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 Terms and Conditions: To the best of the producer's knowledge, the presentation’s academic content is accurate but errors and omissions may be present and Brain-Cells: E.Resources Ltd cannot be held responsible for these or any lack of success experienced by individuals or groups or other parties using this material. The presentation is intended as a support material for GCSE maths and is not a comprehensive pedagogy of all the requirements of the syllabus. The copyright proprietor has licensed the presentation for the purchaser’s personal use as an educational resource and forbids copying or reproduction in part or whole or distribution to other parties or the publication of the material on the internet or other media or the use in any school or college that has not purchased the presentation without the written permission of Brain-Cells: E.Resouces Ltd.

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…

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

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

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

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  0.4444… We could write as a decimal if this is required

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

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…

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

Reciprocals and Multiplicative Inverse

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

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 0.138888… 0.15 6.666666…

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

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 56 x 0.125 = 7 192 x 0.0625 = 12 135 x 0.06666… = 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