Understanding the Age Relationship

This problem involves finding a future time based on the age relationship between a father and his son at different points in time. We are given information about their ages 8 years ago and need to determine how many years from now the father will be twice the son's age.

Setting Up Algebraic Equations

Let's use variables to represent their current ages:

• Let $F$ be the current age of the father.
• Let $S$ be the current age of the son.

Information from 8 years ago:

• Father's age 8 years ago was $F - 8$.
• Son's age 8 years ago was $S - 8$.
• The condition given is: "the age of a father was 21 years more than twice his son's age".
• This translates to the equation: $F - 8 = 2(S - 8) + 21$.

Simplifying the first equation:

1. $F - 8 = 2S - 16 + 21$
2. $F - 8 = 2S + 5$
3. $F = 2S + 5 + 8$
4. $F = 2S + 13$ (Equation 1)

Information about the future:

• Let $x$ be the number of years from now when the father will be twice his son's age.
• Father's age after $x$ years will be $F + x$.
• Son's age after $x$ years will be $S + x$.
• The condition is: "he will be twice his son's age".
• This translates to the equation: $F + x = 2(S + x)$.

Simplifying the second equation:

1. $F + x = 2S + 2x$
2. $F = 2S + 2x - x$
3. $F = 2S + x$ (Equation 2)

Calculating the Number of Years

Now we have two equations for $F$:

• Equation 1: $F = 2S + 13$
• Equation 2: $F = 2S + x$

Since both equations represent the father's current age ($F$), we can set them equal to each other:

$2S + 13 = 2S + x$

To solve for $x$, we can subtract $2S$ from both sides of the equation:

$13 = x$

Therefore, after 13 years, the father will be twice his son's age.