After simplifying (84/210), what type of decimal expansion will it have?
Answer and explanation
Correct answer: Terminating
The governing theorem states that a rational number in lowest terms has a terminating decimal expansion exactly when the prime factors of its denominator are only 2 and/or 5. First simplify the given fraction: 84/210 can be divided by 42, giving 2/5. The denominator 5 is a permitted prime factor, so the decimal terminates; in fact, 2/5 = 0.4. Therefore option B, terminating, is correct. The original denominator 210 contains other factors, but those factors cancel during simplification and must not be used for the final classification. A non-terminating recurring expansion would require a remaining denominator factor other than 2 or 5, while a non-recurring decimal is irrational, and 2/5 is not an integer.
Frequently asked questions
What is the correct answer to this question?
Terminating
Why is this the correct answer?
The governing theorem states that a rational number in lowest terms has a terminating decimal expansion exactly when the prime factors of its denominator are only 2 and/or 5. First simplify the given fraction: 84/210 can be divided by 42, giving 2/5. The denominator 5 is a permitted prime factor, so the decimal terminates; in fact, 2/5 = 0.4. Therefore option B, terminating, is correct. The original denominator 210 contains other factors, but those factors cancel during simplification and must not be used for the final classification. A non-terminating recurring expansion would require a remaining denominator factor other than 2 or 5, while a non-recurring decimal is irrational, and 2/5 is not an integer.
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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