Which of the Following Fractions is Closest to 1/2? GMAT Problem Solving

Question: Which of the following fractions is closest to 1/2?

1. 4/7
2. 5/9
3. 6/11
4. 7/13
5. 9/16

‘Which of the following fractions is closest to 1/2?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude for reasoning and mathematics. The GMAT Quantitative Test consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

Solution and Explanation:
Approach Solution 1:
To determine which answer choice is closest to 1/2, we can subtract 1/2 from each answer choice. The difference that is closest to zero will indicate the fraction that is closest to 1/2.

1. 4/7

=>4/7 - 1/2

Multiplying to make them similar
= 8/14 - 7/14
= 1/14

2. 5/9

=>5/9 - 1/2
Multiplying to make them similar
= 10/18 - 9/18

= 1/18

100. 6/11

=>6/11 - 1/2
Multiplying to make them similar
= 12/22 - 11/22
= 1/22

500. 7/13

=>7/13 - 1/2
Multiplying to make them similar
= 14/26 - 13/26
= 1/26

5. 9/16

=>9/16 - 1/2
Multiplying to make them similar
= 9/16 - 8/16
= 1/16
Since 1/26 is closest to zero, 7/13 is closest to 1/2. Hence, D is the correct answer.

Approach Solution 2:
If we look at the options, there are similarities in all choices- they all EXCEPT E are of type: x/(2x-1).
For x/(2x-1) an integer x is always >1/2 and will become lesser in value , that is will keep getting closer to 1/2 with higher VALUE of x.
There are two ways to check this:
1)

for x as 1 or 2 or 3..
x=1, 1/(2-1) = 1
x=2, 2/(4-1) = 2/3
x=3, 3/(6-1) = 3/5
We can see that each successive value is smaller than previous one.
2)
We will simplify the term
x/(2x-1)
=> 1/ (2x-1)/x
=> 1/ (2x/x-1/x)
=> 1/ (2-1/x)
so the denominator should be closer to 2 so 2-1/x should be close to 2.
Therefore x should be MAX to make 1/x less.
Therefore, here 7 is the highest value in the four choices so D is the closest to ½.
Clearly since 7 and 9 are close to each other but the denominator in E is reduced by a larger number,
Hence, D is the answer

Correct Answer: D

Approach Solution 3:

In the given question, it is stated that the fraction closest to 1/2 will have the least distance from 1/2.
Let us measure the distance of the answer options from ½.
Option A 4/7
Distance of 4/7 from 1/2 = 4/7 - 1/2
= 1/14

Option B 5/9
Distance of 5/9 from 1/2 = 5/9 - 1/2
= 1/18

Option C 6/11
Distance of 6/11 from 1/2 = 6/11 - 1/2
=1/22

Option D 7/13
Distance of from 7/13 from 1/2 = 7/13 - 1/2
= 1/26

Option E 9/16
Distance of 9/16 from 1/2 =9/16 - 1/2
= 1/16

So, all the fractions have the same numerator of 1 and the fraction with the largest denominator will carry the smallest value. Thus 1/26 has the smallest value and is closest to 0 and hence 7/13 is the answer choice that is closest to 1/2.

Correct Answer: D

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