1\dfrac{3}{4}\) is a little less than 2, so the answer should be close to 2, and \(1\dfrac{11}{12}\) is.

Example 2

Work out \(\dfrac{8}{15}\times\dfrac{5}{12}\), cancelling first.

Before multiplying, we look for a numerator and a denominator that share a factor. 8 and 12 share the factor 4, so they become 2 and 3. 5 and 15 share the factor 5, so they become 1 and 3. The calculation is now \(\dfrac{2}{3}\times\dfrac{1}{3}=\dfrac{2}{9}\). Cancelling first keeps the numbers small, and the answer is already in its simplest form.

Example 3

A ribbon is \(2\dfrac{1}{4}\) m long. How many pieces of length \(\dfrac{3}{8}\) m can be cut from it?

“How many pieces fit” is a division question. \(2\dfrac{1}{4}=\dfrac{9}{4}\), so we work out \(\dfrac{9}{4}\div\dfrac{3}{8}=\dfrac{9}{4}\times\dfrac{8}{3}=\dfrac{72}{12}=6\). Exactly six pieces can be cut. Notice that dividing by a fraction smaller than 1 gave a bigger answer, which makes sense because each piece is short.

Example 4

Which is bigger, \(\dfrac{5}{8}\) or \(\dfrac{7}{11}\)?

The denominators 8 and 11 have no common factor, so the lowest common denominator is \(8\times11=88\). We write \(\dfrac{5}{8}=\dfrac{55}{88}\) and \(\dfrac{7}{11}=\dfrac{56}{88}\). Now the pieces are the same size, and 56 pieces are more than 55 pieces, so \(\dfrac{7}{11}\) is bigger, but only just. Another way is to change both fractions into decimals: \(5\div8=0.625\) and \(7\div11=0.636\ldots\), which leads to the same conclusion. Choose whichever method gives easier numbers. When the fractions are very close, as here, write your working carefully, because a guess based on how the fractions look is very likely to be wrong. Remember too that with the same numerator, the fraction with the smaller denominator is bigger, since the whole is cut into fewer, larger pieces.