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Reciprocals

FoundationHigherAQAEdexcelOCR

Reciprocals is a key number topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on working with reciprocals, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. The reciprocal of a fraction is found by swapping its numerator and denominator.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

The reciprocal of a number is \(1\) divided by that number. Multiply any number by its reciprocal and the answer is always \(1\), which is the defining property.

For a fraction, finding the reciprocal simply means turning it upside down: the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). For a whole number, write it over \(1\) first, so the reciprocal of \(4\) is \(\frac{1}{4}\).

Reciprocals matter because dividing by a number is the same as multiplying by its reciprocal. That single fact underpins dividing fractions, and it reappears later in negative indices, where \(x^{-1}\) means exactly the reciprocal of \(x\).

Revision notes

Finding a reciprocal

Turn the fraction upside down. For a whole number, write it as a fraction over \(1\) and then invert.

So the reciprocal of \(\frac{2}{7}\) is \(\frac{7}{2}\), and the reciprocal of \(6\) is \(\frac{1}{6}\). Mixed numbers must be converted to improper fractions first.

The defining property

A number multiplied by its reciprocal gives \(1\), because the numerators and denominators cancel completely.

Check: \(\frac{3}{5} \times \frac{5}{3} = \frac{15}{15} = 1\). This is a quick way to confirm you have found the right reciprocal.

Zero and negatives

Zero has no reciprocal, because nothing multiplied by \(0\) can give \(1\), and dividing by zero is undefined.

A negative number keeps its sign: the reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\), since a negative times a negative would give a positive.

Key points

  • The reciprocal of \(x\) is \(\frac{1}{x}\).
  • Turn a fraction upside down to find its reciprocal.
  • Write a whole number over 1 first.
  • A number times its reciprocal equals 1.
  • Zero has no reciprocal.
  • The reciprocal keeps the sign of the original number.

Worked examples

Example 1

Write down the reciprocal of \(\frac{4}{9}\).

Working

\[\frac{4}{9}\]turn the fraction upside down
\[= \frac{9}{4}\]this is the reciprocal

Example 2

Write down the reciprocal of \(5\) and check your answer.

Working

\[5 = \frac{5}{1}\]write the whole number as a fraction
\[\frac{1}{5}\]invert it
\[5 \times \frac{1}{5} = 1\]check the defining property holds

Example 3

Find the reciprocal of \(1\frac{2}{3}\).

Working

\[\frac{5}{3}\]convert the mixed number to an improper fraction first
\[= \frac{3}{5}\]then invert

Common mistakes

  • Inverting a mixed number directly.

    1⅔ flipped is not 1³⁄₂. Convert to 5/3 first, then invert to 3/5.

  • Thinking a reciprocal is the negative.

    The reciprocal of 4 is 1/4, not −4. The sign is unchanged.

  • Giving 0 as having a reciprocal.

    Dividing by zero is undefined, so zero has no reciprocal.

  • Forgetting to write a whole number over 1.

    Without that step it is unclear what to invert, and the answer is often left as the original number.

Exam tips

  • Check any reciprocal by multiplying it back and confirming you get 1.
  • Convert mixed numbers before inverting.
  • Remember that dividing by a number equals multiplying by its reciprocal.
  • Link this to negative indices, where \(x^{-1}\) means the reciprocal.

Key terms

Reciprocal
The number that multiplies with the original to give 1.
Invert
Turn a fraction upside down.
Undefined
A calculation with no valid result, such as division by zero.
Improper fraction
A fraction whose numerator is greater than its denominator.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.