Terminating Decimals in Data Sufficiency on the GMAT

Quarter Wit, Quarter Wisdom

Last week, we discussed the basics of terminating decimals. Let me review the important points here:

-  To figure out whether the fraction is terminating, bring it down to its lowest form.

-  Focus on the denominator – if it is of the form 2^a * 5^b, the fraction is terminating, else it is not.

 

Keeping this in mind, let’s look at a couple of DS questions on terminating decimals.

Question 1: If a, b, c, d and e are integers and m = 2^a*3^b and n = 2^c*3^d*5^e, is m/n a terminating decimal?

Statement 1: a > c
Statement 2: b > d

Solution:

Given: a, b, c, d and e are integers

Question: Is m/n a terminating decimal?

Or Is (2^a*3^b)/(2^c*3^d*5^e)?

We know that powers of 2 and 5 in the denominator are acceptable for the decimal to be terminating. If there is a power of 3 in the denominator after reducing the fraction, then the decimal in non- terminating. So our question is basically whether the power of 3 in the denominator gets canceled by the power of 3 in the numerator. If b is greater than (or equal to) d, after reducing the fraction to lowest terms, it will have no 3 in the denominator which will make it a terminating decimal. If b is less than d, even after reducing the fraction to its lowest terms, it will have some powers of 3 in the dominator which will make it a non-terminating decimal.

Question: Is b >= d?

Statement 1: a > c

This statement doesn’t tell us anything about the relation between b and d. Hence this statement alone is not sufficient.
Statement 2: b > d

This statement tells us that b is greater than d. This means that after we reduce the fraction to its lowest form, there will be no 3 in the denominator and it will be of the form 2^c * 5^e only. Hence it will be a terminating decimal. This statement alone is sufficient.
Answer (B)

Now onto another DS question.

Question 2: If 0 < x < 1, is it possible to write x as a terminating decimal?

Statement 1: 24x is an integer.

Statement 2: 28x is an integer.

Solution:

Given: 0 < x < 1

Question: Is x a  terminating decimal?

Again, x will be a terminating decimal if it is of the form m/(2^a * 5^b)

Statement 1: 24x is an integer.

24x = 2^3 * 3 * x = m (an integer)

x = m/(2^3 * 3)

Is x a terminating decimal? We don’t know. If m has 3 as a factor, x will be a terminating decimal. Else it will not be. This statement alone is not sufficient.

Statement 2: 28x is an integer.

28x = 2^2 * 7 * x = n (an integer)

x = n/(2^2 * 7)

Is x a terminating decimal? We don’t know. If n has 7 as a factor, x will be a terminating decimal. Else it will not be. This statement alone is not sufficient.

Taking both together,

m/24 = n/28

m/n = 6/7

Since m and n are integers, m will be a multiple of 6 (and thereby of 3 too) and n will be a multiple of 7. So x will be a terminating decimal.

Answer (C)

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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