Question 38 of 58intermediate🔧 ApplyNumerical4 marks

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 720,415\frac{7}{20}, \frac{4}{15} and 13250\frac{13}{250}. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

Correct Answer

(13/250) Terminating, 0.052

(4/15) Repeating, 0.2666...

(7/20) Terminating, 0.35

Exercise: Exercise Set 3.5 | Q: 1 | (Page 21)
For More Understanding

Explanation

The context explains that we can predict whether a rational number has a terminating or repeating decimal by finding the prime factors of the denominator. If the denominator's prime factors consist of only 2s and 5s, the decimal terminates; otherwise, it repeats. For 7/20, the denominator 20 = 222^{2} × 5, so it terminates. For 4/15, the denominator 15 = 3×53 \times 5, so it repeats. For 13/250, the denominator 250 = 2 × 535^{3}, so it terminates. Long division confirms these predictions.

Solution Steps

  1. Step 1: Predict for 7/20. The prime factorisation of 20 is 22×52^2 \times 5. Since the prime factors are only 2 and 5, the decimal expansion is terminating.

  2. Step 2: Predict for 4/15. The prime factorisation of 15 is 3×53 \times 5. Since there is a prime factor other than 2 and 5 (which is 3), the decimal expansion is repeating.

  3. Step 3: Predict for 13/250. The prime factorisation of 250 is 2×532 \times 5^3. Since the prime factors are only 2 and 5, the decimal expansion is terminating.

  4. Step 4: Check 7/20 by long division. 7÷20=0.357 \div 20 = 0.35. The remainder eventually becomes 0, confirming it is a terminating decimal.

  5. Step 5: Check 4/15 by long division. 4÷15=0.2666...=0.26‾4 \div 15 = 0.2666... = 0.2\overline{6}. The remainder repeats, confirming it is a repeating decimal.

  6. Step 6: Check 13/250 by long division. 13÷250=0.05213 \div 250 = 0.052. The remainder eventually becomes 0, confirming it is a terminating decimal.

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