If p/q is in simplest form and q = 2⁹ × 5¹², what is the maximum number of decimal places in the terminating decimal?
Answer and explanation
Correct answer: 12
A fraction in lowest terms has a terminating decimal exactly when the denominator contains no prime factors other than 2 and 5. To express q = 2⁹ × 5¹² as a power of 10 multiplied by a remaining factor, pair nine 2s with nine 5s to form 10⁹. Five 5s remain, so the denominator can be converted to 10¹² by supplying three additional factors of 2 in the numerator’s decimal scaling. In general, the required number of places is the larger exponent, max(9, 12) = 12. Therefore option D is correct. Cancellation in p may reduce the actual number, but 12 is the maximum possible.
Frequently asked questions
What is the correct answer to this question?
12
Why is this the correct answer?
A fraction in lowest terms has a terminating decimal exactly when the denominator contains no prime factors other than 2 and 5. To express q = 2⁹ × 5¹² as a power of 10 multiplied by a remaining factor, pair nine 2s with nine 5s to form 10⁹. Five 5s remain, so the denominator can be converted to 10¹² by supplying three additional factors of 2 in the numerator’s decimal scaling. In general, the required number of places is the larger exponent, max(9, 12) = 12. Therefore option D is correct. Cancellation in p may reduce the actual number, but 12 is the maximum possible.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Decimal representation.
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