Question

The ratio of ages (in years) of A and B , 9 years ago, was 23:12. The ratio of their ages after 15 years will
be 35:24. What is the ratio of present ages of A and B ?
5:4
4:3
3:2
5:3​

Answer 1

Solution :-

Let,

Present age of A = x

Present age of B = y

9 years ago,

Age of A = x - 9

Age of B = y - 9

After 15 years,

Age of A = x + 15

Age of B = y + 15

According to the first condition :-

$\longrightarrow \sf \dfrac{x-9}{y-9}= \dfrac{23}{12} \\\\\longrightarrow 12( x - 9 ) = 23 ( y - 9 ) \\\\\longrightarrow 12x-108 = 23y - 207 \\\\\longrightarrow 12x-23y = - 9 9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .........................(1)$

According to the second condition :-

$\longrightarrow \sf \dfrac{x+15}{y+15} = \dfrac{35}{24} \\\\\longrightarrow 24 ( x+ 15 ) = 35 ( y+ 15 ) \\\\\longrightarrow 24x+360 = 35y + 525 \\\\\longrightarrow 24x-35y = 165 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .........................(2)$

Multiply eq (1) by 2 :-

$\longrightarrow \sf 24x-46y = -198 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .........................(3)$

Eq (2) - Eq (3) :-

$\longrightarrow \sf 11y = 363 \\\\\longrightarrow \bf y = 33$

Substitute the value of y in eq (1) :-

$\longrightarrow \sf 12x - 23 \times 33 = -99 \\\\\longrightarrow 12x - 759 = - 99 \\\\\longrightarrow 12 x = 660 \\\\\longrightarrow \bf x = 55$

Present age of A = 55 years

Present age of B = 33 years

Ratio of present ages of A and B :-

$\longrightarrow \sf 55 \ : \ 33 \\\\\longrightarrow \bf 5 \ : \ 3$