Solving the Age Word Problem

Let the daughter's current age be d years and the man's current age be m years.

Setting Up Age Equations

From the first statement, "A man is 4 times as old as his daughter", we get the first equation:

$m = 4d \quad (1)$

In 20 years, the daughter will be d + 20 years old, and the man will be m + 20 years old. The second statement, "In 20 years, he will be twice as old as she will be then", gives us the second equation:

$m + 20 = 2(d + 20) \quad (2)$

Solving for Daughter's Current Age

Substitute the value of m from equation (1) into equation (2):

$4d + 20 = 2(d + 20)$

Distribute the 2 on the right side:

$4d + 20 = 2d + 40$

Subtract 2d from both sides:

$4d - 2d + 20 = 40$

$2d + 20 = 40$

Subtract 20 from both sides:

$2d = 40 - 20$

$2d = 20$

Divide by 2:

$d = \frac{20}{2}$

$d = 10$

The daughter's current age is 10 years.

Verification

If the daughter is 10 years old now, the man is $4 \times 10 = 40$ years old.

In 20 years:

• Daughter's age will be $10 + 20 = 30$ years.
• Man's age will be $40 + 20 = 60$ years.

Check if the man is twice as old as the daughter then: $60 = 2 \times 30$. This is true.

Therefore, the daughter's current age is 10 years.