Question

After how many places the decimal expansion of 153/450 terminates?​

Answer 1

To Find :- After how many places the decimal expansion of 153/450 terminates ?

Solution :-

(153/450) can be written as ,

$\rightarrow\sf \: \frac{153}{450} \\ \\\rightarrow\sf \: \frac{9 \times 17}{9 \times 50} \\ \\ \rightarrow\sf \: \frac{17}{50} \\ \\ \rightarrow\sf \: \frac{17 \times 2}{50 \times 2} \\ \\ \rightarrow\sf \: \frac{34}{100} \\ \\ \rightarrow\sf \boxed{0.34}$

therefore, we can conclude that, the decimal expansion of given rational number is 0.34 and it will terminate after two decimal places.

Method 2) :-

→ (153/450)

→ (17/50)

now, prime factors of denominators are ,

→ 50 = 2¹ * 5² = 2^m * 5^n

• when m > n , the decimal expansion will terminate after m places .
• when m < n , the decimal expansion will terminate after n places .

since,

→ m < n

→ 1 < 2

therefore, the decimal expansion of given rational number will terminate after two decimal places .

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Answer 2

SOLUTION

TO DETERMINE

After how many places the decimal expansion of $\sf{ \dfrac{153}{450} }$ terminates

CONCEPT TO BE IMPLEMENTED

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

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 }

EVALUATION

Here the given rational number is

$\sf{ \dfrac{153}{450} }$

First we simplify the given number

$\sf{ \dfrac{153}{450} }$

$\sf{ = \dfrac{17 \times 9}{50 \times 9} }$

$\sf{ = \dfrac{17 }{50 } }$

Numerator = 17

Denominator = 50

Now 50 = 2 × 5 × 5 = 2 × 5²

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 = 1

The exponent of 5 = 2

Max { 1 , 2 } = 2

Hence the given rational number terminates after 2 decimal places

FINAL ANSWER

After two decimal places the decimal expansion of $\sf{ \dfrac{153}{450} }$ terminates

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