M31-47

A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. For example, 1/3 = 0.333333....., 2/9 = 0.2222.......

If it's not the case then the decimal is terminating: 0.7, 0.139, 0.999, 0.55, ...

THEORY:

A reduced fraction \(\frac{a}{b}\) (meaning that the fraction is already in its simplest form, so reduced to its lowest term) can be expressed as a terminating decimal if and only if the denominator \(b\) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as the denominator \(250\) equals \(2*5^3\). The fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and the denominator \(10=2*5\).

Note that if the denominator already consists of only 2s and/or 5s, then it doesn't matter whether the fraction is reduced or not.

For example, \(\frac{x}{2^n5^m}\), (where \(x\), \(n\), and \(m\) are integers) will always be a terminating decimal.

(We need to reduce the fraction in case the denominator has a prime other than 2 or 5, to see whether it can be reduced. For example, the fraction \(\frac{6}{15}\) has 3 as a prime in the denominator, and we need to know if it can be reduced.)

Check below for more:
Math: Number Theory
How to Identify Terminating Decimals on the GMAT
Terminating Decimals in Data Sufficiency on the GMAT
Terminating and Recurring Decimals

Even more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.

How do we know that a/b is not an irrational number? Why do we assume it to be rational i.e. if it's not repeating, then it must be terminating? Thanks in advance!!

a/b, where a and b are digits (0, 1, 2, 3, 4, 5, 6, 7, 8 9) cannot be an irrational number. An Irrational Number is a real number that cannot be written as a simple fraction of two integers.

Thanks, Bunuel..

I was thinking of pi (=3.14159...) being an irrational number. But, you are right the 22/7 form of pi is rational (& repeating)!! Thanks, again!!

pi is an irrational number. 22/7 is just an approximation of pi.

Bunuel Yeah, kind of got that idea!! Thanks for your prompt responses; they are much appreciated!!

P.S. - Majority of the questions on GMATClub are highly thought provoking and ensure that all such concepts get fine tuned.. Very thankful for them!!

Hi Bunuel
If recurring or repeating decimals are ones that follow patterns (like 0.333333) then shouldn't non recurring or non repeating decimals be like the values of pi (3.14...)so why have we considered non recurring here as only terminating decimals like 0.4 and 0.5 only?

Rational numbers, numbers that can be expressed as the ratio of two integers, comprise of terminating decimals (-0.11; 17, 12.78, ...) and repeating decimals (0.33..... = 1/3, -0.7 = -7/10, 17/19 = 0.894736842105263157... (period 18)). So, if a number can be expressed as a ratio of two integers, then we have a real number and this real number will be either repeating or terminating decimal (an integer can also be called a terminating decimal).

Irrational numbers, numbers that cannot be expressed as the ratio of two integers, on the other hand, are both non-terminating and non-repeating decimals. For example, \(\sqrt{2}\), \(\pi\), ...

In this problem, both and b are integers, so a/b cannot be an irrational number, like \(\sqrt{2}\) or \(\pi\).

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Bunuel

There is a typo at the end in the Explanation.

It says that a=1 instead of 3

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Fixed the typo. Thank you for spotting it.