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which of the following rational numbers ...

which of the following rational numbers is esxpressible as a terminating decimal ?

A

`124/165`

B

`131/30`

C

`2075/625`

D

`1625/462`

Text Solution

AI Generated Solution

The correct Answer is:
To determine which of the given rational numbers is expressible as a terminating decimal, we need to analyze the factors of the denominators of each rational number. A rational number can be expressed as a terminating decimal if the denominator (when the fraction is in its simplest form) has only the prime factors 2 and/or 5. ### Step-by-Step Solution: 1. **Analyze the First Option: \( \frac{124}{165} \)** - Factor the numerator: \( 124 = 2^2 \times 31 \) - Factor the denominator: \( 165 = 3 \times 5 \times 11 \) - The denominator has factors of 3 and 11, which are not 2 or 5. - **Conclusion**: \( \frac{124}{165} \) is **not expressible** as a terminating decimal. 2. **Analyze the Second Option: \( \frac{131}{30} \)** - Factor the numerator: \( 131 \) is a prime number. - Factor the denominator: \( 30 = 2 \times 3 \times 5 \) - The denominator has a factor of 3, which is not 2 or 5. - **Conclusion**: \( \frac{131}{30} \) is **not expressible** as a terminating decimal. 3. **Analyze the Third Option: \( \frac{2075}{625} \)** - Factor the numerator: \( 2075 = 5^2 \times 83 \) (since \( 2075 = 25 \times 83 \)) - Factor the denominator: \( 625 = 5^4 \) - The denominator has only the factor of 5. - **Conclusion**: \( \frac{2075}{625} \) is **expressible** as a terminating decimal. 4. **Analyze the Fourth Option: \( \frac{1625}{462} \)** - Factor the numerator: \( 1625 = 5^2 \times 13 \) - Factor the denominator: \( 462 = 2 \times 3 \times 7 \times 11 \) - The denominator has factors of 3, 7, and 11, which are not 2 or 5. - **Conclusion**: \( \frac{1625}{462} \) is **not expressible** as a terminating decimal. ### Final Answer: The only rational number expressible as a terminating decimal is \( \frac{2075}{625} \).
To determine which of the given rational numbers is expressible as a terminating decimal, we need to analyze the factors of the denominators of each rational number. A rational number can be expressed as a terminating decimal if the denominator (when the fraction is in its simplest form) has only the prime factors 2 and/or 5. ### Step-by-Step Solution: 1. **Analyze the First Option: \( \frac{124}{165} \)** - Factor the numerator: \( 124 = 2^2 \times 31 \) - Factor the denominator: \( 165 = 3 \times 5 \times 11 \) - The denominator has factors of 3 and 11, which are not 2 or 5. ...
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