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 , 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 , , ,  and  are integers and  and , is  a terminating decimal?

Statement 1: 

Statement 2: 

Solution:

Given: , , ,  and  are integers

Question: Is  a terminating decimal?

or is ?

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 cancelled by the power of 3 in the numerator. If  is greater than (or equal to) , after reducing the fraction to lowest terms, it will have no 3 in the denominator which will make it a terminating decimal. If  is less than , 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 ?

Statement 1: 

This statement doesn’t tell us anything about the relation between  and . Hence this statement alone is not sufficient.

Statement 2: 

This statement tells us that  is greater than . 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  only. Hence it will be a terminating decimal. This statement alone is sufficient.

Answer (B).

Now onto another DS question.

Question 2: If , is it possible to write  as a terminating decimal?

Statement 1:  is an integer.

Statement 2:  is an integer.

Solution:

Given: 

Question: Is  a terminating decimal?

Again,  will be a terminating decimal if it is of the form 

Statement 1:  is an integer.

 (an integer)

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

Statement 2:  is an integer.

 (an integer)

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

Taking both together,

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

Answer (C).