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Number Properties - Terminating Decimals Application

A recap on Terminating Decimals: 

–  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)

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