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Percent Change Problems on the GMAT

This is, among other things, a case study of one of the harder Data Sufficiency problems in the OG.

 

Percents as Multipliers

You can find several good tips on percent problems at this post.  The most relevant here is the idea of a percent as multiplier.  To change a percent into a multiplier:

(a) change it from a percent to a decimal

(b) for a percent increase, keep it positive; for a percent decrease, make it negative.

(c) add that to one.

For example, for a 12% decrease,

(a) 12% > 0.12

(b) it’s a decrease, so -0.12

(c) 1 + (-0.12) = 0.88

That final number, 0.88 is the multiplier for a 12% decrease.  If we want to decrease any quantity by 12%, all we have to do is multiply by that multiplier.

 

The DS Problem in the OG

Here’s the problem from the OG, practice DS #120.

120) The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997, and y percent less in 1999 than in 1998.  Was the annual rent collected by the corporation from the building more in 1999 than in 1997?

1. x > y
2.{xy}/100<x-y

    (A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
    (B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
    (C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
    (D) Each statement alone is sufficient to answer the question.
    (E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

 

The Tempting Wrong Answer

When a percent increase is combined with a percent decrease on the GMAT, the one archetypal mistake the testmakersare trying to educe is idea that you can find the net result by simply adding and subtracting percents.  Increase of 20%, followed by a decrease of 15% — the huge mistake to say 20 – 15 = 5%.  That’s like a huge booby trap the GMAT sets up every time, and test after test the unsuspecting masses predictably fall into it in swarms.  Here, statement #1 is all about that huge mistake.  Many people will mistakenly think (1) is sufficient, but it’s not.

When a percent increase is combined with a percent decrease on the GMAT, you need to multiply the multipliers.  For example, 20% increase means that the multiplier = 1.20; 15% decrease -> multiplier = 0.85.  Combination = 1.20*0.85 = 1.02, a 2% increase.

In particular, increasing by a percent, then decreasing by that same percent, does not leave you in the same place.  Consider an increase of 50% followed by a decrease of 50%.  Start at $100.  Increase by 50% to $150.  Now, decrease by 50%, from $150 to 75%.  The net result of a 50% increase, followed by a 50% decrease, is a 25% decrease.

 

Solution

Call the initial 1997 amount of rent A.  It increased x% from 1997 to 1998.  Notice that x% as a multiplier is (1 + x/100), so the amount of rent in 1998 is:
A(1+{x/100})

Then, it decreased y% from 1998 to 1999.  The multiplier now is (1 - y/100), so the amount of rent in 1999 is:
A(1+{x/100})*(1-{y/100})

So, in order for the rent to be more in 1999 than in 1997, what multiplies A here, the composite factor multiplying A, must be greater than one.
(1+{x/100})*(1-{y/100}) > 1

FOIL out the left side:
1+{x/100}-{y/100}-(x/100)(y/100) > 1

Subtract 1 from both sides:
{x/100}-{y/100}-({xy}/{10000}) > 0
{x/100}-{y/100} > ({xy}/{10000})

Multiply both side by 100, to clear some of the fractions
x-y > {xy}/{100}

This is the condition for the rent in 1999 being larger than the rent in 1997.  Notice this equivalent to Statement #2.  Since statement #2 is another form of the statement that the rent in 1999 is larger than the rent in 1997, it is sufficient, and the answer is B.

 

Here’s a practice PS question exploring some of the same ideas.

http://gmat.magoosh.com/questions/30

 

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16 Responses to Percent Change Problems on the GMAT

  1. Avni May 4, 2016 at 10:59 am #

    Will you always use 1 unless it’s says percent of? I’m struggling when to use 1. Q

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert May 13, 2016 at 5:54 am #

      Hi Avni,

      Happy to help! 🙂

      The 1 stands for the starting value (because 1 = 100%, so we are talking about 100% of what we started with). If I have a 5% increase, for example, that means I have all of what I started with and then an additional 5%, so 105% or 1.05! I hope that makes sense. 🙂

  2. Riya October 26, 2015 at 7:59 pm #

    How is the first one not sufficient? I understand the second one being sufficient, but why is the first one not? Can you please explain it?

    Thank you!

  3. philip March 8, 2015 at 2:53 pm #

    If a number is 20% more than the other, how much percent is the second number less than the first?

    • Rachel Wisuri
      Rachel March 9, 2015 at 10:30 am #

      Hi Philip!

      We don’t answer questions from outside materials on this blog, but I’d suggest posting this question (along with its source!) here: http://gmatclub.com/forum/magoosh-324/

      Thanks!

    • kiran May 3, 2016 at 5:20 am #

      16.66% less than first number…..
      let,x = 1.2 y ,
      => y = 5/6 x
      y = (1- 1/6)x
      so. 1/6 = 16.66%….

  4. sanjoy August 20, 2014 at 12:10 pm #

    can u plz clarify it????? i m not getting it clearly “So, in order for the rent to be more in 1999 than in 1997, what multiplies A here, the composite factor multiplying A, must be greater than one.

    • Mike MᶜGarry
      Mike August 20, 2014 at 5:59 pm #

      Dear Sanjoy,
      I’m happy to help. 🙂 In order for the original rent A to get bigger from 1997 to 1997, we must multiply by a number greater than 1. We multiply A by a composite factor. Therefore, this composite factor must be greater than one.
      Does this make sense?
      Mike 🙂

      • Shraddha Dalvi July 10, 2016 at 7:46 am #

        Hi Mike
        I did not get this concept of the composite factor must be greater than one. would it be possible for you to please clarify on this point
        Thanks! 🙂

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert July 21, 2016 at 7:16 am #

          Hi Shraddha!

          Happy to clarify 🙂 First, it’s key to remember that there is an overall percent increase from 1997 to 1999. To reflect this increase, the multiplier used to go from the value in 1997 to 1999 must be greater than 1. Because there are two different changes (from 1997 to 1998 and from 1998 to 1999), we have two parts to this multiplier: the multiplier is the product (1+x/100)*(1-y/100). And this multiplier is the “composite factor” we’re referring to. So, because there is an overall percent increase, the composite factor (1+x/100)*(1-y/100) must be greater than 1.

          Does that make sense?

          A composite factor refers to a factor

  5. sanjoy July 1, 2014 at 1:51 pm #

    Increasing the original price of a certain item by 25 percent and then increasing the new price by 25 percent is equivalent to increasing the original price by what percent?……..yap this kinda question possible to use successive multipliers ?????except letting 100…….!!!!

    • Mike MᶜGarry
      Mike July 1, 2014 at 4:31 pm #

      Dear Sanjoy,
      Yes, in principle, we would use this by multiplying multipliers — (1.25)*(1.25). In practice, the GMAT will not give you that, because no one expects you to be able to do that quickly without a calculator.
      Does this make sense?
      Mike 🙂

      • sanjoy July 18, 2014 at 12:15 pm #

        awesome thnx!!!!

        • Mike MᶜGarry
          Mike July 18, 2014 at 12:34 pm #

          Sanjoy,
          You are quite welcome. 🙂 Best of luck to you!
          Mike 🙂

  6. domenico May 15, 2012 at 3:27 pm #

    this was tough………….really. 🙁

    Thanks.

    • Mike MᶜGarry
      Mike May 16, 2012 at 11:51 am #

      You’re welcome. Let us know if you have any more questions.
      Mike 🙂


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