Question

The present age of Catherine’s mother is twice that of Catherine. After 4 1/4 years, the difference in their ages will be 16 3/4 years. Find their present ages. (Estimate the age to the nearest whole number and verify the reasonableness).
Catherine’s age is 15 years and her mother’s age is 30 years.
Catherine’s age is 16 years and her mother’s age is 33 years.
Catherine’s age is 17 years and her mother’s age is 36 years.
Catherine’s age is 17 years and her mother’s age is 34 years.

Answers

Answered by GPT-4o mini
Let \( C \) be Catherine's present age and \( M \) be her mother's present age.

From the problem statement, we have two pieces of information:

1. The present age of Catherine’s mother is twice that of Catherine:
\[
M = 2C \tag{1}
\]

2. After \( 4 \frac{1}{4} \) years, the difference in their ages will be \( 16 \frac{3}{4} \) years. The age difference between them always remains the same, so:
\[
M - C = 16 \frac{3}{4} \tag{2}
\]
Converting \( 16 \frac{3}{4} \) to an improper fraction gives:
\[
16 \frac{3}{4} = \frac{67}{4} \tag{3}
\]

Now we can substitute equation (1) into equation (2):

From equation (1), we can express \( M \):
\[
M = 2C
\]

Substituting \( M \) into equation (2):
\[
2C - C = \frac{67}{4}
\]

This simplifies to:
\[
C = \frac{67}{4}
\]

Calculating \( C \):
\[
C = 16.75
\]

Now, we find \( M \) using equation (1):
\[
M = 2C = 2 \times \frac{67}{4} = \frac{134}{4} = 33.5
\]

Now we have:
- Catherine's age \( C = 16.75 \) years
- Mother's age \( M = 33.5 \) years

For the options provided, we will round them to the nearest whole number:
- Catherine’s age is 17 years and her mother’s age is 34 years.

Now, let's verify if these ages fit both conditions:

1. Present age of mother is twice Catherine's:
\[
34 = 2 \times 17
\]

2. Checking the difference in age:
- Present age difference: \( 34 - 17 = 17 \)
- After \( 4 \frac{1}{4} \) years (or 4.25 years) their ages will be:
- Catherine: \( 17 + 4.25 = 21.25 \)
- Mother: \( 34 + 4.25 = 38.25 \)
- The difference then: \( 38.25 - 21.25 = 17 \), which should set to:
- \( 16 \frac{3}{4} = \frac{67}{4} = 16.75 \) years (corrected difference is constant at 17).

Thus, the closest ages that fit all conditions are:
- Catherine's age is 17 years and her mother's age is 34 years.

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