Question

The present ages of Manoj and Amit are in ratio 2:3. After 12 years, their age will be in ratio 11:15. The present age of Amit is:
A. 32 years
B. 48 years
C. 40 years
D. 35 years

Answer 1

Hint:Find the relation connecting the present ages of Manoj and Amit. Similarly find the relation connecting their ages after 12 years. Thus solve by substituting the values.

Complete step-by-step answer:
Let us assume the present age of Manoj as ‘x’.
The present age of Amit can be taken as ‘y’.
It is said that the ratio of their present ages is 2:3,
i.e. x:y = 2:3
Thus we can write it as, \[\dfrac{x}{y}=\dfrac{2}{3}\]
i.e. \[x=\dfrac{2y}{3}\]
Thus we got a relation connecting the ages of Manoj and Amit.
It is also said that after 12 years, the ratio of their ages becomes 11:15.
After 12 years, the age of Manoj becomes x + 12.
After 12 years, the age of Amit becomes y + 12.
Thus the ratio of Manoj and Amit’s age after 12 years is 11:15.
\[\therefore x+12:y+12=11:15.\]
We can write the above expression as,
\[\dfrac{x+12}{y+12}=\dfrac{11}{15}\] [Cross multiply and simplify it]
\[\begin{align}
  & x+12=\dfrac{11\left( y+12 \right)}{15} \\
 & x+12=\dfrac{11y+132}{15} \\
\end{align}\]
Put \[x=\dfrac{2y}{3}\].
\[\begin{align}
  & \dfrac{2y}{3}+12=\dfrac{11y+132}{15} \\
 & \dfrac{2y+36}{3}=\dfrac{11y+132}{15} \\
\end{align}\]
Cross multiply, simplify the expression and find the value of
\[\begin{align}
  & 2y+36=\dfrac{11y+132}{5} \\
 & 5(2y+36)=11y+132 \\
 & 10y+180=11y+132 \\
 & 11y-10y=180-132 \\
 & y=48. \\
\end{align}\]
Thus we got the age of Amit as y = 48 years.
Now let us find the age of Manoj.
\[x=\dfrac{2y}{3}=\dfrac{2\times 48}{3}=32\]
Thus we got the age of Manoj as 32 years and age of Amit as 48 years.
Option B is the correct answer.

Note:You can also take the ratio of age of Manoj and Amit as 2x:3x, i.e. the age of both is taken as x.
Thus after 12 years, their age becomes (2x+12)(3x+12) which is 11:15.
\[\dfrac{2x+12}{3x+12}=\dfrac{11}{15}\] [Cross multiply and get the value of x].
\[\begin{align}
  & 15(2x+12)=1(3x+12) \\
 & \therefore x+16 \\
\end{align}\]
Age of Manoj \[=2x=2\times 16=32\].
Age of Amit \[=3x=3\times 16=48\].