Without performing division, determine whether the decimal expansion of 18/125 is terminating Class 9

Without performing division, determine whether the decimal expansion of 18/125 is terminating Class 9

Question 1.

Without performing division, determine whether the decimal expansion of $\frac{18}{125}$ is terminating or non-terminating. If it terminates, state the number of decimal places. Class 9

Solution:

Factors of denominator 125 = 5 × 5 × 5 = 5³ Since the denominator has only the prime factor 5, the decimal expansion is a terminating decimal.

To find the number of decimal places, making the denominator a power of 10.

Multiplying the numerator and denominator by 2³,

$\frac{18}{125}=\frac{18}{5^3}=\frac{18 \times 2^3}{5^3 \times 2^3}=\frac{18 \times 8}{10^3}=\frac{144}{1000}=0.144$

Since the denominator becomes 10³, there are 3 decimal places.

Question 2.

A rational number in its lowest form has denominator 2³ × 5. How many decimal places will its decimal expansion have? Explain your answer. Class 9

Solution:

Given denominator: 2³ × 5 = 8 × 5 = 40

To find the number of decimal places, we convert the denominator into the form 10ⁿ.

We know that 10 = 2 × 5

So, make the powers of 2 and 5 equal.

Here,

2³ × 5 = 2³ × 5¹

There are 3 factors of 2 and only 1 factor of 5. Multiply by 5² to make the powers equal.

∴ 2³ × 5 × 5² = 2³ × 5³ = (2 × 5)³ = 10³

Thus, the denominator becomes 10³

∴ The decimal expansion will have 3 decimal places.