Question 19 of 58intermediate💡 UnderstandShort Answer2 marks

Can you tell for which rational numbers the decimal will be terminating?

Correct Answer

A rational number in its lowest form, expressed as p/q (where q ≠\neq 0), will have a terminating decimal expansion if the prime factors of its denominator q are only 2, only 5, or both 2 and 5. This is because when the denominator has no prime factors other than 2 and 5, we can make the denominator a power of 10 by multiplying the numerator and denominator by a suitable number. For instance, in 3/20, the prime factorisation of 20 is 2² × 5. Multiplying by 5 makes the denominator 100, yielding the terminating decimal 0.15. If the denominator has other prime factors, the decimal repeats.

Exercise: 3.6.1 Rational Decimals: Terminating and Repeating | Q: 1 | (Chapter: 17)
For More Understanding

Explanation

The textbook context explicitly states the condition for a rational number p/q (in lowest terms) to have a terminating decimal: the prime factors of q must be only 2, only 5, or both. It explains this by showing that such denominators can be converted into a power of 10, which naturally yields a terminating decimal.

Solution Steps

  1. Step 1: Identify the rational number in its lowest form as p/q.

  2. Step 2: Check the prime factorisation of the denominator q.

  3. Step 3: If the prime factors are only 2, only 5, or both, the decimal expansion will be terminating.

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