Multiplying fractions and mixed numbers practice | Year 8

Connect multiplication to “of”, practise simplifying across a product and convert a mixed number before multiplying.

Multiply numerators and denominators and simplify the result.
Find a fraction of a fractional quantity.
Convert mixed numbers before calculating a product.

The idea

Multiplying by a fraction finds that fraction of an amount. Multiply numerators together and denominators together. You may simplify a numerator against a denominator before multiplying because you are removing the same nonzero factor from the overall numerator and denominator.

Before you begin: Recognise equivalent fractions, common factors and improper fractions.

Worked examples

01 Start with the method

Multiply and give the result in simplest form.

\frac{2}{3} \times \frac{9}{10}

Step 1

Write the product as one fraction. The numerator counts the selected parts; the denominator describes the smaller equal parts.

\frac{2}{3} \times \frac{9}{10} = \frac{2 \times 9}{3 \times 10}

Step 2

Simplify across the product: 2 and 10 share a factor of 2; 9 and 3 share a factor of 3.

\frac{2 \times 9}{3 \times 10} = \frac{1 \times 3}{1 \times 5} = \frac{3}{5}

Step 3

Both starting fractions are between zero and one. Taking two thirds of nine tenths should give less than nine tenths; three fifths does.

\frac{3}{5} < \frac{9}{10}

A mistake to avoid

A common denominator is needed for adding fractions, not for multiplying them. Do not cancel two numerators against each other.

02Find a fraction of a fraction

A jug contains two fifths of a litre. You pour out three quarters of that amount. How much do you pour?

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

Step 1

The word “of” asks for multiplication. The starting amount is already a fraction of a litre.

\frac{3}{4} of \frac{2}{5} L = \frac{3}{4} \times \frac{2}{5} L

Step 2

Multiply, then reduce by the common factor 2.

\frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10}

Step 3

Three quarters of the starting amount must be smaller than the starting amount.

\frac{3}{10} < \frac{2}{5}

Result

\frac{3}{10} L

03Convert a mixed number first

Find the exact product.

1 \frac{1}{2} \times \frac{2}{3}

Step 1

One whole and one half make three halves. Rewrite the mixed number as a single fraction.

1 \frac{1}{2} = \frac{3}{2}

Step 2

Multiply the two fractions. Here the numerator and denominator have the same product.

\frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1

Step 3

A reciprocal pair multiplies to one, which confirms the answer.

\frac{3}{2} \times \frac{2}{3} = 1

Result

1

Practice questions with answers

Work these out first, then open each solution to compare your method. These questions are untimed.

Question 1

Simplify before multiplying.

\frac{5}{6} \times \frac{3}{10}

Show solution 1

\frac{5 \times 3}{6 \times 10} = \frac{1}{4}

Cancel 5 with 10 and 3 with 6. The remaining factors give 1 × 1 over 2 × 2.

Question 2

Multiply and simplify.

\frac{7}{8} \times \frac{4}{21}

Show solution 2

\frac{7}{8} \times \frac{4}{21} = \frac{1}{6}

Cancel 7 with 21 and 4 with 8. The remaining denominator is 2 × 3.

Question 3

Find the exact product.

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

Show solution 3

\frac{9}{4} \times \frac{2}{3} = \frac{3}{2} = 1 \frac{1}{2}

Convert 2¼ to 9/4, then simplify the product 18/12.

Question 4

Multiply and simplify.

\frac{3}{5} \times \frac{10}{9}

Show solution 4

\frac{30}{45} = \frac{2}{3}

You can simplify 10 with 5 and 3 with 9 before multiplying. An improper factor is allowed.

Question 5

Find two thirds of three quarters of a litre.

\frac{2}{3} \times \frac{3}{4} L

Show solution 5

\frac{1}{2} L

The product is 6/12 litre, which simplifies to 1/2 litre. Keep the unit in the final answer.

Common questions

Do I need matching denominators?

No. For multiplication, multiply the numerators and multiply the denominators. Matching denominators are useful when adding or subtracting fractions.

Why does cancellation work?

Cancellation divides the overall numerator and denominator by the same nonzero factor. It changes the written form without changing the value.

Does multiplication always make a number bigger?

No. Multiplying a positive amount by a number between zero and one makes it smaller. Multiplying it by a number greater than one makes it bigger.