Which of the following has the greatest value?

Hi All,

Certain questions on the GMAT can be solved rather quickly if you spot the Number Properties involved and you have the ability to pattern-match.

The 'key' to answering this question quickly is the knowledge that 'squaring a positive fraction makes the fraction SMALLER, while square-rooting a positive fraction makes the fraction BIGGER.'

Here is the proof:

(1/2)^2 = (1/2)(1/2) = 1/4

Squaring the positive fraction made the fraction SMALLER.

Root(1/4) = Root(1)/Root(4) = 1/2

Square-rooting the positive fraction made the fraction BIGGER.

With that knowledge, take a look at the first 3 answer choices to this question. The prompt asks which of the 5 answers is biggest (and we'll get to all of the answers in a moment). Which of the first 3 answer choices is biggest? Notice how all 3 have the same fractions - one answer square-roots them all, one leaves them as is, and one squares them all. You should be able to say definitively that Answer A is the largest of those 3.

Next, a bit of estimation will come in handy. Notice how Answer A ADDS 4 fractions that are all fairly close to 1.... so that total has to be a little less than 4.....

What do Answers D and E do....? Without doing the calculation, you should notice how BOTH answers subtract at least one fraction; by extension, there's no way that those 2 totals are close to 4.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

All terms of each option are 1 or less than 1. Options (D) and (E) have negative terms and a quick look shows that they will certainly be less than the other 3 options. So ignore them.

Out of (B) and (C), compare first terms of each: 2/3 (= 6/9) of option (B) is more than 4/9 of option (C). All other terms will follow the same pattern so (B) is greater than (C)

Now compare the first terms of options (A) and (B): \(\sqrt{2}/\sqrt{3} ( = \sqrt{6}/3\) = 2.something / 3) while first term of option (B) is just 2/3. So the first term of option (A) is greater. All other terms will follow the same pattern.

Hence option (A) is greatest.

One of this type question is also in the GMATclub tests as well i didn't remember which one it was. However in order to solve the greatest value for the options available we have to find out the lowest denominator of the options available and use POE method.

A) \(\frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{4}} + \frac{\sqrt{4}}{\sqrt{5}} + \frac{\sqrt{5}}{\sqrt{6}}\)
B) \(\frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\)
C) \(\frac{2^2}{3^2} + \frac{3^2}{4^2} + \frac{4^2}{5^2} + \frac{5^2}{6^2}\)

For all these options it is quite evident that option A has the lowest denominator so it will result in greater value because of the root.

D) \(1 - \frac{1}{3} + \frac{4}{5} - \frac{3}{4}\)
E) \(1 - \frac{3}{4} + \frac{4}{5} + \frac{1}{3}\)
if you quickly solve D and E will reveal that denominator is greater than the option A

So it must be A

I really think this is one of those problems which can throw you off because it seems complicated, but I do believe its one of those questions you can easily do in less than 30 secs ... and its this kind of stuff that will make you save time in the real GMAT

Between A,B,C --> A is clearly the biggest

D is clearly less than B, it has the same terms some positive and some negative

E is also less than B : 4/5 - 3/4 + 2/3 + 2/3 which is less than 2/3-3/4+4/5+5/6, which has to be less than B

No calculations, just a quick manipulation mentally is sufficient
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x^m is greater when

M is more if X >1
M is less if X <1

in this case out of a b c , a will hav max value.

option E > D clearly

in E . 3/4 is greater than 1/3

that means E is at most 1+4/5 = 9/5

now in option 1... \sqrt{2}/\sqrt{3} is smallest .

Thus sum of option 1 is atleast 4*\sqrt{2}/\sqrt{3} which comes more than 2

and 9/5 is less than 2.

lol this method seems to be weird but i calculated it in dat way only

It seems to be easy but time consuming. What is the source of this problem.

This seems to be simple but time consuming problem what is the source?

If its taking more time then try this:

Try to find out the option which has the lowest denominator so it will result in greater value.
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shrouded1

Thanks for sharing approach +1 to you

+1 A

I think that here is not necessary to make many calculations.
You can eliminate some options with just watching.

E is less than B because E = 1/4 + 1/3 + 4/5 < 2/3 + 4/5 => E <B

D vs. E: E > D
\(-\frac{1}{3}+ \frac{4}{5}<\frac{4}{5} + \frac{1}{3}\)

B vs. E: B > E
\(\frac{2}{3} + \frac{3}{4} + \frac{4}{5}>\frac{4}{5} + \frac{1}{3} + \frac{1}{4}\)

B vs. C: B > C
A proper fraction raised to a positive number gets smaller.

B vs. A: A > B
When we take the root of a proper fraction it gets larger.

Ans (A)

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