Which of the following vulgar fractions, when written as a decimal, will not terminate? Options are (a) `(80)/(150)` (b) `(81)/(150)` (c) `(15)/(48)` (d) `(21)/(600)`.

Options

A. `(80)/(150)`
B. `(81)/(150)`
C. `(15)/(48)`
D. `(21)/(600)`

Answer

To determine which of the given vulgar fractions will not terminate when expressed as a decimal, we can follow these steps:

### Step 1: Simplify Each Fraction
We need to simplify each fraction to its lowest terms.

1. **For (80)/(150)**:
- Both numerator and denominator can be divided by 10.
- \( \frac{80}{150} = \frac{8}{15} \)

2. **For (81)/(150)**:
- Both numerator and denominator can be divided by 3.
- \( \frac{81}{150} = \frac{27}{50} \)

3. **For (15)/(48)**:
- Both numerator and denominator can be divided by 3.
- \( \frac{15}{48} = \frac{5}{16} \)

4. **For (21)/(600)**:
- Both numerator and denominator can be divided by 3.
- \( \frac{21}{600} = \frac{7}{200} \)

### Step 2: Factor the Denominators
Next, we need to factor the denominators of the simplified fractions to check for terminating decimals.

1. **For (8)/(15)**:
- Denominator: \( 15 = 3 \times 5 \)

2. **For (27)/(50)**:
- Denominator: \( 50 = 2 \times 5^2 \)

3. **For (5)/(16)**:
- Denominator: \( 16 = 2^4 \)

4. **For (7)/(200)**:
- Denominator: \( 200 = 2^3 \times 5^2 \)

### Step 3: Check for Terminating Conditions
A fraction will terminate if the prime factorization of the denominator (after simplification) contains only the primes 2 and/or 5.

1. **For (8)/(15)**:
- Factors: \( 3 \) is present, so it will **not terminate**.

2. **For (27)/(50)**:
- Factors: Only \( 2 \) and \( 5 \) are present, so it will **terminate**.

3. **For (5)/(16)**:
- Factors: Only \( 2 \) is present, so it will **terminate**.

4. **For (7)/(200)**:
- Factors: Only \( 2 \) and \( 5 \) are present, so it will **terminate**.

### Conclusion
The only fraction that does not meet the terminating condition is \( \frac{8}{15} \) (which comes from the original fraction \( \frac{80}{150} \)).

Thus, the answer is:
**(a) \( \frac{80}{150} \)**

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