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Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
- a)23 / 8
- b)125 / 441
- c)35 / 50
- d)77 / 210
Correct answer is option 'A'. Can you explain this answer?
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Without actually performing the long division, state whether the foll...
Denominator is 8, so the rational has a non-terminating decimal expansion
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Without actually performing the long division, state whether the foll...
Terminating and Non-terminating Decimal Expansions
To determine whether a rational number will have a terminating or non-terminating repeating decimal expansion, we need to analyze its denominator.
a) 23/8:
The denominator of 8 can be factored into 2 * 2 * 2. Since the only prime factors of the denominator are 2 and no factor of 5, the decimal expansion of 23/8 will terminate. This means that it will have a finite number of digits after the decimal point.
b) 125/441:
The denominator of 441 can be factored into 3 * 3 * 7 * 7. Since the prime factors of the denominator include 3 and 7 but no factor of 2 or 5, the decimal expansion of 125/441 will be a non-terminating repeating decimal. This means that it will have a pattern of digits that repeats indefinitely.
c) 35/50:
The denominator of 50 can be factored into 2 * 5 * 5. Since the prime factors of the denominator include both 2 and 5, the decimal expansion of 35/50 will be a terminating decimal. However, it is important to simplify the fraction before drawing this conclusion. Simplifying 35/50 gives us 7/10, which is a terminating decimal.
d) 77/210:
The denominator of 210 can be factored into 2 * 3 * 5 * 7. Since the prime factors of the denominator include both 2 and 5, the decimal expansion of 77/210 will be a terminating decimal. However, it is important to simplify the fraction before drawing this conclusion. Simplifying 77/210 gives us 11/30, which is a terminating decimal.
Therefore, the correct answer is option 'A' (23/8) because its denominator only has the prime factor 2 and no factor of 5, indicating that its decimal expansion will terminate.
To determine whether a rational number will have a terminating or non-terminating repeating decimal expansion, we need to analyze its denominator.
a) 23/8:
The denominator of 8 can be factored into 2 * 2 * 2. Since the only prime factors of the denominator are 2 and no factor of 5, the decimal expansion of 23/8 will terminate. This means that it will have a finite number of digits after the decimal point.
b) 125/441:
The denominator of 441 can be factored into 3 * 3 * 7 * 7. Since the prime factors of the denominator include 3 and 7 but no factor of 2 or 5, the decimal expansion of 125/441 will be a non-terminating repeating decimal. This means that it will have a pattern of digits that repeats indefinitely.
c) 35/50:
The denominator of 50 can be factored into 2 * 5 * 5. Since the prime factors of the denominator include both 2 and 5, the decimal expansion of 35/50 will be a terminating decimal. However, it is important to simplify the fraction before drawing this conclusion. Simplifying 35/50 gives us 7/10, which is a terminating decimal.
d) 77/210:
The denominator of 210 can be factored into 2 * 3 * 5 * 7. Since the prime factors of the denominator include both 2 and 5, the decimal expansion of 77/210 will be a terminating decimal. However, it is important to simplify the fraction before drawing this conclusion. Simplifying 77/210 gives us 11/30, which is a terminating decimal.
Therefore, the correct answer is option 'A' (23/8) because its denominator only has the prime factor 2 and no factor of 5, indicating that its decimal expansion will terminate.
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Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.a)23 / 8b)125 / 441c)35 / 50d)77 / 210Correct answer is option 'A'. Can you explain this answer?
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