If a/b is in lowest form and b = 2³ × 5 × 11, what type of decimal expansion will it have?
Answer and explanation
Correct answer: Non-terminating recurring
The governing theorem states that a rational number a/b in lowest form has a terminating decimal expansion if and only if the prime factors of b are only 2 and/or 5. Here the denominator is 2³ × 5 × 11, and the factor 11 remains because the fraction is already in lowest form. Therefore the decimal cannot terminate. Since a/b is rational, its decimal expansion must eventually repeat, so it is non-terminating recurring. Option A is correct. Option B would be possible only if no prime factor other than 2 or 5 remained. Option C describes irrational numbers, not a rational fraction, and option D is not guaranteed merely from the denominator’s factorization.
Frequently asked questions
What is the correct answer to this question?
Non-terminating recurring
Why is this the correct answer?
The governing theorem states that a rational number a/b in lowest form has a terminating decimal expansion if and only if the prime factors of b are only 2 and/or 5. Here the denominator is 2³ × 5 × 11, and the factor 11 remains because the fraction is already in lowest form. Therefore the decimal cannot terminate. Since a/b is rational, its decimal expansion must eventually repeat, so it is non-terminating recurring. Option A is correct. Option B would be possible only if no prime factor other than 2 or 5 remained. Option C describes irrational numbers, not a rational fraction, and option D is not guaranteed merely from the denominator’s factorization.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
