Father and Son Age Ratio Calculation

Problem Setup

Let the father's current age be $F$ and the son's current age be $S$.

We are given two conditions:

• The sum of their current ages is 50 years: $F + S = 50$.
• 10 years from now, the difference in their ages will be 20 years. In 10 years, their ages will be $F+10$ and $S+10$. The difference remains constant: $(F+10) - (S+10) = F - S$. Thus, $F - S = 20$.

Solving the Age Equations

We need to solve the system of linear equations:

1. $F + S = 50$
2. $F - S = 20$

Add equation (1) and equation (2):

$(F + S) + (F - S) = 50 + 20$

$2F = 70$

$F = \frac{70}{2} = 35$

Substitute the value of $F$ into equation (1):

$35 + S = 50$

$S = 50 - 35 = 15$

The father's current age is 35 years and the son's current age is 15 years.

Determining the Age Ratio

The question asks for the ratio of the son's age to the father's age.

Ratio $= S : F = 15 : 35$

Simplify the ratio by dividing both parts by their greatest common divisor, which is 5:

Ratio $= \frac{15 \div 5}{35 \div 5} = \frac{3}{7}$

The ratio is 3:7.