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 ?
​

Answer 1

Step-by-step explanation:

Let the present age of A be " x "

And present age of B be " y "

9 years ago:-

Age of A = x - 9

Age of B = y - 9

Given,9 years ago, the ratio was 23 : 12

$\rm \dashrightarrow \dfrac{x - 9}{y - 9} = \dfrac{23}{12}$

$\rm \dashrightarrow 12(x - 9) = 23(y - 9)$

$\rm \dashrightarrow 12x - 108 = 23y - 207$

$\rm \dashrightarrow 12x - 23y = -207 + 108$

$\rm \dashrightarrow 12x - 23y = -99.....(i)$

15 years later:-

Age of A = x + 15

Age of B = y + 15

Given,9 years ago, the ratio was 35 : 24

$\rm \dashrightarrow \dfrac{x + 15}{y + 15} = \dfrac{35}{24}$

$\rm \dashrightarrow 24(x + 15) = 35(y + 15)$

$\rm \dashrightarrow 24x + 360 = 35y + 525$

$\rm \dashrightarrow 24x - 35y = 525 - 360$

$\rm \dashrightarrow 24x - 35y = 165.....(ii)$

Multiplying equation (i) with 2

$\rm \dashrightarrow 2(12x - 23y = -99)$

$\rm \dashrightarrow 24x - 46y = -198.....(iii)$

Equation (ii) - Equation (iii)

$\rm \dashrightarrow 24x - 35y - 24x + 46y = 165 + 198$

$\rm \dashrightarrow 11y = 363$

$\rm \dashrightarrow y = \dfrac{363}{11} = 33$

Substituting y = 33 in equation (i)

$\rm \dashrightarrow 12x - 23y = -99$

$\rm \dashrightarrow 12x - 23(33) = -99$

$\rm \dashrightarrow 12x - 759 = -99$

$\rm \dashrightarrow 12x = 759 -99$

$\rm \dashrightarrow 12x = 660$

$\rm \dashrightarrow x = \dfrac{660}{12} = 55$

Therefore:-

Present age of A = 66 years

Present age of B = 33 years

Answer 2

$\huge {\underline {\mathfrak{ \red a \blue n \green \pink s \orange w\purple e\red r\: - }}}$

• Let the present age of A be x

• And present age of B be y

$\underline\bold\blue{9 \: years \: ago:-}$

• Age of A = x - 9

• Age of B = y - 9

Given,9 years ago, the ratio was 23 : 12

$\rm \dashrightarrow \dfrac{x - 9}{y - 9} = \dfrac{23}{12}</p><p> </p><p>$

$\rm \dashrightarrow 12(x - 9) = 23(y - 9)$

$\rm \dashrightarrow 12x - 108 = 23y - 207$

$\rm \dashrightarrow 12x - 23y = -207 + 108$

$\rm \dashrightarrow 12x - 23y = -99.....(i)$

$\underline\bold\pink{15 \: years \: later:-}$

• Age of A = x + 15

• Age of B = y + 15

Given,9 years ago, the ratio was 35 : 24

$\rm \dashrightarrow \dfrac{x + 15}{y + 15} = \dfrac{35}{24}</p><p>$

$\rm \dashrightarrow 24(x + 15) = 35(y + 15)$

$\rm \dashrightarrow 24x + 360 = 35y + 525$

$\rm \dashrightarrow 24x - 35y = 525 - 360$

$\rm \dashrightarrow 24x - 35y = 165.....(ii)$

$\boxed{Multiplying \: equation \: (i) \: with \: 2}$

$\rm \dashrightarrow 2(12x - 23y = -99)$

$\rm \dashrightarrow 24x - 46y = -198.....(iii)$

$\underline\bold{Subtract \: Equation \: (ii) - Equation \: (iii)}</p><p>$

$\rm \dashrightarrow 24x - 35y - 24x + 46y = 165 + 198$

$\rm \dashrightarrow 11y = 363$

$\rm \dashrightarrow y = \dfrac{363}{11} = 33$

$\underline\bold{Substituting \: y \: = \: 33 \: in \: equation \: (i)}$

$\rm \dashrightarrow 12x - 23y = -99$

$\rm \dashrightarrow 12x - 23(33) = -99$

$\rm \dashrightarrow 12x - 759 = -99$

$\rm \dashrightarrow 12x = 759 -99$

$\rm \dashrightarrow 12x = 660$

$\rm \dashrightarrow x = \dfrac{660}{12} = 55$

$\underline\bold\red{Therefore:-}$

☞Present age of A = 66 years

☞Present age of B = 33 years