Question

Ms S Rao has two sons. At present her age is
equal to the sum of the squares of their
Seventeen years hence, she will be twice her
older son's age. Find the present ages of her
sons if their age difference is 1 year.​

Answer 1

GIVEN :

• Mrs. Rao has two sons.
• Her pesent age is equal to a of the squares of both the sons.
• After seventeen years she will be twice her older son's age.
• The difference between her sons age is 1 year.

TO FIND :

• The present ages of her sons.

SOLUTION :

• We know that, there is 1 year difference between the two sons age.

So,

Let's assume that younger son's age is x years.

Then older son's age will be (x + 1) years.

Thereafter,

Mrs. Rao's age will be = x² + (x + 1)² years.

Now,

• It's told that after seventeen years her age will be twice her older son's age.

Hence,

Her younger son's age after 17 years :

(x + 17) years.

Age of her older son after seventeen years :

(x + 1) + 17 years.

Or,

= (x + 18) years.

And,

Ms. Rao's age after seventeen years will be :

x² + (x + 1)² + 17 years.

Now,

• As it is being told that she will now(after 17 years) grew twice as her older son's age.

Therefore,

The equation formed,

$\tt \implies x² + (x + 1)² + 17 = 2(x + 18)$

According to the question,

$\tt \implies \: {x}^{2} + {x}^{2} + 2x + 1 + 17= 2x + 36$

$\tt \implies \: 2 {x}^{2} + 2x + 18 = 2x + 36$

$\tt \implies \: 2 {x}^{2} + 2x - 2x + 18 = 36$

$\tt \implies \: 2 {x}^{2} + 18 = 36$

$\tt \implies \: 2 {x}^{2} = 36 - 18$

$\tt \implies \: 2 {x}^{2} = 18$

$\tt \implies \: {x}^{2} = \frac{18}{2}$

$\tt \implies \: {x}^{2} = 9$

$\tt \implies \: x = \sqrt{9}$

$\bold\red \dag{ \underline{ \boxed{\bf {\green {\therefore \: x = 3 \: years.}}}}} \bold\red \dag$

HENCE,

We get

Her younger son's age i.e., x = 3 years.

And,

Her older son's age i.e., (x + 1) years = 4 years.

$\bold \gray \dag{ \underline{ \boxed{ \bf{ \blue {\therefore Required \: answer : }}}}} \bold \gray \dag$

$\green\odot \: { \underline{ \boxed{ \bf{ \pink{ \therefore Younger\: son's \: age \: = 3 \: years \: \: }}}}} \checkmark$

$\green\odot \: { \underline{ \boxed{ \bf{ \orange { \therefore Older \: son's \: age \: = 4 \: years \: \: }}}}} \checkmark$

Verification :

• We will substitute the value of x in the equation :

$\tt \implies x² + (x + 1)² + 17 = 2(x + 18)$

Substituting the value of x :

$\tt \implies \: {3}^{2} + {(3 + 1)}^{2} + 17 = 2(3 + 18)$

$\tt \implies \: 9 + 9 + 6 + 1 + 17 = 6 + 36$

$\bold \gray \dag{ \underline{ \boxed{ \blue{ \bf \therefore \: 42 = 42}}}} \bold \gray \dag$

AFTERALL,

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

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

Answer 2

_________________________

$\huge{\red{\mathfrak{Answer}}}$

Given :

• Mrs Rao has two sons

• her present age is equal to the square of both the sons

• After seventeen years she will be twice her olders son age

• The difference between her son's age is 1 year

To find :

• present ages of her sons

Solution :

• we know that difference between the ages of two sons is 1 year

➜ so,

let us assume that the younger son age is x years , then the older son will be x + 1 years old

➜ thereafter ,

Mrs Rao age will be x² + (x + 1 )² years

➜ now ,

• it is told that after seventeen years her age will be twice his older son

➜ hence ,

we get ,

Her younger son's age after 17 years :

( x + 17)

Her older son's age after 17 years:

( x + 1 ) + 17

or

( x + 18 )

and ,

Mrs Rao's age after 17 years will be ;

x² + (x + 1) ² + 17

now ,

• As being told that now ( after 17 years ) she will grew as twice as her older son's age.

therefore the equation formed :

⇒$\large\mathtt{\boxed{x²\:+(x+1)²\:+\:17\:=2(x+18)}}$

According to question ,

⇒ $\large\mathtt{x²\:+\:x²\:+2x\:+1\:+17\:=\:2x\:+36}$

⇒ $\large\mathtt{2x²\:+2x\:+18\:=\:2x\:+36}$

⇒ $\large\mathtt{2x²\:+2x\:-2x\:+18\:=\:36}$

⇒ $\large\mathtt{2x²\:+\:18\:=\:36}$

⇒ $\large\mathtt{2x²\:=\:36\:-18}$

⇒$\large\mathtt{2x²\:\:=\:\:18}$

⇒$\large\mathtt{x²\:=\:{\cancel{\frac{18}{2}}}}$

⇒$\large\mathtt{x²\:=\:9}$

⇒$\large\mathtt{x\:=\:{\sqrt{9}}}$

hence ,

we get..

$\large\mathtt{\boxed{\green{Younger\:son\:age\:=\:3yrs}}}$

$\large\mathtt{\boxed{\green{Older\:son\:age\:=\:x+1\:=\:4yrs}}}$

__________________________