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

Hi,

Do not waste time in calculating..
look at the similarity in all choices- they all EXCEPT E are of type \(\frac{x}{2x-1}\)...
and \(\frac{x}{2x-1}\) for 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..

you can check in two ways-
1) for x as 1 or 2 or 3..
x=1, \(\frac{x}{2x-1}\) = 1..
x=2, \(\frac{x}{2x-1}\) = 2/3..
x=3, \(\frac{x}{2x-1}\) = 3/5..
so each successive value is smaller than previous one..

2) simplify the term..- IMPORTANT
\(\frac{x}{2x-1}\)=\(\frac{1}{(2x-1)/x}\) = \(\frac{1}{(2x/x-1/x)}\) = \(\frac{1}{(2-1/x)}\)
so the denominator should be closer to 2 so \(2-\frac{1}{x}\) should be close to 2, and therefore x should be MAX to make \(\frac{1}{x}\) less

Therefore, here 7 is the highest value in the four choices so D is the closest to 1/2.

lets see E... It is the TYPE \(\frac{x}{2(x-1)}\)... clearly since 7 and 9 are close to each other but denominator in E is reduced by a larger number,
D is the answer

\(\frac{1}{2}\) = 50%

Now, Check the options...

A) \(\frac{4}{7}\) = 57.14 %
B) \(\frac{5}{9}\) = 55.56 %
C) \(\frac{6}{11}\) = 54.54 %
D) \(\frac{7}{13}\) = 53.85 %
E) \(\frac{9}{16}\) = 56.25 %

Since the closest answer is (D) , that will be our answer...

Let's check the answer choices...

A) 4/7
3.5/7 = 1/2. So, 4/7 is 0.5/7 away from 1/2
Since 0.5/7 = 1/14, we can say that 4/7 is 1/14 away from 1/2

B) 5/9
4.5/9 = 1/2. So, 5/9 is 0.5/9 away from 1/2
Since 0.5/9 = 1/18, we can say that 5/9 is 1/18 away from 1/2

C) 6/11
Using the same logic, we can say that 6/11 is 1/22 away from 1/2

D) 7/13
Likewise, 7/13 is 1/26 away from 1/2

E) 9/16
And 9/16 is 1/16 away from 1/2

Of the BLUE fractions, 1/26 is the SMALLEST
So, the correct answer is D

Cheers,
Brent

0.5/7, 0.5/9, 0.5/11, 0.5/13, 0.5/8 .. are the differences.
Pick 0.5/13

I approached it in this way:
1/2, each of the answer choices is 1/2 + something, so: 0,5/7, 0,5/9, 0,5/11, 0,5/13, 1/16=0,5/8, the smallest is 0,5/13
Hence D

Sometimes it is easier to go directly with calculation instead of try with equations.

(1/2) - each option. In less than 2 minutes you will find that the smallest difference is (1/2) - (7/13)
Answer is D

There are 4 ways to solve the problem (Which is also basically a summary to above explanations):
1. Calculate the decimal and find the answer: This should be used when the numbers are small or calculation is easy.
2. Subtract the number (1/2 in this case) one by one with all the options to find the distance between those numbers, the smallest value would mean the closest number. (My favorite)
3. Use algebra to solve the arithmetic: To be used cautiously after identifying the pattern enabling algebraic equation formation.
4. Divide the options with the given number to see which number is closest to 1 (1 because we will also be dividing the given number, 1/2 in this case, to the same number)

Of all the explanations I read, I feel the below one is the best one.

Since we need to find out what's closest to 1/2, just subtract that from each of the options.

(A) 4/7 - 1/2 = 1/7
(B) 5/9 - 1/2 = 1/18
(C) 6/11 - 1/2 = 1/22
(D) 7/13 - 1/2 = 1/26
(E) 9/16 - 1/2 = 2/32 => 1/16

From the above since 1/26 is the lowest number, (D) is the correct answer.

My favorite as well!!

Of all the explanations I read, I feel the below one is the best one.

Since we need to find out what's closest to \(\frac{1}{2}\), just subtract that from each of the options.

(A) \(\frac{4}{7}\) - \(\frac{1}{2}\) = \(\frac{1}{7}\)
(B) \(\frac{5}{9}\) - \(\frac{1}{2}\) = \(\frac{1}{18}\)
(C) \(\frac{6}{11}\) - \(\frac{1}{2}\) = \(\frac{1}{22}\)
(D) \(\frac{7}{13}\) - \(\frac{1}{2}\) = \(\frac{1}{26}\)
(E) \(\frac{9}{16}\) - \(\frac{1}{2}\) = \(\frac{2}{32}\) => \(\frac{1}{16}\)

From the above since \(\frac{1}{26}\) is the lowest number, (D) is the correct answer.

ScottTargetTestPrep /VeritasKarishma. Kindly help. If the Options include the following three:

1. 5/11
2. 50/99
3. 50/101.

Then with the above approach I am getting two negative values (For 1 and 3). Should I ignore the negative sign and focus on largest denominator ? So that the largest denominator can be the ans.

Closest to 1/2 could be slightly less than 1/2 or slightly more than 1/2. When you subtract 1/2, the number slightly less than 1/2 will become negative. Still, we need to be as close to 0 as possible i.e. distance from 0 to be as small as possible. Basically, the absolute value of the number obtained should be as small as possible. So you ignore the sign you get and focus on just the absolute value. It should be smallest (with same numerator, the denominator should be greatest).

In the original example, all the given answer choices were greater than 1/2; that’s why we didn’t bother with absolute values and just subtracted 1/2 from each answer choice. In reality, if some of the answer choices were greater than 1/2 and some were less, we would compare the absolute value of the difference of the numbers and 1/2. The reason for that has to do with distances on the real number line; the distance between two real numbers is the absolute value of their difference. So, for the numbers you provided, you should compare the absolute values of the distances and take the one closest to zero. Another way of saying that is for the ones greater than 1/2, subtract 1/2 and for the ones less than 1/2, subtract those numbers from 1/2.

Video solution from Quant Reasoning:

Hi ScottTargetTestPrep

Could you explain the logic behind the strategy that you showed here? Will this work for any "closest to" questions?

Thanking you in advance!

Response:

As a matter of fact, the solution simply applies the definition of distance. Since each fraction is greater than 1/2, subtraction of 1/2 from each fraction will give us the distance between that fraction and 1/2.

If some of the answer choices were less than 1/2. you could still subtract 1/2 from it. However, the distance between that number and 1/2 is the absolute value of the difference. For instance, if 13/28 was an answer choice, then subtracting 1/2 from it will result -1/28. Taking the absolute value of this result, we have 1/28, which is the distance between 13/28 and ½, and it would be the correct answer since 1/28 is less than any of the other distances.

In this question we can bring the idea of distance and measure the distance between the options and 1/2 as one of the strategies to answer the question.
So how should a SMART & LOGICAL GMAT aspirant think on this question?
The fraction closest to 1/2 will have the least distance from 1/2. Let me measure the distance of the answer options from ½.
A 4/7
Distance of 4/7 from 1/2 = 4/7 - 1/2
= 1/14

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

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

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

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

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

Devmitra Sen
GMAT Mentor

Hi All,

We’re asked which of the following fractions is CLOSEST to ½. There are a number of different ways to approach the Arithmetic in this question (and you can use the given fractional answers or convert them to decimals).

Since the numerator of each answer is MORE than half of the denominator, we know that each fraction is GREATER than ½. Thus, we can look for how much greater each fraction is relative to ½ and define which is closest to that fraction.

Answer A: 4/7…. 3.5/7 is exactly ½, so this fraction is .5/7 = 1/14 greater than 1/2
Answer B: 5/9…. 4.5/9 is exactly ½, so this fraction is .5/9 = 1/18 greater than 1/2
Answer C: 6/11…. 5.5/11 is exactly ½, so this fraction is .5/11 = 1/22 greater than 1/2
Answer D: 7/13…. 6.5/13 is exactly ½, so this fraction is .5/13= 1/26 greater than 1/2
Answer E: 9/16…. 8/16 is exactly ½, so this fraction is .1/16 greater than 1/2

The smallest fraction ‘greater than ½’ is the one that’s closest to 1/2...

Final Answer:
GMAT Assassins aren’t born, they’re made,
Rich