Question

The ratio of the ages of a girl and her mother is 1:3. After 6 years, the ratio will be 5:12. If the present ages are x and 3x, find the ages after 6 years.

Answer 1

Answer:

Girl's age : 20

Mother's age : 48

Step-by-step explanation:

It is given that the ratio is 1:3, and the current ages are also given (because of the ratio) as x and 3x. We also know for a fact that after 6 years, their respective ages are in the ratio 5:12.

So, we can write the following equation:

$\frac{x+6}{3x+6} = \frac{5}{12}$

The above relation is because of their respective ratios.

Solving the same,

$(x + 6) * 12 = (3x + 6) * 5\\12x +72 = 15x + 30\\42 = 3x$

Hence, x = 14

But we need to find their respective ages after 6 years:

Girl's age after 6 years : 14 + 6 = 20

Mother's age after 6 years : (3*14) + 6 = 42 + 6 = 48

Answer 2

Answer -

Age of girl after 6 years = 20 years

Age of mother after 6 years = 48 years

Step-by-step explanation -

$\bold{\sf{\green{Method\:1)}}}$

The ratio of the ages of a girl and her mother is 1:3.

Let present age of girl be 1M and present age of mother be 3M.

After 6 years, the ratio will be 5:12.

After 6 years, age of girl will be 1M + 6 and mother will be 3M + 6 and ratio of their ages after 6 years, will be 5:12.

$\implies\:\sf{\dfrac{1M\:+\:6}{3M\:+\:6}\:=\:\dfrac{5}{12}}$

$\implies\:\sf{12(M\:+\:6)\:=\:5(3M\:+\:6)}$

$\implies\:\sf{12M\:+\:72\:=\:15M\:+\:30}$

$\implies\:\sf{12M\:-\:15M\:=\:30\:-\:72}$

$\implies\:\sf{-3M\:=\:-42}$

$\implies\:\sf{M\:=\:14}$

$\therefore$ Age of girl after 6 years = $\sf{1(14)\:+\:6}$

=> $\sf{14\:+\:6}$

=> $\sf{20\:years}$

$\therefore$ Age of mother after 6 years = $\sf{3(14)\:+\:6}$

=> $\sf{42\:+\:6}$

=> $\sf{48\:years}$

$\rule{200}2$

$\bold{\sf{\green{Method\:2)}}}$

The ratio of the ages of a girl and her mother is 1:3.

Let age of girl be 'G' and age of mother be 'M'

$\implies\:\sf{\dfrac{G}{M}\:=\:\dfrac{1}{3}}$

$\implies\:\sf{G\:=\:\dfrac{M}{3}}$ ...(1)

After 6 years, the ratio will be 5:12.

$\implies\:\sf{\dfrac{G\:+\:6}{M\:+\:6}\:=\:\dfrac{5}{12}}$

$\implies\:\sf{12(G\:+\:6)\:=\:5(M\:+\:6)}$

$\implies\:\sf{12G\:+\:72\:=\:5M\:+\:30}$

$\implies\:\sf{12G\:-\:5M\:=\:30\:-\:72}$

$\implies\:\sf{12(\frac{M}{3})\:-\:5M\:=\:-42}$

$\implies\:\sf{\frac{12M\:-\:15M}{3}\:=\:-42}$

$\implies\:\sf{-3M\:=\:3(-42)}$

$\implies\:\sf{M\:=\:42}$

Substitute value of M in equation (1)

$\implies\:\sf{G\:=\:\dfrac{42}{3}}$

$\implies\:\sf{G\:=\:14}$

$\therefore$ Age of girl after 6 years = $\sf{14\:+\:6}$

=> $\sf{20\:years}$

$\therefore$ Age of mother after 6 years = $\sf{42\:+\:6}$

=> $\sf{48\:years}$