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GMAT Shortcut: Adding to the Numerator and Denominator

First, try these practice DS questions:

1) If x and y are positive integers, is {x/y} <{{x+2}/{y+3}}?

Statement #1: y > 20
Statement #2: x < 5

2) If x and y are positive integers, is {x/y} <{{x+5}/{y+5}}?

Statement #1: y = 5
Statement #2: x > y

Throughout this post, assume that I am talking about positive fractions with a positive numerator and positive denominator.  If the fraction is negative, use the information below to figure out what happens to the absolute value of the fraction, and judge from there.

 

Adding and Subtracting

Ironically, it’s a bit easier if we add to part of the fraction and subtract from the other. The BIG idea here: if you increase the numerator and/or decrease the denominator of any positive fraction, that fraction will get bigger; if you decrease the numerator and/or increase the denominator of any positive fraction, that fraction will get smaller.  Add a positive number to the numerator and/or subtract a positive number from the denominator of any positive fraction, and the new fraction will be greater than the starting fraction.  Subtract a positive number from the numerator and/or add a positive number to the denominator of any positive fraction, and the new fraction will be smaler than the starting fraction.  Though not relevant in the two practice problems above, this is a golden rule that will help you in a panoply of fraction and ratio problems.

 

Adding the Same Number to Numerator and Denominator

Suppose we start with the positive fraction x/y and we want to add some positive number b to both the numerator and the denominator.  How does the resultant fraction, (x + b)/(y + b), compare to the starting fraction?

Well, the rule here is a bit subtle.  When you add the same number to numerator and denominator, the resultant fraction is closer to 1 than is the starting fraction.  This means, if the starting fraction x/y is less than 1, then the resultant fraction is closer to one — bigger than the starting fraction.  If the starting fraction x/y is an “improper fraction”, a fraction with a value greater than one, than adding the same number to both the numerator and the denominator will make the resultant fraction closer to 1 — less than the starting fraction.

Here are a couple of examples.

Example #1

Start = 2/3 —- a fraction less than one.

Add five to the numerator and the denominator.

Result = 7/8 — this fraction is closer to one than is 2/3: on the number line —–

Since 1 is bigger than 2/3, when the resultant fraction moved closer to 1, it got bigger than 2/3.  Therefore, we know 2/3 < 7/8

Example #2

Start = 3/2 —- a fraction greater than one.

Add two to the numerator and the denominator.

Result = 5/4 — this fraction is closer to 1 than is 3/2: on the number line —–

Since 1 is less than 3/2, when the result fraction moved closer to 1, it got smaller than 3/2.  Therefore, we know 3/2 > 5/4

 

Adding Different Numbers to the Numerator and Denominator

Actually, this case is simply a generalization of the previous case.  Suppose we start with a fraction x/y, and we add the positive number a to the numerator and the positive number b to the denominator, and we want to know if the resultant fraction is bigger or smaller than the starting fraction.

Well, the general rule is: adding a to the numerator and b to the denominator moves the resultant fraction closer to the fraction a/b.  If x/y < a/b, moving the starting fraction close to a/b will make it bigger.  If x/y > a/b, moving the starting fraction close to a/b will make it smaller.

Here are some example:

Example #3

Start = 2/7

Add 3 to the numerator and 5 to the denominator.

Resultant fraction = 5/12— this fraction is closer to 3/5 than is 2/7 —-

On the number line —–

Because 3/5 is bigger than 2/7, adding 3 to the numerator and 5 to the denominator has the net effect of producing a fraction that is bigger: 2/7 < 5/12

Example #4

Start = 11/12

Add 2 to the numerator and 5 to the denominator.

Resultant fraction = 13/17— this fraction is closer to 2/5 than is 11/12 —-

On the number line —–

Because 2/5 is less than11/12, adding 2 to the numerator and 5 to the denominator has the net effect of producing a fraction that is smaller: 11/12 > 13/17

Now that you know these rules, go back to the practice problems at the beginning and see whether they make more sense now.

 

Practice Problem Solutions

1) Statement #1: We are adding 2 to the numerator and 3 to the denominator, so we know the resultant fraction will move closer to 2/3.  If all we know is that the denominator of the starting fraction is greater than 20, then we have no idea what the size of the starting fraction is: it could be much greater than 2/3, or much smaller than 2/3, depending on the numerator, of which we have no idea.  We can draw no conclusion right now.  This statement, alone, by itself, is insufficient.

Statement #2: Now, all we know is that the numerator of the starting fraction is less than 5 — it could be 4, 3, 2, or 1.  We have no idea of the denominator.  If y = 50, then we get a very small fraction.  But if x = 4 and y = 1, the fraction equals 4, much larger than 2/3.  In this statement, we have no information about the denominator, and since we know nothing about the denominator, we know nothing about the size of the starting fraction: it could be either greater or less than 2/3.  Therefore, we can draw no conclusion.  This statement, alone, by itself, is also insufficient.

Now, combine the statements.  We know y > 20 and x < 5.  Well no matter what values we choose, we are going to have a denominator much bigger than the numerator.  The larger possible fraction we could have under these constraints would be 4/21 (largest possible numerator with smallest possible denominator).  The fraction 4/21 is much smaller than 1/2, so it’s definitely smaller than 2/3.  Any fraction with y > 20 and x < 5 will be less than 2/3.  Therefore, adding 2 to the numerator and 3 to the denominator will move the resultant fraction closer to 2/3, which has the net effect of increasing its value.  Therefore, the answer to the prompt question is “yes.”  Because we can give a definite answer to the prompt, we have sufficient information.

Neither statement is sufficient individually, but together, they are sufficient.  Answer = C.

2) We are adding the same number, 5, to both the numerator and the denominator, so the value of x/y will move closer to 1.  All we need to determine is whether x/y is greater than 1 or less than 1.

Statement #1: y = 5.  Here, we have a definite value for y, but zero information about x.  If y = 5, some fractions (1/5) can be less than one, while others (7/5) will be greater than one.  Either is possible.  Since both are possible, we can’t give a definitive answer to the prompt.  This statement, alone, by itself, is insufficient.

Statement #2: x>y.  Dividing both sides of this inequality by y, we get (x/y) > 1.  This means x/y must be a fraction greater than 1, which means the resultant fraction (x + 5)/(y + 5) must be closer to one, which means the resultant fraction must be smaller.  Therefore, we can definitively say: the answer to the prompt question is, “No.”  Because we can give a definite answer to the prompt, we have sufficient information.  This statement, alone, by itself, is sufficient.

Statement #1 is insufficient and Statement #2 is sufficient.  Answer = B.

 

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16 Responses to GMAT Shortcut: Adding to the Numerator and Denominator

  1. Simeon April 22, 2015 at 6:17 pm #

    There seems to be a typographical error in the third paragraph under the ‘Practice Problem Solutions’ heading :

    “Now, combine the statements. We know y > 20 and x < 5. Well no matter what values we choose, we are going to have a denominator much smaller than the numerator."

    Shouldn't that last sentence read "…we are going to have a denominator much BIGGER THAN the numerator" ?

    • Mike McGarry
      Mike April 23, 2015 at 10:03 am #

      Dear Simeon,
      YES! That’s perfectly correct! Thank you! :-) That was a typo, and I just fixed it. Your sharp eye for detail will serve you very well on the GMAT. Best of luck!
      Mike :-)

  2. ali April 21, 2015 at 11:29 pm #

    I am confused s to how you place numbers like 13/17, 11/12, 2/5 on the number line. I mean how do you position them. Like on a number line between o and 1 where and how will I put 13/17. Do I have to divide the section in 17 parts then count till the 13th part and mark it?. This takes a lot of time. Or is there a simpler way. Thanks – See more at: http://magoosh.com/gmat/2012/gmat-shortcut-adding-to-the-numerator-and-denominator/#comment-1658493

    • Mike McGarry
      Mike April 23, 2015 at 10:41 am #

      Dear Ali,
      I’m happy to respond. My friend, you are thinking too literally about 13/17. You have to use number sense. Number sense is the ability to use the patterns of numbers. It’s the ability to think creatively, outside-of-the-box, to use these patterns for efficient problem-solving.
      Suppose I had to place (13/17) and (11/12) and (2/5) in order, smallest to biggest. Figuring out the exact position of each one separately is a colossal waste of time. First of all, notice that (13/17) and (11/12) are bigger than (1/2) because the numerators are more than half of the denominator, and (2/5) is less than (1/2), because 2 is less than half of 5. Thus, two of the fractions are greater than (1/2), so the one that is less than (1/2) must be the smallest. You should be able to see that very quickly.
      Now, I notice that (11/12) is very close to 1: it is only (1/12) less than one. Meanwhile, (13/17) is (4/17) less than one. Well, it’s easy to compare (1/12) and (4/17). Notice that (4/17) is almost as big as (1/4), which is much larger than (1/12). Another way to say it is
      (4/17) > (4/20) = (1/5) > (1/12)
      So, (11/12) is much closer to one, so it must be bigger than 13/17. Thus,
      (2/5) < (13/17) (12/18) = (2/3)
      Similarly, if we added one to the numerator and subtracted one from the denominator, the fraction would get bigger.
      (13/17) < (14/16) = (7/8)
      This simple trick immediately tells us that (13/17) must be between (2/3) and (7/8). That is super-precise estimation. I thought to do this, adding & subtracting 1, because both the numerator and denominator of (13/17) are odd, so I knew adding and subtracting 1 would produce two even numbers, so something would have to cancel and get simpler.
      This reasoning here contains several examples of number sense. It's all about being familiar with the patterns of numbers so you can apply them creatively in ways that will simply a problem. You may find this blog helpful.
      https://magoosh.com/gmat/2013/how-to-do-gmat-math-faster/
      Does all this make sense?
      Mike :-)

  3. Mayank January 20, 2015 at 7:41 am #

    this is the best possible explanation I have seen for this otherwise confusing and difficult topic. A gem indeed for related DS questions, thanks again !!!

    • Mike McGarry
      Mike January 20, 2015 at 10:26 am #

      Dear Mayank,
      You are quite welcome, my friend! :-) Best of luck to you in the future!
      Mike :-)

  4. Tina November 7, 2014 at 9:45 pm #

    I’ve been confused about this since so long. Thanks this helped

    • Mike McGarry
      Mike November 8, 2014 at 8:41 am #

      Dear Tina,
      You are quite welcome! :-) I am very glad that we cleared up your confusion! Best of luck to you in the rest of your test prep!
      Mike :-)

  5. Laura October 16, 2013 at 6:38 pm #

    Hi,
    Isn’t the opposite? Since the fraction we are considering is x/y, x20, we are going to have a denominator much BIGGER than the numerator?

    “We know y > 20 and x < 5. Well no matter what values we choose, we are going to have a denominator much smaller than the numerator. The larger possible fraction we could have under these constraints would be 4/21 (largest possible numerator with smallest possible denominator).'

    Thank you,
    Laura

    • Mike McGarry
      Mike October 17, 2013 at 10:56 am #

      Laura,
      Yes, the numerator would be much smaller than the denominator, the denominator would be much bigger, which means the fraction overall would have to be small. What I said was —of those small possible values for the fraction, the largest among them is 4/21. The fraction equals 4/21 or smaller. By giving a *largest* possible value, I am indicating how *small* the fraction is, not how large the fraction is. Does this make sense?
      Mike :-)

  6. Adeel Yasin August 25, 2013 at 4:56 am #

    Hi Mike,

    Excellent work!

    However, I feel that there is a typo in the paragraph Adding and Subtracting. Here is the original content,

    Subtract a positive number from the numerator and/or add a positive number to the denominator of any positive fraction, and the new fraction will be greater than the starting fraction.

    Here is what it should be,

    Subtract a positive number from the numerator and/or add a positive number to the denominator of any positive fraction, and the new fraction will be SMALLER than the starting fraction.

    Although, it is clear in the examples, but I think this typo may confuse others. Do you agree?

    • Mike McGarry
      Mike August 25, 2013 at 12:16 pm #

      Dear Adeel,
      You are 100% correct. I just fixed that typo. Thanks for pointing it out!
      Mike :-)

  7. John May 14, 2013 at 7:26 am #

    I am confused. Lets see

    4/6
    decreasing nominator makes it smaller = 3/6
    decreasing denominator makes it bigger = 4/5

    but why decreasing both denominator and nominator at the same time makes it smaller?

    • Mike McGarry
      Mike May 14, 2013 at 10:08 am #

      John,
      The fraction 4/6 = 2/3 is less than one. If we *add* the same number to both the numerator & denominator, that moves it closer to 1, to the right on the number line, which makes it bigger. Thus, if we *subtract* the same number from both the numerator & denominator, that moves it further from 1, to the left on the number line, which makes it smaller.
      Another way to say it — start with 3/5 — add one to both the numerator & denominator, and it becomes 4/6 —- 4/6 is closer to 1 than 3/5, so it is bigger than 3/5, which means 3/5 is smaller than 4/6.
      Does all this make sense?
      Mike :-)

  8. Faruk August 6, 2012 at 12:59 pm #

    Excellent article Mike…Very helpful indeed…:)

    • Mike McGarry
      Mike August 7, 2012 at 8:57 am #

      Thank you for your kind words, sir. Best of luck to you!
      Mike :-)


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