Question

Decimal expansion of which of the following is non-terminating recurring? Choose the correct alternative answer for the question given below.
(A) 2/5
(B) 3/16
(C) 3/11
(D) 137/25

Answer 1

Answer:

(C) 3/11

Step-by-step explanation:

In the question,

Non-terminating and Recurring number : It is the fraction which if expressed in decimal has not an exact value after the decimal and it goes onto infinite number of terms after the decimal.

And, it is termed as recurring because some of the digits after the decimals keeps repeating.

Therefore, on checking the options we get,

(A) 2/5 = 0.25

(B) 3/16 = 0.1875

(C) 3/11 = 0.27272727.........

(D) 137/25 = 5.48

Here, we can see that the fraction 3/11 is a non-terminating and recurring number as the digits 27 after the decimal keep on repeating.

Hence, the correct option is given by,

(C) 3/11

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

SIMPLE PROCESS TO CHECK A FRACTION FOR TERMINATING / NON TERMINATING

If the denominator of a fraction consists of 2 and 5 as its prime factors only , then the fraction can be written as terminating decimal.

Otherwise the fraction is non terminating decimal.

TO CHOOSE THE CORRECT OPTION

TO choose the Decimal expansion of which fraction is non-terminating recurring

(A) 2/5

(B) 3/16

(C) 3/11

(D) 137/25

CALCULATION

CHECKING FOR OPTION (A)

$\displaystyle \: \sf{For \: the \: fraction \: \: \: \frac{2}{5} \: }$

The denominator = 5

Since 5 is the only number which is prime is present in the denominator

Hence the decimal expansion is terminating

CHECKING FOR OPTION ( B)

$\displaystyle \: \sf{For \: the \: fraction \: \: \: \frac{3}{16} \: }$

The denominator is 16

$\sf{16 = 2 \times 2 \times 2 \times 2 \: \: }$

Since 2 is the only number which is prime , is present in the prime factorisation of the denominator

Hence the decimal expansion is terminating

CHECKING FOR OPTION ( C )

$\displaystyle \: \sf{For \: the \: fraction \: \: \: \frac{3}{11} \: }$

The denominator is 11

Since 11 is the only number which is not prime is present in the denominator

Hence the decimal expansion is non terminating

CHECKING FOR OPTION ( D )

$\displaystyle \: \sf{For \: the \: fraction \: \: \: \frac{137}{25} \: }$

The denominator is 25

$\sf{25 = 5 \times 5 \: \: }$

Since 5 is the only number which is prime , is present in the prime factorisation of the denominator

Hence the decimal expansion is terminating

RESULT

Hence the required fraction which has non-terminating recurring is

$\displaystyle \: \sf{ \frac{3}{11} \: }$

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