Five years ago, the ratio of Aman's age to his father's age was 1:4, and five years from now, the ratio will be 2:5. What was his father's age when Aman was born?

Solving Aman's Age Ratio Problem

Let Aman's current age be **$A\ast \ast$ years and his father's current age be **$F\ast \ast$ years.

Formulating Equations from Ratios

- Five years ago: Aman's age was $A-5$, Father's age was $F-5$. The ratio was $1:4$. $\frac{A-5}{F-5}=\frac{1}{4}$ Cross-multiplying gives: $4(A-5)=1(F-5)$ $4A-20=F-5$ $4A-F=15\cdots (1)$
- Five years from now: Aman's age will be $A+5$, Father's age will be $F+5$. The ratio will be $2:5$. $\frac{A+5}{F+5}=\frac{2}{5}$ Cross-multiplying gives: $5(A+5)=2(F+5)$ $5A+25=2F+10$ $5A-2F=-15\cdots (2)$

Calculating Current Ages

We need to solve the system of linear equations (1) and (2) to find $A$ and $F$.

1. From equation (1), express $F$ in terms of $A$: $F=4A-15$
2. Substitute this expression for $F$ into equation (2): $5A-2(4A-15)=-15$ $5A-8A+30=-15$ $-3A=-15-30$ $-3A=-45$ $A=\frac{-45}{-3}=15$ So, Aman's current age is 15 years.
3. Substitute the value of $A$ back into the expression for $F$: $F=4(15)-15$ $F=60-15=45$ So, the father's current age is 45 years.

Determining Father's Age at Aman's Birth

The father's age when Aman was born is the difference between the father's current age and Aman's current age.

Father's age at Aman's birth = $F-A=45-15=30$ years.