Question

5. After 8 years, ratio of ages of A and B will be 9:7
respectively. Four years ago, the ratio of their ages was
15:11 respectively. Find present age of A.
fa) 64 years
(b) 48 years
(c) 56 years
(d) 60 years
(e) None of these​

Answer 1

$\large\underline{\bold{Solution-}}$

Concept Used :-

Writing Systems of Linear Equations from Word Problems

Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

Carry out the plan and solve the problem.

Let's now solve the problem!!

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{present \: age \: of \: A \: be \: x \: years} \\ &\sf{present \: age \: of \: B \: be \: y \: years} \end{cases}\end{gathered}\end{gathered}$

$\bf{ \underline{According \: to \: Ist \: condition - }}$

• After 8 years, ratio of ages of A and B will be 9:7

So,

• The ages after 8 years will be

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{ \: age \: of \: A \: be \: x + 8\: years} \\ &\sf{ \: age \: of \: B \: be \: y + 8\: years} \end{cases}\end{gathered}\end{gathered}$

Therefore,

$\sf \: \dfrac{x + 8}{y + 8} = \dfrac{9}{7}$

$\sf \: 7x + 56 = 9y + 72$

$\therefore \: \: \boxed{ \bf{7x - 9y = 16}} - - - (1)$

$\bf{ \underline{According \: to \: 2nd \: condition - }}$

• Four years ago, the ratio of their ages was 15:11

So,

• The age 4 years ago wil be

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{\: age \: of \: A \: be \: x - 4 \: years} \\ &\sf{ \: age \: of \: B \: be \: y - 4\: years} \end{cases}\end{gathered}\end{gathered}$

Therefore,

$\sf \: \dfrac{x - 4}{y - 4} = \dfrac{15}{11}$

$\sf \:11x - 44 = 15y - 60$

$\therefore \: \: \boxed{ \bf{11x - 15y = - 16 }} - - - - (2)$

Now, solving equation (1) and (2) using elimination method.

Multiply equation (1) by 5 and equation (2) by 3, we get

$\sf \:35x - 45y = 80 - - - (3)$

and

$\sf \:33x - 45y = - 48 - - - - (4)$

Now,

On Subtracting equation (4) from equation (3), we get

$\sf \:2x = 128$

$\therefore \: \: \boxed{ \bf{x \: = \: 64 \: years }}$

$\: \: \boxed{ \bf{Hence \: age \: of \: A \: is \: 64 \: years.}}$

$\therefore \: \: \boxed{ \bf{Hence, \: Option \: (a) \: is \: correct}}$

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