Question

the number of decimal places after which the decimal expansion of the rational number 9/2×2×2×2×5 will terminate is.... a.2 b.3 c. 4 d.5

Answer 1

after 4places of decimal the given fraction terminates

Answer 2

The decimal expansion of the rational number 9/2×2×2×2×5 will terminate after 4 decimal places

Given :

The rational number 9/2×2×2×2×5

To find :

The number of decimal places after which the decimal expansion of the rational number 9/2×2×2×2×5 will terminate is

a. 2

b. 3

c. 4

d. 5

Concept :

$\displaystyle\sf{Fraction = \frac{Numerator}{Denominator} }$

A fraction is said to be terminating if prime factorisation of the denominator contains only prime factors 2 and 5

If the denominator is of the form

$\sf{Denominator = {2}^{m} \times {5}^{n} }$

Then the fraction terminates after N decimal places

Where N = max { m , n }

Solution :

Step 1 of 2 :

Write down the given rational number

The given rational number is

$\displaystyle \sf{ \frac{9}{2 \times 2 \times 2 \times 2 \times 5} }$

$\displaystyle \sf = \frac{9}{ {2}^{4} \times {5}^{1 } }$

Step 2 of 2 :

Find the number of decimal places after which the decimal form will terminate

Numerator = 9

Denominator = 2⁴ × 5¹

Since the prime factorisation of the denominator contains only prime factors as 2 and 5

So the given rational number is terminating

The exponent of 2 = 4

The exponent of 5 = 1

Now max{4 , 1} = 4

∴ The decimal expansion of the rational number 9/2×2×2×2×5 will terminate after 4 decimal places

Hence the correct option is c. 4

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