Question

A father is six times as old as his son. after 4 years he will be four times as old as his son will be. what are their present ages ​

Answer 1

Let present age of father be "F" years and present age of son be "S" years.

A father is six times as old as his son.

According to question,

$\implies\:F\:=\:6\:\times\:S$

$\implies\:F\:=\:6S$ ____ (eq 1)

After four years, father will be four times as old as his son.

Now,

• Age of father = (F + 4) years
• Age of son = (S + 4) years.

According to question,

$\implies\:(F\:+\:4)\:=\:4(S\:+\:4)$

$\implies\:F\:+\:4\:=\:4S\:+\:16$

$\implies\:F\:-\:4S\:=\:16\:-\:4$

$\implies\:6S\:-\:4S\:=\:12$

[From (eq 1)]

$\implies\:2S\:=\:12$

$\implies\:S\:=\:6$

Put value of S in (eq 1)

$\implies\:F\:=\:6(6)$

$\implies\:F\:=\:36$

∴ Present age of father is 36 years and present age of son is 6 years.

______________________________

✯ Verification :

From above calculations we have F = 36 and S = 6

According to question, we have equation :

1) F = 6 × S

2) (F + 4) = 4(S + 4)

Put value of F and S in (eq 1)

→ 36 = 6 × 6

→ 36 = 36

Similarly, Put value of F and S in (eq 2)

→ (36 + 4) = 4(6 + 4)

→ 40 = 4(10)

→ 40 = 40

Answer 2

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

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

• A father is six times as old as his son
• After 4 years Father will be four times as old as his Son

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

• Present age of Father
• Present age of Son

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

Let the present age of Father be x years.

Let the present age of Son be y years.

Let the present age of Son be

$\bf{\underline{\underline{\sf{\blue{As\:per\:first\:condition:}}}}}$

• A father is six times as old as his son

Representing mathematically we obtain our first equation.

=> x = 6y ----> 1

$\bf{\underline{\underline{\sf{\blue{As\:per\:second\:condition:}}}}}$

• After 4 years Father will be four times as old as his Son.

Ages after 4 years :

Father = x + 4 years

Son = y + 4 years

Representing the second condition mathematically,

=> x + 4 = 4 ( y + 4)

=> x + 4 = 4y + 16

=> x - 4y = 16 - 4

=> x - 4y = 12 ----> 2

Substitute value of x from equation 1 in equation 2,

=> 6y - 4y = 12

=> 2y = 12

=> y = $\frac{12}{2}$

=> y = 6

Substitute y = 6 in equation 2,

=> x - 4y = 12 - - -> 2

=> x - 4 ( 6) = 12

=> x - 24 = 12

=> x = 12 + 24

=> x = 36

•°•Present age of Father = x = 36 years

Present age of Son = y = 6 years

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

For first case :-

• A father is six times as old as his son

Present age of Father = 36 years = x

Present age of Son = 6 years = y

=> x = 6y

=> 36 = 6 ( 6)

=> 36 = 36

LHS = RHS.

For second case :-

• After 4 years he will be four times as old as his son

Ages after 4 years :-

Father = x + 4 = 36 + 4 = 40 years

Son = y + 4 = 6 + 4 = 10 years

=> x + 4 = 4 ( y + 4)

=> 40 = 4 ( 10)

=> 40 = 40

LHS = RHS.