Question

Find the fraction which becomes `1/2` when the denominator is increased by 1 and is equal to `2/3` when both the numerator and denominator are increased by 4.

Answer 1

Here, let the fraction be `x/y`

According to the given conditions:

(1) When the denominator is increased by 1, the fraction becomes `1/2`:

`x/(y + 1) = 1/2`     ...(i)

(2) When both numerator and denominator are increased by 4, the fraction becomes `2/3`:

`(x + 4)/(y + 4) = 2/3`     ...(ii)

Now, cross-multiplying both equations:

From equation (i): `x/(y + 1) = 1/2`

2x = y + 1     ...(iii)

From equation (ii): `(x + 4)/(y + 4) = 2/3`

3(x + 4) = 2(y + 4)

3x + 12 = 2y + 8     ...(iv)

Substituting equation (iii) into equation (iv):

From equation (iii): y = 2x − 1

3x + 12 = 2(2x − 1) + 8

3x + 12 = 4x − 2 + 8

3x + 12 = 4x + 6

12 − 6 = 4x − 3x

∴ x = 6

Then, substitute x = 6 into equation (iii):

y = 2x − 1

y = 2(6) − 1

y = 12 − 1

∴ y = 11

Hence, the fraction is `6/11`.