If the numbers 17/24, 1/2, 3/8, 3/4, and 9/16 were ordered from greate

SOLUTION

If the numbers 17/24, 1/2, 3/8, 3/4, and 9/16 were ordered from greatest to least, the middle number of the resulting sequence would be

(A) 17/24
(B) 1/2
(C) 3/8
(D) 3/4
(E) 9/16

The least common denominator for all the fractions above is 48:

\(\frac{17}{24}=\frac{34}{48}\), \(\frac{1}{2}=\frac{24}{48}\), \(\frac{3}{8}=\frac{18}{48}\), \(\frac{3}{4}=\frac{36}{48}\), \(\frac{9}{16}=\frac{27}{48}\).

The numerators in descending order are: 36, 34, 27, 24, 18. The middle number is 27, which corresponds to \(\frac{9}{16}=\frac{27}{48}\).

Answer: E.
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Yes, that's true when everything is positive -- the idea is, if you add a to the numerator and b to the denominator of a fraction, its overall value moves towards a/b. That can be a great way to compare fractions quickly when other techniques (common denominator, benchmarking, converting to decimal, etc) would be awkward. Here, I'd use that principle, but only once -- we can almost immediately see here that 3/8 < 1/2 < 9/16 < 3/4, whether by comparing each fraction to 1/2, getting a common denominator of 16, or using knowledge of decimal conversions. To find which number is the median, the only question is whether 17/24 is less than or greater than 9/16. And going from 9/16 to 17/24, we're adding 8 to the numerator and 8 to the denominator, so the value of the fraction would move towards 8/8 = 1, and therefore must get bigger since we're starting with a fraction less than 1, and 9/16 < 17/24 must be true. So 9/16 is in the middle.

For your second question, again assuming everything is positive, subtraction will work in the opposite way as addition (e.g. subtracting 8 on the top and on the bottom, as you were doing, will move you away from 8/8 = 1). But there's never any reason to think about subtraction, because you can always think only about addition in situations like this. So rather than start, in this question, from 17/24, and notice you subtract 8 from the top and bottom to get to 9/16, you can instead start from the fraction with the smaller numbers, 9/16, and notice you add 8 to the top and bottom to get to 17/24. Then you're in a situation that is easier to think about, conceptually.

[quote="Bunuel"]The Official Guide For GMAT® Quantitative Review, 2ND Edition

If the numbers 17/24, 1/2, 3/8, 3/4, and 9/16 were ordered from greatest to least, the middle number of the resulting sequence would be

(A) 17/24
(B) 1/2
(C) 3/8
(D) 3/4
(E) 9/16

17/24, 1/2, 3/8, 3/4, and 9/16 - Multiply all the fractions by 48 we get: 34, 24, 18, 36, 27

So 36 is the greatest which is 3/4
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Take LCM of denominators and get common denominator to all fractions so that you can compare the numerators and arrange in the increasing or decreasing order

LCM of denominators is 48

fractions can be rewritten as
34/48,24/48,18/48,36/48,27/48.
now arrange them
18/48,24/48,27/48,34/48,36/48
hence the middle element is 27/48=9/16

Easiest way to solve this type of question is to find a common multiplier for denominators of all the given number.
As we can see that the Common Factor for all the denominators- 24,2,4,8,16 is 48 .
Multiply each number by 48-
(17/24)*48 = 34
(1/2)*48 = 24
(3/8)*48 = 18
(3/4)*48 = 36
(9/16)*48 =27

So the given number becomes - 34,24,18,36,27

In descending order - 36,34,27,24,18

So mid-number is 27 .
Now again to get the required answer divide it by 48 -> 27/48 = 9/16

So answer is (E)

Note: Although the explanation may look lengthy but when we do it pen-paper it will not take more that 15 sec.

Approach 1:
Here, taking LCM for all the denominators, as others suggested, would be a good idea.

Approach 2:
Alternatively, if we are little good at approximation, looking at the numbers, we can say that 3/8 is the least and 3/4 is the greatest. Comparing 17/24 and 9/16 will be the slightly trickiest thing in this problem.

9/16 will be a little more than half(1/2).
17/24 will be a little less than 3/4 as 18/24 will give 3/4.

So, if we arrange from greatest to least, the order will be 3/4, 17/24, 9/16, 1/2, 3/8. Since we are asked for the middle element, 9/16 will be the answer.

Ans is (E).

Ans should be (e)..9/16
We can see the other nos relative to 1/2 i.e. 50%.
Just by seeing numerator and denominator 3/4 and 3/8 can be placed on either side of 1/2. 17/24 and 9/16 are definitely greater than 1/2.
So, to check which no comes in 3rd place, we can individually compute their values and see that 9/16 is smaller than 17/24.
So 9/16 must be at the 3rd spot and thus, the middle no.

Correct Sequence in descending order should be
3/4 ,17/24,9/16,1/2,3/8
hence middle term is 9/16

answer E
Find the common denominator then compare.
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Answer is E, I spent a little bit too much time on this though.

I made a number line and put in the obvious ones. The "problem" was deciding between 9/16 and 17 24 -> I realized I could reach a common denominator and it became clear which was bigger.

Finding a common denominator is the easiest way to solve this problem:

All we need to do is get to the third number of the sequence and we have our answer (because the third is the median of the set)

18/48, 24/48, 27/48...

So the middle of the set is 27/48 = 9/16

My least common denominator is coming out to be 96. How did we arrive at 48?

Thank You!

Isn't 48 divisible by 24, 2, 8, 4, and 16?

Basically we need to find the LCM of 24 = 2^3*3 and 16 = 2^4, which is 3*2^4 = 48.
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We can express each fraction using a common denominator. The LCD here is 48, so we have:

17/24 = 34/48, 1/2 = 24/48, 3/8 = 18/48, 3/4 = 36/48 and 9/16 = 27/48

In ascending order, we have:

18/48, 24/48, 27/48, 34/48, 36/48

So the middle number is 27/48, or 9/16.

Answer: E
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Fraction to percentage would be quicker

We just need to know the following for any decimal to fraction to percentage conversion

\(\frac{1}{2}\) = (50%, .05) ,\(\frac{1}{3}\) = (33.3%, .3333) ,\(\frac{1}{4}\) = (25%, .25) ,\(\frac{1}{5}\) = (20%, .2) \(\frac{1}{6}\) = (16.66%, .1666) ,\(\frac{1}{7}\) =(14.28%, .1428) ,\(\frac{1}{8}\) =(12.5%, .125) ,\(\frac{1}{9}\) = (11.1%, .111) ,\(\frac{1}{10}\) = (10%,.1),\(\frac{1}{11}\) = (9.1%, .0909) ,\(\frac{1}{13}\) = (7.7%, .077) ,\(\frac{1}{17}\) = (5.9%, .059) ,\(\frac{1}{19}\) = (5.3%, .053) ,\(\frac{1}{23}\) = (4.3%, .0435) ,\(\frac{1}{29}\) = (3.40%, .0345) ,\(\frac{1}{31}\) = (3.2%, .0323)

\(\frac{17}{24}\) = 68%
\(\frac{1}{2}\) = 50%
\(\frac{3}{8}\)= 36%
\(\frac{3}{4}\)= 75%
\(\frac{9}{16}\)= 54%

75>68>54>50>36
So D>A>E>B>C
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[quote="Bunuel"]The Official Guide For GMAT® Quantitative Review, 2ND Edition

If the numbers 17/24, 1/2, 3/8, 3/4, and 9/16 were ordered from greatest to least, the middle number of the resulting sequence would be

(A) 17/24
(B) 1/2
(C) 3/8
(D) 3/4
(E) 9/16


One quick way to look at the fractions is to quickly convert it into decimal / percentage ...

i ) 17/24 = 0.70
ii ) 1/2 = 0.50
iii) 3/8 = 0.37 ( you dont have to find the exact decimal, 3 is less than half of 8)
iv) 3/4 = 0.75
v) 9/16 = 0.56 ( you dont have to find the exact decimal, 9 is just little above half of 16 )

arrange in ascending order...

Ans : E

IanStewart in your book in which you cover the concept of ratios you mentioned a particular logic:

If \(\frac{ x}{y}\) < \(\frac{a}{b }\) Then \( \frac{x}{y}\) < \(\frac{ x+a}{y+b}\) < \(\frac{a}{b}\)

And

If\(\frac{ x}{y }\) > \(\frac{a}{b}\) Then \(\frac{x}{y} \) > \(\frac{x+a}{y+b }\) > \(\frac{a}{b}\\
\)

Question: Is it possible if you could show how we could apply this concept in the above question?

Question: Also, what about a situation in which we are subtracting? I tried comparing 17/24 and 9/16 and saw that by subtracting 8 from 17 and 24 we get 9 and 16 respectively i.e. \(\frac{17-8}{24-8}\) = \(\frac{9}{16}\) But in this case we can't say 17/24 < 8/8 OR 17/24 < 1 and hence \(\frac{17}{24}\) < \(\frac{17-8}{24-8 }\) < \(\frac{9}{16}\)

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