Question

write whether the rational number 64/455 will have a terminating decimal expansion or a non-terminating repeating decimal expansion

Answer 1

64/455     
      Factorize the denominator we get
                        455     =5 x 7 x 13 
       There are 7 and 13 also in denominator so denominator is not in form of 2n×5m.Hence  64/455 is not terminating.

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Answer 2

The rational number 64/455 will have a non-terminating repeating decimal expansion

Given :

The rational number 64/455

To find :

State whether the rational number 64/455 will have a terminating decimal expansion or a non-terminating repeating decimal expansion

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 3 :

Write down the given rational number

The given rational number is 64/455

Step 2 of 3 :

Prime factorise the denominator

For the rational number 64/455

Numerator = 64

Denominator = 455

On prime factorisation of the denominator we get

455 = 5 × 7 × 13

Step 3 of 3 :

Check the number has terminating decimal expansion or a non-terminating repeating decimal expansion

The prime factorisation of the denominator has the prime factors 5 , 7 , 13

Since the prime factorisation of the denominator contains prime factors other than 2 and 5

Hence the rational number 64/455 will have a non-terminating repeating decimal expansion

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