Which of the following will yield a recurring decimal ?

Options

A. A) `(24)/(60)`
B. B) `(24)/(30)`
C. C) `(24)/(90)`
D. D) `(24)/(120)`

Answer

To determine which of the given fractions yields a recurring decimal, we need to analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Recurring Decimals**: A fraction will yield a recurring decimal if, after simplification, the denominator has prime factors other than 2 and 5. If the denominator only has the prime factors of 2 and 5, it will yield a terminating decimal.

2. **Option A: 24/60**
- Factor the numerator and denominator:
- \( 24 = 12 \times 2 \)
- \( 60 = 12 \times 5 \)
- Simplifying gives:
- \( \frac{24}{60} = \frac{12 \times 2}{12 \times 5} = \frac{2}{5} \)
- The denominator (5) has only the prime factor 5.
- **Result**: This fraction yields a terminating decimal (0.4).

3. **Option B: 24/30**
- Factor the numerator and denominator:
- \( 24 = 6 \times 4 \)
- \( 30 = 6 \times 5 \)
- Simplifying gives:
- \( \frac{24}{30} = \frac{6 \times 4}{6 \times 5} = \frac{4}{5} \)
- The denominator (5) has only the prime factor 5.
- **Result**: This fraction yields a terminating decimal (0.8).

4. **Option C: 24/120**
- Factor the numerator and denominator:
- \( 24 = 12 \times 2 \)
- \( 120 = 12 \times 10 \)
- Simplifying gives:
- \( \frac{24}{120} = \frac{12 \times 2}{12 \times 10} = \frac{2}{10} = \frac{1}{5} \)
- The denominator (5) has only the prime factor 5.
- **Result**: This fraction yields a terminating decimal (0.2).

5. **Option D: 24/90**
- Factor the numerator and denominator:
- \( 24 = 6 \times 4 \)
- \( 90 = 15 \times 6 \)
- Simplifying gives:
- \( \frac{24}{90} = \frac{6 \times 4}{15 \times 6} = \frac{4}{15} \)
- The denominator (15) has prime factors 3 and 5.
- Since the denominator has a prime factor other than 2 and 5, this fraction yields a recurring decimal.
- **Result**: This fraction yields a recurring decimal (0.26666...).

### Conclusion:
The fraction that yields a recurring decimal is **Option D: 24/90**.