Question

5 years ago, the age of father was 2.25 times the age of his son. 2 years hence, the age of father becomes 2.6 times the age of his daughter. If the son is 7 years elder to daughter, find the present age of Father​

Answer 1

$\bold\red{\underline{\underline{Answer:}}}$

$\bold{The \ present \ age \ of \ father \ is \ 50 \ years.}$

$\bold\orange{Given:}$

$\bold{=>5, \ years \ ago \ father's \ age \ was \ 2.25}$

$\bold{times \ the \ son's \ age.}$

$\bold{=>2, \ years \ hence \ the \ age \ of \ father}$

$\bold{becomes \ 2.6 \ times \ the \ age \ of \ daughter.}$

$\bold{=>Son \ is \ 7 \ years \ elder \ than \ daughter.}$

$\bold\pink{To \ find:}$

$\bold{=>Present \ age \ of \ father.}$

$\bold\green{\underline{\underline{Solution}}}$

$\bold{Let \ the \ son's \ age \ be \ x \ and \ father's }$

$\bold{age \ be \ y \ years \ respectively.}$

$\bold{According \ to \ first \ condition}$

$\bold{y-5=2.25(x-5)}$

$\bold{y-5=2.25x-11.25}$

$\bold{2.25x-y=-5+11.25}$

$\bold{2.25x-y=6.25}$

$\bold{Multiply \ equation \ by \ 10 \ throughout}$

$\bold{22.5x-10y=62.5...(1)}$

$\bold{Son \ is \ 7 \ years \ elder \ than \ daughter}$

$\bold{\therefore{Daughter's \ age \ will \ be \ (x-7) \ years.}}$

$\bold{According \ to \ second \ condition.}$

$\bold{y+2=2.6(x-7+2)}$

$\bold{y+2=2.6(x-5)}$

$\bold{y+2=2.6x-13}$

$\bold{2.6x-y=2+13}$

$\bold{2.6x-y=15}$

$\bold{Multiply \ equation \ by \ 10 \ throughout}$

$\bold{26x-10y=150...(2)}$

$\bold{Subtract \ equation (1) \ from \ equation (2)}$

$\bold{26x-10y=150}$

$\bold{-}$

$\bold{22.5x-10y=62.5}$

$\bold{3.5x=87.5}$

$\bold{x=\frac{87.5}{3.5}}$

$\bold{x=25}$

$\bold{Substitute \ x=25 \ in \ equation (1), \ we \ get}$

$\bold{22.5(25)-10y=62.5}$

$\bold{562.5-10y=62.5}$

$\bold{10y=562.5-62.5}$

$\bold{10y=500}$

$\bold{y=\frac{500}{10}}$

$\bold{y=50}$

$\bold\purple{\tt{\therefore{The \ present \ age \ of \ father \ is \ 50 \ years.}}}$

Answer 2

Given :

• The son is 7 years elder to daughter.
• 5 years ago, the age of father was 2.25 times the age of his son.
• 2 years hence, the age of father becomes 2.6 times the age of his daughter.

To find :

• Present age of father.

Solution :

Consider,

• Present age of father = x years.
• Present age of son = y years.
• Present age of daughter = z years.

According to 1st condition :-

• The son is 7 years elder to daughter.

$\implies\sf{y=z+7.............eq(1)}$

According to 2nd condition :-

• 5 years ago, the age of father was 2.25 times the age of his son.

5 years ago,

• Age of father = (x-5) years
• Age of son = (y-5) years

$\implies\sf{x-5=2.25(y-5)..............eq(2)}$

According to 3rd condition :-

• 2 years hence, the age of father becomes 2.6 times the age of his daughter.

After 2 years,

• Age of father = (x+2) years
• Age of daughter = (z+2) years

$\implies\sf{x+2=2.6(z+2)}$

$\implies\sf{x+2=2.6z+5.2}$

$\implies\sf{x-2.6z=3.2................eq(3)}$

Now taking eq(2) and put y=z+7 from eq(1).

$\implies\sf{x-5=2.25(z+7-5)}$

$\implies\sf{x-5=2.25(z+2)}$

$\implies\sf{x-5=2.25z +4.5}$

$\implies\sf{x-2.25z=9.5.................eq(2)}$

★ By elimination of eq(2) and eq(3) , we get,

x - 2.25z = 9.5

x - 2.6z = 3.2

(-). (+) (-)

______________

→ 0.35 z = 6.3

→ z = 18

• Present age of daughter = 18 years.

Now put z = 18 in eq(1) for getting the value of y.

$\implies\sf{y=z+7}$

$\implies\sf{y=18+7}$

$\implies\sf{y=25}$

• Present age of son = 25 years.

Now put z = 18 in eq(3) for getting the value of x.

$\implies\sf{x-2.6z=3.2}$

$\implies\sf{x-2.6\times\:18=3.2}$

$\implies\sf{x-46.8=3.2}$

$\implies\sf{x=3.2+46.8}$

$\implies\sf{x=50}$

Therefore, the present age of father is 50 years.