Reciprocals
A reciprocal is the number you multiply by to get 1. Reciprocals are important in fractions, division, algebra and negative indices.
The reciprocal of a number is 1 divided by that number.
A number × its reciprocal = 1
Reciprocal of a whole number
To find the reciprocal of a whole number, put 1 over the number.
Example 1: Reciprocal of a whole number
Find the reciprocal of 5.
reciprocal of 5 = 1/5
Check:
5 × 1/5 = 1
Reciprocal of a fraction
To find the reciprocal of a fraction, swap the numerator and denominator.
Example 2: Reciprocal of a fraction
Find the reciprocal of 3/7.
Swap the top and bottom:
3/7 → 7/3
The reciprocal is 7/3.
Reciprocal of a mixed number
Convert the mixed number to an improper fraction first, then swap the numerator and denominator.
Example 3: Reciprocal of a mixed number
Find the reciprocal of 2 1/3.
Convert to an improper fraction:
2 1/3 = 7/3
Now swap top and bottom:
7/3 → 3/7
The reciprocal is 3/7.
Reciprocal of a decimal
To find the reciprocal of a decimal, write 1 divided by the decimal. Sometimes it helps to convert the decimal to a fraction first.
Example 4: Reciprocal of a decimal
Find the reciprocal of 0.25.
Since 0.25 = 1/4, its reciprocal is:
4
Check:
0.25 × 4 = 1
Using reciprocals for division
Dividing by a fraction is the same as multiplying by its reciprocal.
Example 5: Divide using a reciprocal
Calculate 6 ÷ 2/3.
Multiply by the reciprocal of 2/3:
6 × 3/2 = 18/2 = 9
A common mistake is changing the first fraction instead of the second fraction when dividing. Only the number you are dividing by is replaced by its reciprocal.
To check a reciprocal, multiply the original number by your answer. If the result is 1, your reciprocal is correct.
Video explanation
A short Worthing Maths Tutor video explanation for reciprocals can be embedded here later to improve student engagement and time on page.
Practice questions
- Find the reciprocal of 8.
- Find the reciprocal of 4/9.
- Find the reciprocal of 1 2/5.
- Find the reciprocal of 0.5.
- Calculate 10 ÷ 5/2.
Answers
- 1/8
- 9/4
- 5/7
- 2
- 4
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