Multiplication and Division of Fractions

Worked example

1. 1

   $\frac{5}{6}\times \frac{9}{4}$ — Keep $\frac{5}{6}$, flip the second fraction, and change ÷ to ×.

2. 2

   $=\frac{5}{2}\times \frac{3}{4}$ — 6 and 9 both divide by 3, so cancel them to 2 and 3. Smaller numbers are easier without a calculator.

3. 3

   $=\frac{5\times 3}{2\times 4}=\frac{15}{8}$ — Multiply the tops, then multiply the bottoms.

4. 4

   $=1\frac{7}{8}$ — The question asks for a mixed number. 15 ÷ 8 = 1 remainder 7, so 1 whole and 7 eighths.

Worked example

1. 1

   $\frac{6}{7}÷4$ — The question. You can only flip a fraction, so first make 4 a fraction.

2. 2

   $\frac{6}{7}÷\frac{4}{1}$ — Any whole number is itself over 1, so 4 = 4/1.

3. 3

   $\frac{6}{7}\times \frac{1}{4}$ — Keep $\frac{6}{7}$, flip $\frac{4}{1}$ to $\frac{1}{4}$, and change ÷ to ×. Dividing by 4 is the same as taking a quarter.

4. 4

   $=\frac{6}{28}$ — Multiply the tops: 6 × 1 = 6. Multiply the bottoms: 7 × 4 = 28.

5. 5

   $=\frac{3}{14}$ — 6 and 28 both divide by 2. 3 and 14 share no factor, so this is the simplest form.

Worked example

1. 1

   $\frac{1}{4}\text{ of }360=360÷4=90$ — To find a quarter, cut 360 into 4 equal parts. One part is 90 who walk.

2. 2

   $\frac{1}{5}\text{ of }360=360÷5=72$ — For $\frac{2}{5}$, first find one fifth: cut 360 into 5 equal parts.

3. 3

   $\frac{2}{5}\text{ of }360=72\times 2=144$ — Two fifths is 2 of those parts, so times by the top. 144 take the bus.

4. 4

   $90+144=234$ — Add the two groups you found. 234 students walk or take the bus.

5. 5

   $360-234=126$ — The rest is everyone not in those groups, so take 234 from the total. 126 students cycle.

Worked example

1. 1

   $96÷8=12$ — To find $\frac{3}{8}$ of 96, first find one eighth: cut 96 into 8 equal parts.

2. 2

   $12\times 3=36$ — Three eighths is 3 of those parts. Her sister gets 36.

3. 3

   $96-36=60$ — Take her sister’s share away. 60 sweets are left, and this is the new amount.

4. 4

   $60÷5=12$ — Now find $\frac{2}{5}$ of 60, not of 96, because the question says “of the sweets that are left”. First, one fifth of 60.

5. 5

   $12\times 2=24$ — Two fifths is 2 of those parts. Her friend gets 24.

6. 6

   $60-24=36$ — Take the friend’s share away. Lina has 36 sweets now.

Worked example

1. 1

   $54+42+12=108$ — Add the animals you already know about.

2. 2

   $240-108=132$ — The rest are chickens, so take 108 from the total. You need the part before you can write the fraction.

3. 3

   $\frac{132}{240}$ — Part on top, total on the bottom. Write this line down: it earns a method mark even before you simplify.

4. 4

   $=\frac{33}{60}$ — 132 and 240 both divide by 4. Dividing top and bottom by the same number keeps the fraction’s value.

5. 5

   $=\frac{11}{20}$ — 33 and 60 both divide by 3. 11 and 20 share no factor now, so this is the simplest form.

Worked example

1. 1

   $5\text{ parts}=35$ — The bottom, 8, cuts the number into 8 equal parts. The top says 5 of those parts make 35.

2. 2

   $1\text{ part}=35÷5=7$ — Share 35 equally between the 5 parts to find one part.

3. 3

   $8\text{ parts}=7\times 8=56$ — The whole number is all 8 parts, so times one part by 8.

4. 4

   $\frac{5}{8}\text{ of }56=56÷8\times 5=35✓$ — Check by working forwards from your answer. You get 35 back, so 56 is right.

Worked example

1. 1

   $\frac{1}{3}=\frac{5}{15},\frac{2}{5}=\frac{6}{15}$ — Give both fractions the same bottom number, 15, so you can add them.

2. 2

   $\frac{5}{15}+\frac{6}{15}=\frac{11}{15}$ — Add the tops. He gives away 11 fifteenths of his beads.

3. 3

   $1-\frac{11}{15}=\frac{4}{15}$ — The whole bag is $\frac{15}{15}$. Take away what he gave, so 4 parts are left.

4. 4

   $4\text{ parts}=8$ — The 8 beads he still has are those 4 parts.

5. 5

   $1\text{ part}=8÷4=2$ — Share 8 between the 4 parts to find one part.

6. 6

   $15\text{ parts}=2\times 15=30$ — The whole bag is all 15 parts, so he had 30 beads at the start.

Worked example

1. 1

   $\frac{2}{3}\times \frac{3}{4}$ — “$\frac{2}{3}$ of these adults” means $\frac{2}{3}$ of the $\frac{3}{4}$, and “of” means multiply.

2. 2

   $=\frac{2}{\cancel{3}}\times \frac{\cancel{3}}{4}$ — A 3 on top and a 3 on the bottom divide to 1, so cancel them. This keeps the numbers small.

3. 3

   $=\frac{2\times 1}{1\times 4}=\frac{2}{4}$ — Multiply the tops, then multiply the bottoms.

4. 4

   $=\frac{1}{2}$ — 2 and 4 both divide by 2. Half of all the members are women.

Worked example

1. 1

   $1\frac{3}{5}=\frac{8}{5}$ — Change to an improper fraction first: 1 × 5 + 3 = 8 fifths. You cannot flip a mixed number.

2. 2

   $2\frac{2}{3}=\frac{8}{3}$ — Do the same to the second: 2 × 3 + 2 = 8 thirds.

3. 3

   $\frac{8}{5}÷\frac{8}{3}=\frac{8}{5}\times \frac{3}{8}$ — Keep the first, flip the second, and change ÷ to ×.

4. 4

   $=\frac{\cancel{8}}{5}\times \frac{3}{\cancel{8}}$ — An 8 on top and an 8 on the bottom divide to 1, so cancel them.

5. 5

   $=\frac{1\times 3}{5\times 1}=\frac{3}{5}$ — Multiply what is left. It matches the given answer, which is what “show that” needs.

Worked example

1. 1

   $1\frac{1}{2}=\frac{3}{2}$ — Improper first: 1 × 2 + 1 = 3 halves.

2. 2

   $2\frac{2}{3}=\frac{8}{3}$ — 2 × 3 + 2 = 8 thirds.

3. 3

   $\frac{3}{2}\times \frac{8}{3}$ — Multiplying needs no flip. Multiply the two improper fractions as they are.

4. 4

   $=\frac{\cancel{3}}{2}\times \frac{8}{\cancel{3}}=\frac{8}{2}$ — The 3s cancel, leaving 8 on top and 2 on the bottom.

5. 5

   $=4$ — 8 ÷ 2 = 4. A whole number answer is fine.

Worked example

1. 1

   $10\times \frac{3}{4}$ — Each door uses $\frac{3}{4}$ of a tin and there are 10 doors, so multiply.

2. 2

   $=\frac{10}{1}\times \frac{3}{4}=\frac{30}{4}$ — Write 10 as $\frac{10}{1}$. Tops: 10 × 3 = 30. Bottoms: 1 × 4 = 4.

3. 3

   $=7\frac{2}{4}$ — 30 ÷ 4 = 7 remainder 2, so 7 whole tins and 2 quarters.

4. 4

   $=7\frac{1}{2}$ — Simplify $\frac{2}{4}$ to $\frac{1}{2}$. He needs seven and a half tins of paint.

5. 5

   $8\text{ tins}$ — 7 tins are not enough paint, and you cannot buy half a tin, so round up.