Question

Sum of the present ages of two friends are 23 years five years ago product of their ages was 42. Find their ages 5 years hence.​

Answer 1

Answer:

Given :

• Sum of the present ages of two friends are 23 years.
• Five years age product of their ages was 42.

To Find :-

• What is their ages 5 years hence.

Solution :-

Let,

$\mapsto \bf Present\: Age_{(1^{st}\: Friend)} =\: x\: years$

$\mapsto \bf Present\: Age_{(2^{nd}\: Friend)} =\: (23 - x)\: years$

☆ Five years ago :

$\leadsto \sf Age_{(1^{st}\: Friend)} =\: (x - 5)\: years$

$\small \leadsto \sf Age_{(2^{nd}\: Friend)} =\: (23 - x - 5) =\: (18 - x)\: years$

According to the question :

$\bigstar$ The product of their ages was 42.

So,

$\footnotesize \implies \bf \bigg\{Age_{(1^{st}\: Friend)}\bigg\} \times \bigg\{Age_{(2^{nd}\: Friend)}\bigg\} =\: 42$

$\implies \sf (x - 5)(18 - x) =\: 42$

$\implies \sf 18x - x^2 - 90 + 5x =\: 42$

$\implies \sf - x^2 + 18x + 5x - 90 =\: 42$

$\implies \sf - x^2 + 23x - 90 =\: 42$

$\implies \sf - x^2 + 23x - 90 - 42 =\: 0$

$\implies \sf - x^2 + 23x - 132 =\: 0$

$\implies \sf - x^2 + (11 + 12)x - 132 =\: 0$

$\implies \sf - x^2 + 11x + 12x - 132 =\: 0$

$\implies \sf - (x^2 - 11x - 12x + 132) =\: 0$

$\implies \sf x^2 - 11x - 12x + 132 =\: 0$

$\implies \sf x(x - 11) - 12(x - 11) =\: 0$

$\implies \sf (x - 11)(x - 12) =\: 0$

$\implies \bf x - 11 =\: 0$

$\implies \sf\bold{\purple{x =\: 11}}$

Or,

$\implies \bf x - 12 =\: 0$

$\implies \sf\bold{\purple{x =\: 12}}$

Hence, the required present ages of two friends are :

❒ When x = 11 :

$\footnotesize \dashrightarrow \sf\bold{\green{Present\: Age_{(1^{st}\: Friend)} =\: x\: years =\: 11\: years}}$

$\footnotesize \dashrightarrow \sf\bold{\green{Present\: Age_{(2^{nd}\: Friend)} =\: (23 - x) =\: (23 - 11) =\: 12\: years}}$

❒ When x = 12 :

$\footnotesize \dashrightarrow \sf\bold{\green{Present\: Age_{(1^{st}\: Friend)} =\: x\: years =\: 12\: years}}$

$\footnotesize \dashrightarrow \sf\bold{\green{Present\: Age_{(2^{nd}\: Friend)} =\: (23 - x) =\: (23 - 12) =\: 11\: years}}$

Hence, their ages 5 years hence :

When x = 11 :

$\footnotesize \leadsto \sf\bold{\red{Present\: Age_{(1^{st}\: Friend)} =\: (11 + 5)\: years =\: 16\: years}}$

$\footnotesize \leadsto \sf\bold{\red{Present\: Age_{(2^{st}\: Friend)} =\: (12 + 5)\: years =\: 17\: years}}$

When x = 12 :

$\footnotesize \leadsto \sf\bold{\red{Present\: Age_{(1^{nd}\: Friend)} =\: (12 + 5)\: years =\: 17\: years}}$

$\footnotesize \leadsto \sf\bold{\red{Present\: Age_{(2^{nd}\: Friend)} =\: (11 + 5)\: years =\: 16\: years}}$