Sum of Reciprocals

Introduction:-

The  reciprocal for a number x, denoted by  `1/x`  or x ^−1, is a number which when multiplied by x yields the multiplicative identity 1. The reciprocal of a fraction `a/b` is `b/a` . For the reciprocal of a real number, divide 1 by the number. For example, the reciprocal of 8 is  one fifth (`1/8` or 0.2), and the reciprocal of 0.25 is (`"1/0.25` or 4).                                                                      Source: - Wikipedia

Sum of Reciprocals

The following are the steps used to find sum of reciprocals:-

Step 1 The first step in finding the sum of reciprocal of a two numbers is to find the reciprocal of the given numbers.

Step 2 Then we need to add the reciprocal of these two numbers.

Find the Sum of reciprocal of real numbers a and b.

As per step 1we need to find the reciprocal of a and b.

The reciprocal of a is `1/ a.`

The reciprocal of b is `1/ b.`

Now according to step 2 we add two reciprocals 1/ a and 1/ b.

Sum of reciprocals of a and b is `1/ a + 1/ b` .

Find the Sum of reciprocal of fractions` a/ b` and `c/ d.`

As per step 1we need to find the reciprocal of `a/b` and `c/ d.`

The reciprocal of `a/b` is `b/ a.`

The reciprocal of `c/d` is `d/ c` .

Now according to step 2 we add two reciprocals `b/ a` and `d/ c.`

Sum of reciprocals of `a/b` and `c/ d` is `b/ a + d/ c.`

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Solved Problems:-

Problem 1:-

Find the sum of reciprocal of 2 and 3.

Solution:-

The given two numbers are 2 and 3.

We need to find the sum of their reciprocals.

To find the sum of reciprocals we first need find the reciprocal of given two numbers.

The reciprocal of 2 is `1/2.`

The reciprocal of 3 is `1/ 3.`

Now we can we need to add these two reciprocal values

`= ` `1/ 2+ 1/ 3.`

By taking LCM of 2 and 3 we get 6

`= (3 + 2 )/6`

`= 5/ 6.`

The sum of reciprocal of 2 and 3 is `5/ 6`

Problem 2:-

Find the sum of reciprocal of two fractions  `1/ 3 ` and `3/ 4.`

Solution:-

The given two numbers are `1/ 3` and `3/ 4.`

We need to find the sum of their reciprocals.

To find the sum of reciprocals we first need find the reciprocal of given two numbers.

The reciprocal of `1/ 3` is `3/ 1.`

The reciprocal of `3/ 4` is ` 4 / 3.`

Now we can we need to add these two reciprocal fraction values

= `3/ 1+ 4/ 3.`

By taking LCM of 1 and 3 we get 3.

`= (9 + 4 )/3`

`= 13/ 3.`

The sum of reciprocal of `1/ 3` and `3/ 4.` is `13 / 3`