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 Post subject: GMAT Arithmetics
PostPosted: Tue Apr 09, 2013 11:28 am 
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Which of the following fractions is the largest?

A. 11/14
B. 4/5
C. 17/21
D. 29/35
E. 5/6

(E) One way to solve this type of problem is to find the lowest common denominator for all of the fractions.

To do this, first factor all of the denominators:
11/14: 14 factors into 2 and 7
4/5: 5 is prime
17/21: 21 factors into 3 and 7
29/35: 35 factors into 5 and 7
5/6: 6 factors into 2 and 3.

Then, to get a common denominator, we make sure that each of the factors is represented exactly once (since no factor is raised to a power in any denominator):
2 × 3 × 5 × 7 = 210.
Therefore, 210 is the lowest common denominator.

Now we can express all the fractions in terms of the common denominator of 210 to see which is largest:
11/14 × 15/15 = 165/210
4/5 × 42/42 = 168/210
17/21 × 10/10 = 170/210
29/35 × 6/6 = 174/210
5/6 × 35/35 = 175/210.
Therefore, 5/6 is the largest. The correct answer is choice (E).
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Is there a faster way to solve this problem rather than trying to find the common denominator? That method takes too long. Any help is appreciated.


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 Post subject: Re: GMAT Arithmetics
PostPosted: Tue Apr 09, 2013 11:29 am 
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Posts: 454
A speed of a solution in many situations depends on what techniques you are better at. But in this case the task is pretty simple: compare fractions. Still there are two similar options:
- compare all the fractions at once;
- compare the fractions in pairs.

The first approach is already described in the explanation. So I'm providing the second one.

The method is the same: we find the common denominator and compare. It can be the LEAST common denominator, or just the product of denominators.

11/14 ?? 4/5
(11 × 5)/(14 × 5) ?? (4 × 14)/(5 × 14)
11 × 5 ?? 4 × 14
55 < 56
Therefore 11/14 < 4/5.

As you can see, when we compare fractions, as soon as we found the common denominator all we care are the new numerators. You can think of it as similar to "cross-multiply". So the notation can be simplified to:

11/14 ?? 4/5
11 × 5 ?? 4 × 14
55 < 56
Therefore 11/14 < 4/5.

Since we're looking for the largest fraction, we compare 4/5 to the next fraction.
4/5 ?? 17/21
4 × 21 ?? 17 × 5
84 < 85
So 4/5 < 17/2. 17/2 is the largest one so far. Let's continue.

17/21 ?? 29/35 (21 = 3 × 7, 35 = 5 × 7. So the LCM is 3 × 5 × 7)
17 × 5 ?? 29 × 3
85 < 87
So 17/21 < 29/35.

29/35 ?? 5/6
29 × 6 ?? 5 × 35
174 < 175
Therefore 5/6 is the largest.


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 Post subject: Re: GMAT Arithmetics
PostPosted: Tue Apr 09, 2013 11:29 am 
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Joined: Tue Apr 13, 2010 8:48 am
Posts: 478
This seems to just be a long question in general. The easiest way i tackled it was to just divide the fractions out to get the percents and saw which percent was highest. I ball-parked a little to not waist time dividing all of them out in full. It took me 1 1/2 minutes and i found the right answer. good luck.


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 Post subject: Re: GMAT Arithmetics
PostPosted: Tue Apr 09, 2013 11:32 am 
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Posts: 454
a speed of solving this question depends on what techniques you are better at. Converting the fractions into decimals is also an option if you are good at dividing. (Since you can NOT use a calculator).
As you see, all the fractions are pretty close to each other. So for most of the students multiplication would be easier. But if you chose converting to decimals, you should start with the easiest one and compare the choices in pairs.

---------
B. It's the easy, well-known one: 4/5 = 8/10 = 0.8
A. 11/14
0.7...
Image
As we already see, choice B, 0.7..., is greater than A, 0.8. So there is no need to continue conversion of A.

C. 17/21
0.809...
Image

So C is greater than B.

D. 29/35
0.82...
Image
So D is greater than C.

D. 5/6
0.83...
Image
So E is greater than D and therefore the greatest one.
---------

As you see, this way to solve the question is also an option. But it is good to use, only if you are good at long division. For many students, multiplication would me more preferable than dividing 290 by 35 or dividing 170 by 21.
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