1. Reciprocals Objectives Understand the idea of a reciprocal
Find the reciprocal of numbers
Find an algebraic reciprocal
Understand that reciprocals are numbers used for multiplicative inverse
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2. What’s a reciprocal? The reciprocal of 4 is 1/4  1 ÷ 4 = 0.25 1. 5
It’s 1 over the number which is the number divided into 1 like this…
The reciprocal of 4 is
1/4  1 ÷ 4 = 0.25
1. 5
2. 8
3. 2
4. 3
5. 6
= 1/5  1 ÷ 5 = 0.2
= 1/8  1 ÷ 8 = 0.125
What are the reciprocals of these numbers?
= 1/2  1 ÷ 2 = 0.5
= 1/3  1 ÷ 3 = 0.333…
= 1/6  1 ÷ 6 = 0.166…

3. The reciprocal of the fraction 5/8 is 1 over a 5/8
We can write this as…
1/5/8
 1 ÷ 5/8
We deal with the division by a fraction in this way…
 1 x 8/5
Notice that the reciprocal is the fraction turned up-side-down
 8/5
 13/5
 1.6
This can written as a mixed a decimal number

4. 3/5  5/3 = 12/3  1.67 to 2 dp What is the reciprocal of 3/5?
We could give the answer as a decimal
3/5
 5/3
= 12/3
 1.67 to 2 dp
The reciprocal of any fraction is the fraction inverted (turned up-side-down)
This is an improper fraction and needs to be turned into a mixed number

5. Giving the answers as: i) mixed numbers and ii) decimals to 2 dp, what are the reciprocals of:1.
5/8 2.
3/4 3.
5/9 4.
2/7 5.
8/9 6.
3/10 7.
4/9 8.
8/11
 8/5
= 1 3/5
 1.6
 4/3
= 1 1/3
 1.33
 9/5
= 1 4/5
 1.8
All to
2 dp
 7/2
= 3 1/2
 3.5
 9/8
= 1 1/8
 1.125
 10/3
= 3 1/3
 3.33
 9/4
= 2 1/4
 2.25
 11/8
= 1 3/8
 1.38

6. 21/4  9/4  4/9  0.4444… The reciprocals of mixed numbers
Change the mixed number into an improper fraction
21/4
 9/4
 4/9
Now invert (turn up-side-down) the improper fraction
 …
We could write as a decimal if this is required

7. Write as improper fractions
Change to a decimal (2 dp)
Giving the answers as: i) mixed numbers and ii) decimals to 2 dp, what are the reciprocals of: 1.
15/8 2.
12/3 3.
13/4 4.
25/6 5.
33/5 6.
35/6 7.
17/10 8.
54/9
= 13/8
 8/13
 0.62
= 5/3
 3/5
 0.6
Invert to get the reciprocal
= 7/4
 4/7
 0.57
= 17/6
 6/17
 0.35
= 18/5
 5/18
 0.28
= 23/6
 6/23
 0.26
= 17/10
 10/17
 0.59
= 49/9
 9/49
 0.18

8. b/a What is the reciprocal of a/b?
The reciprocals of algebraic fractions
…occasionally on the GCSE this question is asked…
The reciprocal of any fraction is the fraction inverted
This is true for algebraic fractions and…
What is the reciprocal of a/b?
b/a
The answer is…

9. y/x 5/x z/xy x/3 3. What is the reciprocal of x/5?
1. What is the reciprocal of x/y?
3. What is the reciprocal of x/5?
y/x
5/x
2. What is the reciprocal of 3/x?
4. What is the reciprocal of xy/z?
z/xy
x/3

10. Reciprocals and Multiplicative Inverse

11. The reciprocal of the multiplier is the inverse multiplier
INVERSE takes us back to our starting point
x 2 3 6 3
x 0.5 6
6 x 0.5 = 3
The multiplier is 2
Reciprocal of 2 is
1 ÷ 2 = 0.5

12. It will probably look like this…
If you have a scientific calculator, there will be a button that find reciprocals
It will probably look like this…
or this…
1/x
x-1
Check that you can use your calculator by obtaining the same answers as the ones in the table
Number
Reciprocal 20
0.05
0.5 2
7.2 …
0.15 …

13. Find the reciprocal of the multiplier 1. 7 x 8 = 56 2. 12 x 16 = 192 3.
9 x 15 = 135 4.
12 x 25 = 300 5.
19 x 20 = 380 6.
1.2 x 0.4 = 0.48 7.
2.8 x 1.8 = 5.04 8.
7.9 x 12.5 = 98.75
Use your calculator to find the reciprocal of the multiplier and then check that it is the multiplicative inverse

14. Find the reciprocal of the multiplier1.
7 x 8 = 56 2.
12 x 16 = 192 3.
9 x 15 = 135 4.
12 x 25 = 300 5.
19 x 20 = 380 6.
1.2 x 0.4 = 0.48 7.
2.8 x 1.8 = 5.04 8.
7.9 x 12.5 = 98.75
56 x
0.125
= 7
192 x
0.0625
= 12
135 x …
= 9
300 x
0.04
= 12
380 x
0.05
= 19
0.48 x
2.5
= 1.2
5.04 x
0.5555…
= 2.8
98.75 x
0.08
= 7.9