Question

1. Four years ago, a father's age was square of
his daughter's age. Nine years hence, his age
will be 2 more than thrice his daughter's age
Find their present ages.​

Answer 1

GIVEN :

• Four years ago father's age was square of his daughter's age.
• Nine years later his age will be 2 more than thrice than his daughter's age.

TO FIND :

• Their present age i.e., Father's and his daughter's age.

SOLUTION :

Let his daughter's age is x years.

4 years ago i.e., (x - 4) years

Squaring his father's age :

= (x - 4)²

Again,

• It is told that Nine years later his age will be 2 more than thrice of his daughter's age.

So,

• his daughter's age after 9 years : (x + 9) years.

Hence,

His father's age after 9 years :

= 2 + 3(x + 9) years

= 2 + 3x + 27

= 3x + 29 years

Which will be equal to (x - 4)² + 9 + 4

As it is total 13 years later.

According to the question,

$\bf \implies \: 3x + 29 = (x - 4)² + 9 + 4$

$\bf \implies \: 3x + 29 = {x}^{2} + 16 - 8x + 13$

$\bf \implies \: 3x + 2 9= {x}^{2} - 8x + 29$

$\bf \implies \: 3x + 8x + 29 - 29 = {x}^{2}$

$\bf \implies \: {x}^{2} = 11x$

$\bf \implies \: {x}^{2} - 11x = 0$

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

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

$\bold \gray \dag{ \underline{ \boxed{\orange {\bf \therefore \: x = 11years. \blue\checkmark }}}} \bold \gray \dag$

HENCE,

His father's age is x = 11 years.

Her father's age is (x - 4)² + 4

$\implies \bf (11 - 4)² + 4$

$\implies\bf 7² + 4$

$\implies \bf 49 + 4$

$\implies \bf 53 years.$

$\purple \bigstar{ \underline{ \boxed{ \bf {\blue{ \: Required \: \blue { answer : }}}}}} \purple \bigstar$

$\bold \gray \dag{ \underline{ \boxed{ \bf { \orange { \therefore fathers \: age \: = 53 years. \: \blue \checkmark}}}}} \bold \gray \dag$

$\bold \gray \dag{ \underline{ \boxed{ \bf { \green { \therefore daughters \: age \: = 11 years. \: \red \checkmark}}}}} \bold \gray \dag$

Verification :

• Verification needs the L. H. S. = R. H. S. by substituting the value of x.

In the equation :

$\bf \implies \: 3x + 29 = (x - 4)² + 9 + 4$

$\bf \implies \: (3 \times 11) + 29 = {(11 - 4) }^{2} + 13$

$\bf \implies \: 33 + 29 = {7}^{2} + 13$

$\bold \gray \dag { \underline{ \boxed{ \green{\bf \therefore\: 62 = 62 \: \red \checkmark}}}} \bold \gray \dag$

FINALLY,

L. H. S. = R. H. S.

${\underline{\boxed{ \green{\bf Hence,\purple V \red e \orange r \pink i \blue f \green i \red e \pink d \red \checkmark }}}}$