Question

Monica, who is 18 years old, is one-third the age of her father. The age at which she will be half the age of her father is ________

Hint:

For age-related problems, always define variables for the current ages first. Then, express future or past ages in terms of the current ages and the time difference. Be careful to answer what is specifically asked (e.g., Monica's age, not the number of years).

Answer 1

Correct Answer: 36

Step 1: Understanding the Concept:
This is a word problem involving ages that can be solved by setting up and solving linear equations based on the given information.
Step 2: Key Formula or Approach:
1. Translate the word statements into mathematical equations.
2. Solve the equations to find the required value.
Step 3: Detailed Explanation:
Let Monica's current age be M and her father's current age be F.
Given: M = 18 years.
Also given: Monica's age is one-third her father's age. \[ M = \frac{1}{3} F \] Substituting M = 18, we get: \[ 18 = \frac{1}{3} F \implies F = 18 \times 3 = 54 \] So, the father's current age is 54 years.
Let 'x' be the number of years from now when Monica will be half her father's age.
In x years:
Monica's age will be \( M' = M + x = 18 + x \).
Father's age will be \( F' = F + x = 54 + x \).
According to the condition, at that time: \[ M' = \frac{1}{2} F' \] \[ 18 + x = \frac{1}{2} (54 + x) \] To solve for x, we multiply both sides by 2: \[ 2(18 + x) = 54 + x \] \[ 36 + 2x = 54 + x \] \[ 2x - x = 54 - 36 \] \[ x = 18 \] The question asks for Monica's age at that time, not the number of years from now.
Monica's age will be \( 18 + x = 18 + 18 = 36 \) years.
To verify, the father's age at that time would be \( 54 + x = 54 + 18 = 72 \). Is 36 half of 72? Yes, \( 36 = \frac{1}{2} \times 72 \). The answer is correct.
Step 4: Final Answer:
Monica will be 36 years old when she is half the age of her father.