Question

A fraction’s value is `4/5`. When its numerator is increased by 9, the new fraction equals the reciprocal of the value of the original fraction. Find the original fraction.

Answer 1

Here, let the original fraction be `x/y`.

Using the given value of the fraction:

`x/y = 4/5`

5x = 4y     ...(i)

Applying the given second condition,

When the numerator is increased by 9, the new fraction becomes equal to the reciprocal of the original fraction:

`(x + 9)/y = 5/4`

4(x + 9) = 5y     ...(ii)

Let’s solve equations:

From equation (i): 5x = 4y

`y = (5x)/4`

Substitute `y = (5x)/4` into equation (ii):

`4(x + 9) = 5((5x)/4)`

`4x + 36 = (25x)/4`

Multiplying by 4,

`4(4x + 36) = 4((25x)/4)`

16x + 144 = 25x

25x − 16x = 144

9x = 144

x = `144/9`

∴ x = 16

Now, substitute x = 16 into y = `(5x)/4`:

`y = (5xx16)/4`

`y = 80/4`

∴ y = 20

Hence, the original fraction is `16/20 = 4/5`.