Teach Me Maths

Multiplying and Dividing Fractions

Question 11 mark

Work out \frac{3}{4}\times8

Hint

Find \frac{1}{4} of 8 first.

Worked solution

Work out \frac{3}{4}\times8

  1. \frac{1}{4} of 8 is 8\div4=2.
  2. So \frac{3}{4} of 8 is 3\times2=6.
  3. Answer: 6.

Question 21 mark

Work out \frac{2}{3}\times\frac{4}{5}

Hint

Multiply the numerators together and multiply the denominators together.

Worked solution

Work out \frac{2}{3}\times\frac{4}{5}

  1. Numerators: 2\times4=8.
  2. Denominators: 3\times5=15.
  3. Answer: \frac{8}{15}.

Question 32 marks

Which of these is \frac{3}{4}\times\frac{2}{9} as a fraction in its simplest form?

Hint

Multiply the numerators and the denominators, then simplify.

Worked solution

Which of these is \frac{3}{4}\times\frac{2}{9} as a fraction in its simplest form?

  1. \frac{1}{6}
  2. \frac{6}{36}
  3. \frac{5}{13}
  4. \frac{27}{8}
  1. \frac{3}{4}\times\frac{2}{9}=\frac{3\times2}{4\times9}=\frac{6}{36}.
  2. Divide the numerator and denominator by 6: \frac{6}{36}=\frac{1}{6}.
  3. Answer: \frac{1}{6}.

Question 42 marks

A plank of wood is 3 metres long.

It is cut into pieces that are each \frac{1}{4} of a metre long.

How many pieces are there?

Hint

Work out how many quarters there are in one metre first.

Worked solution

A plank of wood is 3 metres long.

It is cut into pieces that are each \frac{1}{4} of a metre long.

How many pieces are there?

  1. There are 4 quarters in 1 metre.
  2. So there are 3\times4=12 quarters in 3 metres.
  3. This is 3\div\frac{1}{4}=3\times4=12.
  4. Answer: 12 pieces.

Question 52 marks

Work out \frac{5}{6}\div\frac{2}{3}

Hint

Turn the second fraction upside down and multiply.

Worked solution

Work out \frac{5}{6}\div\frac{2}{3}

  1. \frac{5}{6}\div\frac{2}{3}=\frac{5}{6}\times\frac{3}{2}.
  2. \frac{5\times3}{6\times2}=\frac{15}{12}.
  3. \frac{15}{12}=\frac{5}{4}=1\frac{1}{4}.
  4. Answer: 1\frac{1}{4}.

Question 62 marks

\frac{3}{4} of a cake is left.

The cake that is left is shared equally between 6 people.

What fraction of the whole cake does each person get?

Hint

Divide \frac{3}{4} into 6 equal shares.

Worked solution

\frac{3}{4} of a cake is left.

The cake that is left is shared equally between 6 people.

What fraction of the whole cake does each person get?

  1. \frac{3}{4}\div6=\frac{3}{4}\times\frac{1}{6}=\frac{3}{24}.
  2. \frac{3}{24}=\frac{1}{8}.
  3. Answer: \frac{1}{8} of the cake.

Question 72 marks

Work out \frac{14}{15}\times\frac{10}{21}

Hint

Cancel before you multiply: 14 and 21 have a common factor, and so do 10 and 15.

Worked solution

Work out \frac{14}{15}\times\frac{10}{21}

  1. Divide 14 and 21 by 7: they become 2 and 3.
  2. Divide 10 and 15 by 5: they become 2 and 3.
  3. \frac{2}{3}\times\frac{2}{3}=\frac{4}{9}.
  4. Answer: \frac{4}{9}.

Question 82 marks

This question is about reciprocals.

(a)

Write down the reciprocal of 7

1 mark

(b)

Which of these is the reciprocal of \frac{2}{3}?

1 mark

Hint

The reciprocal of a number is 1 divided by that number.

Worked solution

This question is about reciprocals.

Part (a)

Write down the reciprocal of 7

  1. The reciprocal of 7 is 1\div7=\frac{1}{7}.
  2. Answer: \frac{1}{7}.

Part (b)

Which of these is the reciprocal of \frac{2}{3}?

  1. -\frac{2}{3}
  2. \frac{3}{2}
  3. \frac{1}{3}
  4. 1\frac{2}{3}
  1. To find the reciprocal of a fraction, turn it upside down.
  2. The reciprocal of \frac{2}{3} is \frac{3}{2}. Check: \frac{2}{3}\times\frac{3}{2}=1.

Question 92 marks

Work out 2\frac{1}{4}\times\frac{2}{3}

Hint

Write 2\frac{1}{4} as an improper fraction first.

Worked solution

Work out 2\frac{1}{4}\times\frac{2}{3}

  1. 2\frac{1}{4}=\frac{9}{4}.
  2. \frac{9}{4}\times\frac{2}{3}=\frac{18}{12}.
  3. \frac{18}{12}=\frac{3}{2}=1\frac{1}{2}.
  4. Answer: 1\frac{1}{2}.

Question 103 marks

Work out 2\frac{1}{3}\div1\frac{1}{6}

Hint

Change both mixed numbers to improper fractions. Then turn the second fraction upside down and multiply.

Worked solution

Work out 2\frac{1}{3}\div1\frac{1}{6}

  1. 2\frac{1}{3}=\frac{7}{3} and 1\frac{1}{6}=\frac{7}{6}.
  2. \frac{7}{3}\div\frac{7}{6}=\frac{7}{3}\times\frac{6}{7}=\frac{42}{21}.
  3. \frac{42}{21}=2.
  4. Answer: 2.

Question 112 marks

The diagram shows a rectangle.

Work out the area of the rectangle.

Hint

Area of a rectangle = length × width. Change the mixed numbers to improper fractions first.

Worked solution

The diagram shows a rectangle.

Work out the area of the rectangle.

  1. 3\frac{1}{3}=\frac{10}{3} and 2\frac{1}{4}=\frac{9}{4}.
  2. \frac{10}{3}\times\frac{9}{4}=\frac{90}{12}.
  3. \frac{90}{12}=\frac{15}{2}=7\frac{1}{2}.
  4. Answer: 7\frac{1}{2} cm².

Question 122 marks

A tap drips \frac{3}{8} of a litre of water every hour.

How much water drips from the tap in 12 hours?

Hint

Multiply \frac{3}{8} by the number of hours.

Worked solution

A tap drips \frac{3}{8} of a litre of water every hour.

How much water drips from the tap in 12 hours?

  1. \frac{3}{8}\times12=\frac{36}{8}.
  2. \frac{36}{8}=\frac{9}{2}=4\frac{1}{2}.
  3. Answer: 4\frac{1}{2} litres.

Question 132 marks

A shop has a 6\frac{1}{4} kg sack of rice.

It puts all the rice into small bags. Each small bag holds \frac{5}{8} kg of rice.

How many small bags does the shop fill?

Hint

Work out how many lots of \frac{5}{8} there are in 6\frac{1}{4}.

Worked solution

A shop has a 6\frac{1}{4} kg sack of rice.

It puts all the rice into small bags. Each small bag holds \frac{5}{8} kg of rice.

How many small bags does the shop fill?

  1. Number of bags =6\frac{1}{4}\div\frac{5}{8}.
  2. 6\frac{1}{4}=\frac{25}{4}, so \frac{25}{4}\times\frac{8}{5}=\frac{200}{20}.
  3. \frac{200}{20}=10.
  4. Answer: 10 bags.

Question 142 marks

Work out the missing fraction.

\frac{3}{4}\times\square=\frac{9}{20}

Hint

Think about what the numerator 3 must be multiplied by to give 9, and the same for the denominators.

Worked solution

Work out the missing fraction.

\frac{3}{4}\times\square=\frac{9}{20}

  1. The numerators: 3\times3=9.
  2. The denominators: 4\times5=20.
  3. So the missing fraction is \frac{3}{5}. Check: \frac{3}{4}\times\frac{3}{5}=\frac{9}{20}.
  4. Answer: \frac{3}{5}.

Question 153 marks

Zara works out \frac{2}{5}\div\frac{3}{4}

Here is her working.

\frac{2}{5}\div\frac{3}{4}=\frac{5}{2}\times\frac{3}{4}=\frac{15}{8}

Her answer is wrong.

(a)

What mistake did Zara make?

1 mark

(b)

Work out the correct answer.

2 marks

Hint

To divide by a fraction, multiply by its reciprocal.

Worked solution

Zara works out \frac{2}{5}\div\frac{3}{4}

Here is her working.

\frac{2}{5}\div\frac{3}{4}=\frac{5}{2}\times\frac{3}{4}=\frac{15}{8}

Her answer is wrong.

Part (a)

What mistake did Zara make?

  1. She should have found a common denominator first.
  2. She turned the first fraction upside down instead of the second fraction.
  3. She should not have turned either fraction upside down.
  4. She made a mistake when she worked out 5\times3.
  1. To divide by \frac{3}{4}, you multiply by \frac{4}{3}.
  2. Zara turned \frac{2}{5} upside down instead of \frac{3}{4}.

Part (b)

Work out the correct answer.

  1. \frac{2}{5}\div\frac{3}{4}=\frac{2}{5}\times\frac{4}{3}.
  2. \frac{2\times4}{5\times3}=\frac{8}{15}.
  3. Answer: \frac{8}{15}.

Question 16Challenge3 marks

Work out 1\frac{1}{6}\times2\frac{2}{5}

Give your answer as a mixed number.

Hint

Change both mixed numbers to improper fractions, then look for numbers you can cancel.

Worked solution

Work out 1\frac{1}{6}\times2\frac{2}{5}

Give your answer as a mixed number.

  1. 1\frac{1}{6}=\frac{7}{6} and 2\frac{2}{5}=\frac{12}{5}.
  2. \frac{7}{6}\times\frac{12}{5}=\frac{84}{30}.
  3. \frac{84}{30}=\frac{14}{5}=2\frac{4}{5}.
  4. Answer: 2\frac{4}{5}.

Question 17Challenge4 marks

Lena earns £1440 in one month.

She spends \frac{5}{8} of the £1440 on rent and bills.

She saves \frac{2}{3} of the money that is left.

(a)

How much money does Lena save?

3 marks

(b)

What fraction of the £1440 does Lena save?

1 mark

Hint

First work out what fraction of the £1440 is left after rent and bills.

Worked solution

Lena earns £1440 in one month.

She spends \frac{5}{8} of the £1440 on rent and bills.

She saves \frac{2}{3} of the money that is left.

Part (a)

How much money does Lena save?

  1. After rent and bills, 1-\frac{5}{8}=\frac{3}{8} of the £1440 is left.
  2. \frac{1}{8} of £1440 is 1440\div8=180, so \frac{3}{8} is 3\times180=540. That is £540.
  3. \frac{2}{3} of £540: 540\div3=180 and 2\times180=360.
  4. Answer: £360.

Part (b)

What fraction of the £1440 does Lena save?

  1. She saves £360 out of £1440: \frac{360}{1440}=\frac{1}{4}.
  2. Or: \frac{2}{3}\times\frac{3}{8}=\frac{6}{24}=\frac{1}{4}.
  3. Answer: \frac{1}{4}.

Question 18Challenge3 marks

The diagram shows a rectangle.

The width of the rectangle is \frac{3}{4} m.

The area of the rectangle is \frac{3}{5} m².

Work out the length of the rectangle. Give your answer in centimetres.

Hint

Length = area ÷ width. Then change your answer from metres to centimetres.

Worked solution

The diagram shows a rectangle.

The width of the rectangle is \frac{3}{4} m.

The area of the rectangle is \frac{3}{5} m².

Work out the length of the rectangle. Give your answer in centimetres.

  1. Length =\frac{3}{5}\div\frac{3}{4}=\frac{3}{5}\times\frac{4}{3}=\frac{12}{15}=\frac{4}{5} m.
  2. 1 m =100 cm, so \frac{4}{5} m =\frac{4}{5}\times100=80 cm.
  3. Answer: 80 cm.

Question 19Challenge3 marks

Two machines make cloth.

Machine P makes 2\frac{1}{2} metres of cloth in \frac{3}{4} of an hour.

Machine Q makes 3\frac{1}{4} metres of cloth in 1 hour.

Each machine always works at the same speed.

(a)

How many metres of cloth does machine P make in 1 hour?

2 marks

(b)

Which machine makes more cloth in 1 hour?

1 mark

Hint

Work out how much cloth machine P makes in \frac{1}{4} of an hour first.

Worked solution

Two machines make cloth.

Machine P makes 2\frac{1}{2} metres of cloth in \frac{3}{4} of an hour.

Machine Q makes 3\frac{1}{4} metres of cloth in 1 hour.

Each machine always works at the same speed.

Part (a)

How many metres of cloth does machine P make in 1 hour?

  1. In \frac{1}{4} of an hour, machine P makes 2\frac{1}{2}\div3=\frac{5}{2}\times\frac{1}{3}=\frac{5}{6} m.
  2. In 1 hour it makes 4\times\frac{5}{6}=\frac{20}{6}=3\frac{1}{3} m.
  3. Or in one step: 2\frac{1}{2}\div\frac{3}{4}=\frac{5}{2}\times\frac{4}{3}=\frac{20}{6}=3\frac{1}{3} m.
  4. Answer: 3\frac{1}{3} metres.

Part (b)

Which machine makes more cloth in 1 hour?

  1. Machine P
  2. Machine Q
  3. They both make the same amount
  1. Machine P: 3\frac{1}{3}=3\frac{4}{12} m. Machine Q: 3\frac{1}{4}=3\frac{3}{12} m.
  2. 3\frac{4}{12} is more than 3\frac{3}{12}, so machine P makes more cloth in 1 hour.

Question 20Challenge4 marks

Ella has a roll of ribbon 8\frac{1}{3} m long.

She cuts as many pieces of ribbon as she can from the roll. Each piece is 1\frac{1}{4} m long.

(a)

How many pieces does Ella cut?

2 marks

(b)

How much ribbon is left over? Give your answer in metres.

2 marks

Hint

Divide the length of the roll by the length of one piece, then think about whole pieces.

Worked solution

Ella has a roll of ribbon 8\frac{1}{3} m long.

She cuts as many pieces of ribbon as she can from the roll. Each piece is 1\frac{1}{4} m long.

Part (a)

How many pieces does Ella cut?

  1. 8\frac{1}{3}\div1\frac{1}{4}=\frac{25}{3}\div\frac{5}{4}=\frac{25}{3}\times\frac{4}{5}=\frac{100}{15}=6\frac{2}{3}.
  2. She can only cut whole pieces, so she cuts 6 pieces.
  3. Answer: 6 pieces.

Part (b)

How much ribbon is left over? Give your answer in metres.

  1. 6 pieces use 6\times1\frac{1}{4}=6\times\frac{5}{4}=\frac{30}{4}=7\frac{1}{2} m.
  2. Left over: 8\frac{1}{3}-7\frac{1}{2}=\frac{50}{6}-\frac{45}{6}=\frac{5}{6} m.
  3. Answer: \frac{5}{6} m.

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