Question

The sum of the present ages of Parth and Shivam is 80 years. Ten years ago, Parth's age was 4 times the age of Shyam. Find the difference between their present ages (in years).​

Answer 1

Let the present age of Shivam be x

Then, Parth's age will be 80-x

10 years ago their age were,

x - 10 , 80-x-10=70-x

A.T.Q,

=> 70 - x = 4(x - 10)

=> 70 -x = 4x - 40

=> 70 + x = 4x - 40

=> 70 + 40 = 4x + x

=> 110 = 5x

=> x = 110/5

=> x = 22

Therefore,

Shivam's present age = x = 22

Parth's present = 80-x = 80 - 22 = 58

Hence,

Difference between their present ages

= 58 - 22

= 36.

Answer 2

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

Given that,

The sum of the present ages of Parth and Shivam is 80 years.

Let assume that

Present age of Parth be x years

and

Present age of Shivam be y years.

So,

$\red{\begin{gathered}\begin{gathered}\bf\:\rm :\longmapsto\:\begin{cases} &\sf{Present \: age \: of \: Parth = x \: years} \\ \\ &\sf{Present \: age \: of \: Shivam = y \: years} \end{cases}\end{gathered}\end{gathered}}$

$\rm :\longmapsto\:\boxed{ \tt{ \: \: x + y = 80 \: \: }} - - - - (1)$

According to statement again,

Ten years ago, Parth's age was 4 times the age of Shivam

So,

Parth age 10 years ago is x - 10 years

Shivam age 10 years ago is y - 10 years

Hence,

$\rm :\longmapsto\:x - 10 = 4(y - 10)$

$\rm :\longmapsto\:x - 10 = 4y - 40$

$\rm :\longmapsto\:x - 4y = - 40 + 10$

$\rm :\longmapsto\:\boxed{ \tt{ \: \: x - 4y \: = \: - \: 30 \: \: }} - - - (2)$

On Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\:(x + y) - (x - 4y) = 80 - ( - 30)$

$\rm :\longmapsto\:x + y - x + 4y= 80 + 30$

$\rm :\longmapsto\:5y= 110$

$\purple{\rm \implies\:\boxed{ \tt{ \: y \: = \: 22 \: }}} - - - (3)$

On substituting the value of y in equation (1), we get

$\rm :\longmapsto\:x + 22 = 80$

$\rm :\longmapsto\:x = 80 - 22$

$\purple{\rm \implies\:\boxed{ \tt{ \: x \: = \: 58\: }}} - - - (4)$

Hence,

$\red{\begin{gathered}\begin{gathered}\bf\:\rm :\longmapsto\:\begin{cases} &\sf{Present \: age \: of \: Parth = 58 \: years} \\ \\ &\sf{Present \: age \: of \: Shivam = 22 \: years} \end{cases}\end{gathered}\end{gathered}}$

So, Difference in their present ages = 58 - 22 = 36 years.