Question

The difference between the present age of a father and that of the son is 30years.if 14years ago their ages were in ratio 5:2 .how old is the father

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\red{Answer:}}}}}}$

Given :-

• The difference between the present age of a father and that of the son is 30 years
• 14 years ago their ages were in ratio 5:2

To find :-

• Present age of Father

Solution :-

Let the present age of Father be x years

Let the present age of Son be y years.

As per the first condition :-

• The difference between the present age of a father and that of the son is 30 years

Representing it mathematically we get our first equation.

x - y = 30 -----> 1

As per second condition :-

• 14 years ago their ages were in ratio 5:2

Ages 14 years ago :-

Father = x - 14 years

Son = y - 14 years

Ratio = 5 : 2

Let's solve it to get our second equation.

$\bf\implies$$\bf\large\frac{Father}{Son}$ =$\bf\large\frac{5}{2}$

$\bf\implies$$\bf\large\frac{x - 14}{y - 14}$ = $\bf\large\frac{5}{2}$

Cross multiplying,

2 ( x - 14) = 5 ( y - 14)

2x - 28 = 5y - 70

2x - 5y = - 70 + 28

2x - 5y = - 42 -----> 1

Multiply equation 1 by 2,

x - y = 30 ----> 1

2 × x - 2 × y = 2 × 30

2x - 2y = 60 ------> 3

Solve equations 2 and 3 simultaneously by elimination method.

Subtract equation 3 from 2,

...+ 2x - 2y = + 60

- (+ 2x - 5y = - 42 )

-------------------------------

3y = 102

y = $\large\frac{102}{3}$

y = 34

Substitute y = 34 in equation 1,

x - y = 30 ----> 1

x - 34 = 30

x = 30 + 34

x = 64

•°• Present age of Father = x = 64 years.

Present age of Son = y = 34 years.

$\bf{\huge{\underline{\boxed{\rm{\blue{Verification:}}}}}}$

For first case :-

• The difference between the present age of a father and that of the son is 30 years

Father's present age = 64 years = x

Son's present age = 34 years = y

Difference = 30

x - y = 30

64 - 34 = 30

30 = 30

LHS = RHS.

For second case :-

• 14 years ago their ages were in ratio 5:2

Ages 14 years ago :-

Father = x - 14 = 64 - 14 = 50 years

Son = y - 14 = 34 - 14 = 20 years

Ratio = 5 : 2

$\bf\implies$$\bf\large\frac{Father}{Son}$ = $\bf\large\frac{5}{2}$

$\bf\implies$$\bf\large\frac{50}{20}$ = $\bf\large\frac{5}{2}$

Dividing LHS by 10,

$\bf\implies$$\bf\large\frac{5}{2}$ = $\bf\large\frac{5}{2}$

LHS = RHS.

Hence verified.