Question

The ages of A and B are in ratio 9:4. Seven years hence, the ratio of their ages will be 6:3. Find their ages

Answer 1

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\large{\underline{\red{\rm{Let:}}}}$

• $\sf{The\: ratio\:of\: their\:age\:be\:x}$
• $\sf{The\:age\:of\:A=9x}$
• $\sf{The\:age\:of\:B=4x}$

$\large{\underline{\red{\rm{To\: Find:}}}}$

• $\sf{The\:age\:of\:A\:and\:B}$

$\large{\underline{\red{\rm{Given:}}}}$

• The ages of A and B are in ratio 9:4. Seven years hence, the ratio of their ages will be 6:3.

$\small{\underline{\purple{\rm{According\:to\:the\: Question:}}}}$

$\rm{\implies \dfrac{9x+7}{4x+7}=\dfrac{6}{3}}$

$\rm{\implies 3(9x+7)=6(4x+7)}$

$\rm{\implies 27x+21=24x+42}$

$\rm{\implies 27x-24x=42-21}$

$\rm{\implies \cancel{3}x=\cancel{21}}$

$\rm\green{\implies x=7}$

Hence,

$\sf\purple{The\:age\:of\:A=9x=9*7=63\: year's}$

$\sf\green{The\:age\:of\:B=4x=4*7=28\: year's}$

Answer 2

Let the ages of A and B be x and y respectively.

Given

The ages of A and B are in ratio 9:4.

$\implies \dfrac{x}{y} = \dfrac{9}{4}$

$\implies 4x=9y$

$\implies 4x-9y=0 --(1)$

Seven years hence, the ratio of their ages will be 6:3.

$\implies \dfrac{x+7}{y+7} = \dfrac{6}{3}$

$\implies 3(x+7)=6(y+7)$

$\implies 3x+21=6y+42$

$\implies 3x-6y=42-21$

$\implies 3x-6y=21$

$\implies x-2y=7--(2)$

Solving, (1) and (2), we get,

$4x-9y=0\\\underline{4x-8y=28} \ (\bold { By \ multiplying \ the \ second \ equation \ by \ 4})\\\underline{\underline{-y=-28}}\\\implies y = 28$

$\implies x = 7+2y\\\implies x = 7+56\\\implies x = 63$

$\boxed{\boxed{\bold{Therefore \ the \ ages \ of \ A \ and \ B \ are \ 63 \ and \ 28 \ years \ respectively}}}}}$