Dividing fractions practice with worked answers | Year 8

Interpret the quotient as a number of portions, then use the same reciprocal method for both proper fractions and mixed numbers.

Invert the divisor and multiply to divide nonzero fractions.
Explain why a quotient can be larger than the starting amount.
Check a division result by multiplying it by the original divisor.

The idea

Dividing by a nonzero fraction asks how many of that fractional amount fit into the starting amount. Multiplying by the reciprocal gives the same result: a/b ÷ c/d = a/b × d/c. Only the divisor is inverted. The result can be larger than the starting amount when the divisor is between zero and one.

Before you begin: Multiply and simplify fractions, and convert mixed numbers to improper fractions.

Worked examples

01 Start with the method

Divide exactly and give a simplified fraction.

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

Step 1

The divisor is two fifths. Its reciprocal is five halves; the first fraction stays unchanged.

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

Step 2

Multiply the numerators and denominators, then simplify if possible.

\frac{3 \times 5}{4 \times 2} = \frac{15}{8} = 1 \frac{7}{8}

Step 3

Multiply the quotient by the original divisor. Recovering three quarters checks the division.

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

A mistake to avoid

Invert the divisor, not both fractions. Dividing by zero is undefined, so the fraction being divided by must be nonzero.

02Count how many small portions fit

How many one-sixth portions fit into two thirds?

\frac{2}{3} \div \frac{1}{6}

Step 1

Express two thirds in sixths. This makes the grouping interpretation visible.

\frac{2}{3} = \frac{4}{6}

Step 2

Four one-sixth portions fit. The reciprocal method gives the same result.

\frac{2}{3} \times 6 = 4

Step 3

Check by multiplying the number of portions by the size of each portion.

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

Result

4 portions

03Divide a mixed number

A ribbon is one and a half metres long. How many three-quarter-metre lengths can be cut?

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

Step 1

Convert the mixed number before doing any fraction calculation.

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

Step 2

Invert only the divisor, then multiply and simplify.

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

Step 3

Two lengths of three quarters of a metre use exactly one and a half metres.

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

Result

2 lengths

Practice questions with answers

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

Question 1

How many one-third portions fit into five sixths?

\frac{5}{6} \div \frac{1}{3}

Show solution 1

\frac{5}{6} \times 3 = \frac{5}{2} = 2 \frac{1}{2}

Two full one-third portions use four sixths. The remaining one sixth is half of another portion.

Question 2

Divide and simplify.

\frac{2}{5} \div \frac{4}{7}

Show solution 2

\frac{2}{5} \times \frac{7}{4} = \frac{7}{10}

The divisor’s reciprocal is 7/4. The first fraction stays 2/5.

Question 3

What fraction of a three-quarter portion is three eighths?

\frac{3}{8} \div \frac{3}{4}

Show solution 3

\frac{3}{8} \times \frac{4}{3} = \frac{1}{2}

Three eighths is half of three quarters. Multiplication by 4/3 confirms that comparison.

Question 4

Divide exactly.

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

Show solution 4

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

Convert the mixed number to 9/4, then use the reciprocal 2/3.

Question 5

Share four fifths equally between two people. How much does each receive?

\frac{4}{5} \div 2

Show solution 5

\frac{4}{5} \times \frac{1}{2} = \frac{2}{5}

Write 2 as 2/1 and use its reciprocal 1/2. Each person receives half of the original amount.

Common questions

Why multiply by the reciprocal?

Multiplying the divisor by its reciprocal gives one. Multiplying both the dividend and divisor by that reciprocal leaves a division by one, which gives the same quotient.

Can the answer be a fraction of a group?

Yes. A quotient can describe part of a portion, such as half of a three-quarter-metre length. If the question asks only for complete objects, interpret that fractional result in context.

Can I divide by a fraction with zero on top?

No. Such a fraction equals zero, and division by zero is undefined. A denominator of zero is also invalid.