Question

Mother is 25 year older than her son. Find son’s age if after 8 years the ratio of son’s age to mother’s age will be \[\dfrac{4}{9}\].

Answer 1

Hint: Here, we will first assume that the present age of the son be \[x\] years and we will add the age of the son with 25 to find the age of the mother's. Now after 8 years, the age of the son will be added with 8 to find the new age and the age of the mother will be added with 8 to find the new age. Then we will take the ratio of son’s new age to mother’s new age equal to \[\dfrac{4}{9}\].

Complete step-by-step answer:
We are given that the Mother is 25 year older than her son and after 8 years the ratio of son’s age to mother’s age will be \[\dfrac{4}{9}\].
Let us assume that the present age of the son be \[x\] years.
Since we know that the mother is 25 year older than her son, so we will add the age of the son with 25, we get
\[ \Rightarrow \left( {x + 25} \right){\text{ years}}\]
Now after 8 years, the age of the son will be added with 8 to find the new age, we get
\[ \Rightarrow \left( {x + 8} \right){\text{ years}}\]
So after 8 years, the age of the mother will be added with 8 to find the new age, we get
\[
   \Rightarrow \left( {x + 25 + 8} \right){\text{ years}} \\
   \Rightarrow \left( {x + 33} \right){\text{ years}} \\
 \]
We will first the take the ratio of son’s new age to mother’s new age equal to \[\dfrac{4}{9}\], we get
\[ \Rightarrow \dfrac{{x + 8}}{{x + 33}} = \dfrac{4}{9}\]
Cross-multiplying the above equation, we get
\[
   \Rightarrow 9\left( {x + 8} \right) = 4\left( {x + 33} \right) \\
   \Rightarrow 9x + 72 = 4x + 132 \\
 \]
Subtracting the above equation by 72 on both sides, we get
\[
   \Rightarrow 9x + 72 - 72 = 4x + 132 - 72 \\
   \Rightarrow 9x = 4x + 60 \\
 \]
Subtracting the above equation by \[4x\] on both sides, we get
\[
   \Rightarrow 9x - 4x = 4x + 60 - 4x \\
   \Rightarrow 5x = 60 \\
 \]


Dividing the above equation by 5 on both sides, we get
\[
   \Rightarrow \dfrac{{5x}}{5} = \dfrac{{60}}{5} \\
   \Rightarrow x = 12 \\
 \]
Thus, the present age of the son is 12 years.

Note: Whenever we face such type of questions on age problems, we always first suppose the age ratio with some variable \[x\] and then apply the conditions of the question on that variable to find out the value of that variable to find the result. Do not assume the value of \[x\] is the required answer, we have to substitute the value of \[x\] in one of the ages to find the final age, or else the answer will be incomplete.