Question

The rational number which is equal to the number \[2.\overline {357} \] with recurring decimal is
(a)\[\dfrac{{2355}}{{1001}}\]
(b) \[\dfrac{{2370}}{{999}}\]
(c) \[\dfrac{{2355}}{{999}}\]
(d) \[\dfrac{{2359}}{{991}}\]

Answer 1

Hint: In this question we have to find out the fractional value of the given rational number from the recurring decimal.
We need to follow the steps to convert recurring decimal to fraction.
Steps:
(i)Let, x = recurring decimal.
(ii)Let, n = the number of recurring digits.
(iii)Multiply the recurring decimal by \[{10^n}\].
(iv)Subtract (i) from (iii) to eliminate the recurring part.
(v)Solve for x, expressing your answer as a fraction in its simplest form.

Complete step-by-step solution:
The given recurring decimal is \[2.\overline {357} \].
We need to convert recurring decimal to the rational number that is to fractional value.
Following the above steps of conversion, we get,
Let, \[x = 2.357357...\]
Here, n= the number of recurring digits is \[3\].
\[{10^3}x = 1000 \times 2.357357... = 2357.357\]
Now applying step (iii) to remove the recurring part we get,
\[1000x - x = 2357.357... - 2.357357...\]
Or,\[999x = 2355\]
Or, \[x = \dfrac{{2355}}{{999}}\]
Hence, the rational number which is equal to the number \[2.\overline {357} \] with recurring decimals is \[\dfrac{{2355}}{{999}}\].
Therefore, (c) is the correct option.

Note: Rational number:
A rational number is the number that can be expressed as the ratio of two integers(p/q form), where the denominator should not be equal to zero.
It can be expressed as the quotient or fraction p/q of two integers, a numerator p and a nonzero denominator q.
Recurring decimal:
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are repeated periodically and the portion with infinitely repetition is not zero. It can be shown that a number is rational if its decimal representation is repeating or terminating and vise-versa.