Question 55 of 58intermediate🔧 ApplyShort Answer2 marks

A rational number in its lowest form has denominator 23×52^3 \times 5. How many decimal places will its decimal expansion have? Explain your answer.

Correct Answer

3

Exercise: END-OF-CHAPTER EXERCISES | Q: 12 | (Page 25)
For More Understanding

Explanation

The context explains that a rational number has a terminating decimal expansion if the prime factors of its denominator are only 2, only 5, or both. To find the number of decimal places, we convert the denominator into a power of 10 by multiplying both numerator and denominator by a suitable number. For a denominator of 232^{3} × 5, the highest power of the prime factors is 3. By multiplying by 525^{2}, the denominator becomes 232^{3} × 535^{3} = 10310^{3}, resulting in 3 decimal places.

Solution Steps

  1. Step 1: Identify the prime factorisation of the denominator, which is given as 23×52^3 \times 5.

  2. Step 2: Note that the decimal expansion will terminate because the prime factors are only 2 and 5.

  3. Step 3: To make the denominator a power of 10, match the highest power of 2 and 5. The highest power is 3.

  4. Step 4: Multiply the numerator and denominator by 525^2 to get 535^3 in the denominator, making it 23×53=1032^3 \times 5^3 = 10^3.

  5. Step 5: Since the denominator becomes 10310^3, the decimal expansion will have 3 decimal places.

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