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Percent Increase/Decrease

fontainedfontained Joined: 07/22/2017Posts: 23
Hi there,

Could you provide some "rules" to doing percent increase/decrease? I feel like it is such a simple topic but for some reason, I struggle to figure out the answers to these types of problems because I plug in the wrong numbers to the formulas. And any general advice as to how to tackle these problems would be awesome!

Thanks!

Posts

  • edited Thu Aug 24, 2017 11:12 am
    Jonathan EvansJonathan Evans PowerScore Staff Joined: 10/31/2016Posts: 97
    Diane, 

    Great question about a topic that many students struggle with. In the same manner as we try to simplify combinatorics and probability by stripping away some of the accouterments of formal math, percent increase and decrease also benefits from a simplified, systematic approach. However, as I'm sure you know by now, I'm not a fan of just telling students to plug and chug numbers through inscrutable formulas because it's also important to understand the underlying concept to ensure proper application on relevant problems.

    With these goals of simplifying the topic but also improving understanding in mind, let's tackle percent change first conceptually and then in an applied manner. 

    What is percent change?

    Percent change is a measurement of a change in values, going from an initial ("old") value to a final ("new") value. The result of a percent change calculation gives the amount by which we have changed this old value expressed as a fraction, the percent of the old value we changed to arrive at the new value.

    This might sound complicated, so let's use an example to illustrate.

    Say we started with 50 ding-dongs and went to the convenience store to buy 25 more. By what percent did the number of ding-dongs we have increase, assuming we ate no ding-dongs?

    Solution: 

    Start with the initial value of 50 ding-dongs. Now consider the final value, 75 ding-dongs. How many more ding-dongs do we have now? 25 more. This value of 25 is the change in ding-dongs. It is the actual number. We need to convert this actual number into a percent relative to the initial value of 50. Why? Because that's what we started with!

    Express the change in number of ding-dongs (25) as the dividend/numerator of our fraction (this is the part that changed). 

    Express the initial number of ding-dongs (50) as the divisor/denominator of our fraction (this is the whole that we began with). 

    25 ÷ 50 = ½

    Well, ½ isn't a percent yet. How do we express the fraction ½ as a percent? Multiply by 100!

    ½ = 0.50 

    0.50 x 100 = 50

    Thus we have a percent increase of 50%.

    How do we know it's a percent increase (as opposed to percent decrease)? We know this because our initial/old number was smaller. That value went up to reach our final/new number. 

    Important observations

    From this example, let's move on to a couple general rules:
    1. The formula to use for percent change is: (Difference in Numbers)/(Initial Number) x 100
    2. The initial number is the one you begin with. In the event that it's percent increase, the initial number is always smaller. In the event that it's percent decrease, the initial number is always bigger.
    3. If there is more that one change in values, you must "reset" your calculations each time, changing initial and final values.

    Let's pause and consider point 3 for a moment: If there is more than one change in values, you must "reset" your calculations each time, changing initial and final values.

    What does this mean in practice? Let's return to our ding-dong example.

    We started out with 50 ding-dongs. We went to the store and bought 25 more. Then we ate 15 ding dongs. By what percent did our ding-dongs decrease after our purchase? 

    Solution: 

    Now we ask, is our initial value 50 ding-dongs or 75 ding-dongs? Read exactly what the question asks. Our initial value for the purpose of this percent change is the number of ding-dongs after our purchase (75). Thus, the formula is:

    (Difference in Numbers)/(Initial Number) x 100

    15 ÷ 75 = 1/5 = 0.20 = 20%

    Notice we now had a percent decrease. The initial number (75) was bigger than the final number (60). We had a 20% decrease in ding-dongs.

    Put it all together

    The GRE will sometimes combine all the above concepts into a longer, multi-step question. Let's take a look at how this might appear on the GRE:

    Josie had 50 ding-dongs and went to the convenience store to buy 25 more. She subsequently went on a ding-dong eating bonanza and consumed 30 ding-dongs before being rushed to the emergency room. When all was said and done, what was the percent change in Josie's ding-dongs relative to what she started out with?

    Solution:

    Start with her 50 ding-dongs.

    Now we go up to 75.

    Now we go back down to 45. 

    We have to decide what our initial and final values are. The initial value according the the question is what she started out with, 50. The final value is what she ended up with, 45. The difference in these numbers is 5. Let's use our formula:

    5 ÷ 50 = 1/10 = 0.10 = 10%

    I hope this helps!

     
  • fontainedfontained Joined: 07/22/2017Posts: 23
    Thank you so much!

    Helps a lot :)
  • fontainedfontained Joined: 07/22/2017Posts: 23
    Actually, just looking over it again...for the second ding dong example, where did you calculate the difference between 75 and 15? I think my confusion stems from the #2 of the important observations. When you say that  "In the event that it's percent increase, the initial number is always smaller. In the event that it's percent decrease, the initial number is always bigger" what are you comparing the initial numbers to? For example, in the 2nd ding dong example, it is a percent decrease, so in my mind, I want to do the (difference)/inital number and since it's % decrease, I want to use the bigger number (75) so I do 60/75*100 which is not right (80%) and not 20%. Sorry my I can't wrap my brain around this!!
  • fontainedfontained Joined: 07/22/2017Posts: 23
    Oh wait! I got it----I just didn't do the math right. Since we ate 15 ding dongs, we are then left with 60 instead of 75, thus the difference is 75-60 = 15 and divide that by the BIGGER number (75) to get 20%!! Okay, sorry for all the posts!
  • Jonathan EvansJonathan Evans PowerScore Staff Joined: 10/31/2016Posts: 97
    Not at all! Great job going through the process here. This is exactly the kind of "small" stuff that we have to very clear on with percent change calculations, that is, we have to make extra sure that we're using the correct values in our calculations.
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