Question 37 of 58intermediate💡 UnderstandShort Answer2 marks

The decimal expansion of pq\frac{p}{q} will be terminating precisely when the prime factors of qq are only 2, only 5 or both 2 and 5. Can you explain why?

Correct Answer

The decimal expansion of pq\frac{p}{q} will be terminating when the prime factors of qq are only 2, only 5, or both because we can easily convert such fractions into decimals.

In these cases, we can make the denominator a power of 10 by multiplying both the numerator and the denominator by a suitable number. Since powers of 10 always result in a terminating decimal, the fraction itself will terminate.

For example, consider 320\frac{3}{20}. The prime factorisation of 20 is 22×52^2 \times 5. By multiplying the numerator and denominator by 5, the denominator becomes 22×52=1002^2 \times 5^2 = 100, giving the terminating decimal 0.15.

Exercise: Think and Reflect | Q: 1 | (Page 18)
For More Understanding

Explanation

The textbook context explains that if a denominator's prime factors are limited to 2 and 5, it is possible to multiply the fraction by a suitable number to turn the denominator into a power of 10 (like 10, 100, etc.). Since any fraction with a denominator that is a power of 10 naturally results in a terminating decimal, this property allows us to predict termination. The example of 3/20 demonstrates this by multiplying by 5 to get 15/100, which equals 0.15.

Solution Steps

  1. Step 1: Identify that prime factors of 2 and 5 allow the denominator to be multiplied into a power of 10.

  2. Step 2: Multiply both numerator and denominator by the suitable number to get a power of 10 in the denominator.

  3. Step 3: Convert the resulting fraction into a terminating decimal.

PreviousNext