Question

3. The average age of a father and his son is 29 years. The ratio of their ages 4 years ago was 21: 4 respectively. What is the present age of the son? (a) 12 years (b) 8 years (C) 10 years (d) 14 years​

Answer 1

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

$\bf{\underline{\sf{\purple{Given:}}}}$

• The average age of a father and his son is 29 years.
• The ratio of their ages 4 years ago was 21: 4.

$\bf{\underline{\sf{\purple{To\:find\::}}}}$

• Present age of Son.

$\bf{\underline{\sf{\purple{Solution:}}}}$

Let the present age of Father be x years.

Let the present age of Son be y years.

As per first condition :-

• The average age of a father and his son is 29 years.

$\bf\implies$ $\large\sf\frac{x+y}{2}$ = 29

Cross multiplying,

x + y = 2 ( 29)

x + y = 58 ---> 1

As per second condition :-

• The ratio of their ages 4 years ago was 21: 4

Ages 4 years ago :-

Father = x - 4 years.

Son = y - 4 years

Ratio = 21 : 4

Let's represent it mathematically.

$\bf\implies$ $\large\sf\frac{x-4}{y-4}$ = $\large\sf\frac{21}{4}$

Cross multiplying,

$\bf\implies$ 4 ( x - 4) = 21 ( y - 4 )

$\bf\implies$ 4x - 16 = 21y - 84

$\bf\implies$ 4x - 21y = - 84 + 16

4x - 21y = - 68 ----> 2

Multiply equation 1 by 4,

x + y = 58 ----> 1

4 × x + 4 × y = 4 × 58

4x + 4y = 232 ---> 3

Solve equations 2 and 3 simultaneously by elimination method.

Subtract equation 3 from 2,

.....+ 4x + 4y = + 232 ---> 3

- ( + 4x - 21y = - 68) -----> 2

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

25y = 300

y = $\bf\large\sf\frac{300}{25}$

y = 12

Substitute y = 12 in equation 1,

x + y = 58 ----> 1

x + 12 = 58

x = 58 - 12

x = 46

•°• Present age of Father = x = 46 years

Present age of Son = y = 12 years.

$\bf{\huge{\underline{\boxed{\sf{\red{Verification:}}}}}}$

For first case :-

• The average age of a father and his son is 29 years.

Present age of Father = x = 46 years

Present age of Son = y = 12 years

Average = 29

$\bf\implies$$\large\sf\frac{x+y}{2}$ = 29

$\bf\implies$ $\large\sf\frac{46+12}{2}$ = 29

$\bf\implies$ $\large\sf\frac{58}{2}$ = 29

$\bf\implies$ 29 = 29

LHS = RHS.

For second case :-

• The ratio of their ages 4 years ago was 21: 4

Ages 4 years ago :-

Father = x - 4 = 46 - 4 = 42 years

Son = y - 4 = 12 - 4 = 8 years

Ratio = 21 : 4

$\bf\implies$ $\large\sf\frac{x-4}{y-4}$ = $\large\sf\frac{21}{4}$

$\bf\implies$$\large\sf\frac{42}{8}$ = $\large\sf\frac{21}{4}$

Dividing LHS by 2,

$\bf\implies$$\large\sf\frac{21}{4}$ = $\large\sf\frac{21}{4}$

LHS = RHS.

Hence verified.