The ages of four sisters form a series of consecutive composite numbers. The oldest sister's age is a multiple of 25 and less than 100. Additionally, the age of the second youngest sister, eight years ago, was twice a positive integer $x$. What will be the age of the youngest sister after seven years?

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**Solution:**

1. **Understanding composite numbers:** A composite number is a positive integer that has more than two factors.

2. **Given conditions:**
   - The ages of the four sisters are consecutive composite numbers.
   - The oldest sister's age is a multiple of 25 and less than 100. Possible values: 25, 50, 75.
   - The age of the second youngest sister, 8 years ago, was $2x$, where $x$ is a positive integer.

3. **Finding possible ages for the second youngest sister:**
   - Present age of second youngest sister = $2x + 8$.
   - Since $x$ is positive integer, $2x$ can be 2, 4, 8, 16, 32, 64, etc.
   - Corresponding present ages: 10, 12, 16, 24, 40, 72.

4. **Checking consecutive composite sequences ending with multiples of 25:**
   - For oldest sister = 25, consecutive composites before 25 are 21, 22, 24 (but 23 is prime, so not consecutive composite).
   - For oldest sister = 50, consecutive composites before 50 are 46, 48, 49.
   - For oldest sister = 75, consecutive composites before 75 are 70, 72, 74.

5. **Matching second youngest sister's age with possible values:**
   - Sequence (70, 72, 74, 75): second youngest sister is 72.
   - 72 - 8 = 64, which is $2x$ with $x=32$, a positive integer.

6. **Finding youngest sister's age:**
   - Youngest sister's age in this sequence is 70.
   - After 7 years, youngest sister's age = $70 + 7 = 77$.

**Answer:** 77

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