Question

Three years ago, father was five times as old as his daughter. Seven years hence father's age will be three times the age of his daughter. Find their present ages.

Answer 1

Let age of daughter be x and Father's age be y

$\bigstar{\pmb{\underline{\sf{ According \ to \ 1^{st} \ Condition: }}}}$

3 Years ago,

• Daughter's Age be (x - 3) years
• Father's age be (y - 3) years

$\colon\implies{\sf{ 5(x-3)=(y-3) }} \\ \\ \colon\implies{\sf{ 5x-15 = y-3}} \\ \\ \colon\implies{\sf{ 5x-y = -3+15}} \\ \\ \colon\implies{\pmb{\sf{ 5x-y=12 \ \ \ \ \ \ \ \cdots(1)}}}$

$\bigstar{\pmb{\underline{\sf{ According \ to \ 2^{nd} \ Condition: }}}}$

7 Years Hence,

• Daughter's age be (x + 7) years
• Father's age be (y + 7) years

$\colon\implies{\sf{ 3(x+7)=(y+7) }} \\ \\ \colon\implies{\sf{ 3x+21 = y+7}} \\ \\ \colon\implies{\sf{ 3x-y = 7-21}} \\ \\ \colon\implies{\pmb{\sf{ 3x-y= -14 \ \ \ \ \ \ \ \cdots(2)}}}$

After Equalising both Equations as to compare them :-

$\begin{cases} {\pmb{\sf{ \ \ 5x- \cancel{y} =12 }}} \\ {\pmb{\sf{ -3x+ \cancel{y} = 14 }}} \\ \\ {\pmb{\sf{ 2x = 26 }}} \end{cases}$

After Calculating, we've;

$\colon\mapsto{\sf{ 2x = 26 }} \\ \\ \colon\mapsto{\sf{ x = \cancel{ \dfrac{26}{2} } }} \\ \\ \colon\mapsto{\sf{ x = 13 }}$

So, Present age of the daughter is 13 years.

After putting value of x in any Equation to get the value of y as:-

$\colon\mapsto{\sf{ 5x-y = 12 }} \\ \\ \colon\mapsto{\sf{ 5 \times 13 - y = 12 }} \\ \\ \colon\mapsto{\sf{ 65-y = 12 }} \\ \\ \colon\mapsto{\sf{ 65 -12 = y }} \\ \\ \colon\mapsto{\sf{ y = 53 \ Years }}$

Hence,

• Present age of Daughter = 13 years
• Present age of Father = 53 years