Question

Without actually performing the long division , state whether the following rational no. will have a terminating decimal expansion or non-terminating repeating decimal expansion.
(a)6/15​

Answer 1

SOLUTION:

$\frac{6}{15} = \frac{2 \times \not{3}}{\not{3} \times 5} = \frac{2}{5}$

Here, prime factors of 5 = 5¹ × 2⁰ which is in form $2^{n}\times 5^{m}$

So, it will have a terminating decimal expansion.

.

$\underline{\sf \red{More}\ \blue{Information}}$

Here is a theorem on the decimal expansion of a rational number.

Let 'x' be a rational number whose decimal expansion terminates.

Then, 'x' can be expressed in the form (p/q), where 'p' and 'q' are coprime and the prime factorization of 'q' is in the form $2^{n}\times 5^{m}$, where n and m are non-integers.

Here is the decimal expansion of  $\frac{6}{15}$

$\frac{6}{15} = \frac{2 \times \bf{ 2^1}}{5^{1} \times \bf 2^1} = \frac{4}{10} = 0.4$

It terminates after one decimal place.

Answer 2

To check it whether 6/15 is a terminating decimal place or not.

We will so prime factorization,

6 = 2*3

and 15 = 3*5

3 will be cancelled

the fraction becomes simple

2/5 here 5 is the denominator which is in the form 2^n * 5^m, where 2^0 is there.

SO it is a terminating decimal place