First, here are eight practice problems exploring typical GMAT percent themes.
1. The original price of a suit is $200. The price increased 30%, and after this increase, the store published a 30% off coupon for a oneday sale. Given that the consumers who used the coupon on sale day were getting 30% off the increased price, how much did these consumers pay for the suit?

(A) $182
(B) $191
(C) $200
(D) $209
(E) $219
2. The profits of QRS company rose 10% from March to April, then dropped 20% from April to May, then rose 50% from May to June. What was the percent increase for the whole quarter, from March to June?

(A) 15%
(B) 32%
(C) 40%
(D) 62%
(E) 80%
3. Bert and Rebecca were looking at the price of a condominium. The price of the condominium was 80% more than Bert had in savings, and separately, the same price was also 20% more than Rebecca had in savings. What is the ratio of what Bert has in savings to what Rebecca has in savings.

(A) 1:4
(B) 4:1
(C) 2:3
(D) 3:2
(E) 3:4
4. Company KW is being sold, and both Company A and Company B were considering the purchase. The price of Company KW is 50% more than Company A has in assets, and this same price is also 100% more than Company B has in assets. If Companies A and B were to merge and combine their assets, the price of Company KW would be approximately what percent of these combined assets?

(A) 66%
(B) 75%
(C) 86%
(D) 116%
(E) 150%
5. There are 300 seniors at Morse High School, and 40% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 15% of all the students at Morse have cars, how many students are in those other three lower grades?

(A) 600
(B) 900
(C) 1200
(D) 1350
(E) 1500
6. A scientific research study examined a large number of young foxes, that is, foxes between 1 year and 2 years old. The study found that 80% of the young foxes caught a rabbit at least once, and 60% caught a songbird at least once If 10% of the young foxes never caught either a rabbit or a songbird, then what percentage of young foxes were successful in catching at least one rabbit and at least one songbird?

(A) 40%
(B) 50%
(C) 60%
(D) 80%
(E) 90%
7. A book store bought copies of a new book by a popular author, in anticipation of robust sales. The store bought 400 copies from their supplier, each copy at wholesale price W. The store sold the first 150 copies in the first week at 80% more than W, and then over the next month, sold a 100 more at 20% more than W. Finally, to clear shelf space, the store sold the remaining copies to a bargain retailer at 40% less than W. What was the bookstore’s net percent profit or loss on the entire lot of 400 books?

(A) 30% loss
(B) 10% loss
(C) 10% profit
(D) 20% profit
(E) 60% profit
8. At a certain symphonic concert, tickets for the orchestra level were $50 and tickets for the balcony level were $30. These two ticket types were the only source of revenue for this concert. If R% of the revenue for the concert was from the sale of balcony tickets, and B% of the tickets sold were balcony tickets, then which of the following expresses B in terms of R?
Solutions will be given at the end of this article.
Thoughts on percents
First of all, here are some previous blogs on percents:
(1) Fundamentals of percents, including using multipliers for percent increase and decreases.
(2) Solution of a percent with variables problem, from the OG
(3) Another solution of percent with variables problem, from the OG
(4) Scale factors & percent change
(5) Solution & Mixing problems
Especially in those first three, you can find some useful hints for the foregoing problems.
The BIG percent mistake
Folks who are rusty at math and/or returning to it after a long absence may not appreciate something known well to anyone who writes math problems: certain mistakes are as predictable as clockwork. Anyone who writes problems knows — in suchandsuch kind of problem, the vast majority of testtakers will make suchandsuch very predictable mistake. Of course, on any multiple choice standardized test, that predictable mistake consistently will be among the answer choices: it’s as if the testmaker sets up a huge butterfly net, and the unwitting testtakers run into this trap like lemmings running to the sea. I know this sound cruel, but the purpose of a good standardized test question is to distinguish those who know their stuff from those who don’t, and highly predictable errors are great ways to draw such a distinction very clearly.
Obviously, as a testtaker, it is very much to your advantage to learn to spot these very predicable mistake patterns. Just avoiding these will put you well ahead of the pack. In many different articles on this blog, I discuss these predictable mistakes.
The big predictable mistake with percents has a few variations on the same pattern.
(a) if something increases by P%, then decreases by Q%, the net change is not (P – Q)%
(b) if something increases by P%, then increases by Q%, that’s not an increase of (P + Q)%
(c) [special case] if something increase by P%, then decrease by P%, we do not return to the original value.
(d) [more general case] if something increase by P%, then decreases by Q%, then increase by R%, the total change in percent is not (P – Q + R)%
What all of these have in common is a deep root error — you cannot figure out the total percent change of a series of individual percent changes by adding or subtracting the individual percents. That’s the mistake. People see “start at $100, 40% increase, followed by a 40% decrease“, and scores upon score of people, as predictably as the sunrise, will believe and insist without a shadow of a doubt that the end result must be $100. This large crowd will be in unison and they will be wrong.
What’s the correct thing to do? The correct way to treat any series of percent changes is to express each change as a multiplier, and then multiply all the multipliers. That first blog above talks about creating the multipliers for percent increases & decreases. Once we have that, we can figure out the real change. For example, the multiplier for a 40% increase would be: 1 + 0.40 = 1.4, and the multiplier for a 40% decrease would be 1 – 0.40 = 0.6; now, 1.4*0.6 = = 0.84, so the final price in fact would be $84, which is a 16% decrease.
Summary
If the foregoing discussion gave you any insights, you may want to reread the practice problems before reading the solution. Here’s another practice problem from the Magoosh Product:
9. http://gmat.magoosh.com/questions/2584
If you have any observations or questions, please let us know in the comments section.
Practice problem explanations
1) Given the foregoing discussion, it may be obvious now the trapmistake answer is (C). Even if you can’t remember the correct thing to do, at the very least, learn to spot the trap!
The multiplier for a 30% increases is 1 + 0.30 = 1.3, and the multiplier for a 30% decrease is 1 – 0.30 = 0.70, so the combined change is 1.3*0.7 = 0.91, 91% percent of the original, or a 9% decreases. Now, multiply $200*0.91 = $182. Answer = (A).
2) Given the foregoing discussion, it may be obvious now the trapmistake answer is (C), which results from simply adding and subtracting the percents. We need multipliers.
multiplier for a 10% increases = 1 + 0.10 = 1.1
multiplier for a 20% decreases = 1 – 0.20 = 0.8
multiplier for a 50% increases = 1 + 0.50 = 1.5
Now, multiply these. First, multiply (0.8) and (1.5), using the doubling & halving trick. Half of 0.80 is 0.40, and twice 1.5 is 3
(0.8)*(1.5) = (0.4)*(3) = 1.2
Now, multiply this by 1.1
1.2*1.1 = 1.32
Thus, the three percent changes combined produce a 32% increase. Answer = (B).
3) The trap answer here would be to take the ratio of 80% and 20% — those don’t represent actually amounts that other person has, just the differences between amounts owned and the cost of the condo. Think of this in terms of multipliers. Use the variables:
B = amount Bert has in savings
R = amount Rebecca has in savings
P = price of the condominium
Then in terms of multipliers, the information given tells us that P = 1.8*B, and P = 1.2*R. Set these equal.
1.8*B = 1.2*R
Answer = (C)
4) There are a few ways to solve this. This is plugin approach. Suppose Company A has $100 in assets. (Yes, unrealistic, but a convenient choice.) Then Company KW is being sold for 50% more = $150. Now, this $150 is 100% more than what company B has in assets — i.e., $150 is double what company B has in assets, so company B has $75 in assets. Now, suppose companies A & B pool their resources — together, they have $175 in assets.
Notice, first of all, combined they have more in assets than the cost of KW, the price of KW would be a percent less than 100%. Even if nothing else, we could eliminate (D) & (E), and practice solution behavior.
Part = price of KW = $150
Whole = combined assets = $175
Answer = (C)
5) Let x = the number of students other than seniors (freshmen + sophomores + junior). We know 40% of the 300 seniors have cars. Well, 10% of 300 is 30, so 40% is 4 times this — 4*30 = 120 seniors have cars. We know 10% of the other students have cars, so that would be 0.1*x. The total number of students with cars is 120 + 0.1x. That’s the PART.
The total number of students = 300 + x. That’s the WHOLE.
PART/ WHOLE x 100% = 15%, which means that PART/ WHOLE = 0.15, which means PART = 0.15*(WHOLE). That can be our equation.
(120 + 0.1x) = 0.15(300 + x)
120 + 0.1x = 45 + 0.15x
75 + 0.1x = 0.15x
75 = 0.15x – 0.10x = 0.05x
150 = 0.10x
1500 = x
Answer = (E)
6) This is less about percents and more about probability, particularly the probability ORrule. Let R = the event that a young fox catches at least one rabbit, and let S = the event that a young fox catches at least one songbird. Using algebraic probability notation, we know P(R) = 0.8 and P(S) = 0.6. We know P((not R) and (not S)) = 0.1, and the complement of [(not R) and (not S)] would be [R or S], so by the complement rule, P(R or S) = 1 – 0.1 = 0.9. The question is asking for P(R and S). The OR rule tells us
P(R or S) = P(R) + P(S) – P(R and S)
0.9 = 0.6 + 0.8 – P(R and S)
0.9 = 1.4 – P(R and S)
0.9 + P(R and S) = 1.4
P(R and S) = 0.5
Answer = (B)
7) First of all, amount paid = 400*W. That was the bookstore’s total expenditures. The total revenue came in three stages
150 copies @ 80% more than W = 150*1.8W = 300*.9W = 270W
100 copies @ 20% more than W = 100*1.2W = 120W
150 copies @ 40% less than W = 150*0.6W = 300*0.3W = 90W
Notice, all the percent changes were converted to multipliers. Also, notice the use of the doubling & halving trick in the first and third lines.
Total revenue = 270W + 120W + 90W = 480W
Well, the store took in more revenue than they spent, so they made a profit, not a loss. Notice that 10% more than 400 would be 440, so 480 would be a 20% increase.
Answer = (D)
8) There are a few different methods of solution. I will show a numerical approach. Let’s try some simple cases — suppose they sold nothing but balcony tickets: then B = 100 and R = 100. If we plug in R = 100 …
Right away, with one choice, we know that (A) and (C) are out. At this point, even if we could do nothing else, we could still use solution behavior.
Now, suppose the revenue they took in was half and half, so that R = 50%. Well, the prices of balcony to orchestra tickets are in a ratio of 3:5, so in order for the revenue from balcony tickets to equal the revenue from orchestra tickets, they would have to be sold in the reciprocal ratio of 5:3 (if you think about it, this just uses the fact that A*B = B*A). This means that 5/8 of the tickets sold would be balcony tickets. (For this logic, see the Ratio blog for information on portioning.) Thus, if we plug in R = 50, we should get B = (5/8)*100 — notice, we don’t actually need to calculate that out: the fraction is fine. In the calculations below, notice the use of the doubling & halving trick in the denominator, to get the factor of 100.
Backsolving gets us to answer (D) very efficient.
Now, here’s an algebraic approach, which is longer, but some folks want to see this anyway for the algebra practice:
Let N = the total number of tickets sold. Then:
Therefore,
In the big fraction, in each term: cancel the N’s, cancel a factor of 10, and cancel the “divided by 100”:
Multiply both sides by the denominator.
R*(500 – 2B) = (3B)*100
500R – 2RB = 300B
Get all the B’s on one side
500R = 2RB + 300B
B*(2R + 300) = 500R
Answer = (D)
That’s the fullblown algebraic solution, although why anyone would want to slog through all that instead of using backsolving is beyond me!!
In the second practice question why didn’t we take the 1.1 and decreased it by 20%? instead we subtracted 0.2 from 1
Actually you’re correct, Wael. For Step 2, you could simply decrease 1.1 by 20%. This would be calculated as 1.1 – (1.1*0.2), since (1.1*0.2) is 20% of 1.1. The result of 1.1 – (1.1 * 0.2) is 0.88. You could then multiply 0.88 by the final multiplier, thus getting the correct answer of 1.32. However, this approach is a lot harder to do without a calculator. It’s fairly challenging to multiply 1.1 by 0.2 in your head. And to roll that tricky mental calculation into a subtraction problem is even harder. For the most efficient mental math on the “no calculator” GMAT Quants section, it really is easier to multiply .8 by 1.5 with doubling and having, and then mentally multiply the result– 1.2– by 1.1. This will also being you to the correct result of 1.32.
For #8, I was under the impression that B% = number of balcony tickets / total number of tickets sold. This was how it was stated in the question. However, in the solution I see that B is referred simply to as the number of balcony tickets sold and not percent of balcony tickets sold out of total number of tickets sold.
Was there a mistake in the question? Thanks for clearing this up.
Dear Jenny:
I’m happy to respond. 🙂 I think you might be getting a little confused about the closely related concepts of a ratio and a percent. The ratio of balcony tickets is (number of balcony tickets)/(total number of tickets). The percent of balcony tickets is that ratio times 100. If we sold 250 balcony tickets and 250 orchestra tickets, then the ratio would be 1/2 and the percent would be 50%—in other words, B = 50.
In the solution, B is NOT a number of tickets. B is a percent. When I say B = 100, I don’t mean that 100 balcony tickets were sold. I mean that, of all the tickets sold, however many there were, every single one of them was a balcony ticket. If 831 Balcony tickets were sold, and zero Orchestra tickets, then B = 100.
Does all this make sense?
Mike 🙂
Hi Mike,
Thanks for the reply.
If B is a percent, more specifically the percent of balcony tickets sold, shouldn’t # of balcony tickets sold be N*B, where N is total number of tickets sold?
In the first equation in the algebraic answer, it says N*(B/100). I’m confused because if B is already a %, why is the % divided by 100?
Thank you!
Dear Jenny,
Great question! 🙂 This is subtle and tricky stuff. The question says: “… B% of the tickets sold were balcony tickets …” What exactly does this mean? Well, say, if B = 75, then 75% of the ticket sold were balcony tickets. So B is the number that goes into the percent, but it is not identical to the percent. It’s the difference between 75 and 75% — B equals the former, and not the latter. When we want 75% of something, we can multiply by 75%, but we can’t multiply by 75, which is what B equals. We would have to multiply by 75/100, or B/100.
You may remember that the percent sign, %, originated as an abbreviation for “divided by 100.” Fundamentally, that’s what percent really means. Even the word, “percent,” comes from the Latin per centum, that is, “per 100.” We put the number of the percent over 100 to change any percent into a fraction, and for algebraic purposes, it’s fractions that we need to perform calculations.
Does all this make sense?
Mike 🙂
Thanks Mike! Everything clear now. :)!
Hi Mike
I would Q do Q8 as follows:
Total revenue= B/100 * $30 + (100% _ B/100) * $50
= 3B/10 + 50 (1 – B/100)
= 3B/10 + (500 – 5B)/10
Total revenue = (500 – 2B)/ 10;
balcony revenue is R/100 of total revenue,
Also, balcony revenue is B/100 *$30
B/100 *$30 = R/100 * (500 2B)/ 10
Can easily solve for B
B = 500R / ( 300 +2R)…….choice D
Hi Mike
For Q1 and Q2 types, I think the solutions you are providing will not work, given the 2 minute time limit. I propose a short cut method for % increase/decrease problems.
If x% is the initial increase / decrease and y% is next increase/ decrease, then total % change = x+y + ( x*y)/100.
In Q1. $200 original price, 30% increase and then 30% off
30 + (30) + 30 * 30…..negative sign for decrease and + for increase
0 – 9 % = 9% decrease ( as sign is negative)
$200 * 9% = $182 final price
Q2. Initial 10% increase, then 20% decrease, and then 50% increase
1020 + 10 * 20= 12%
12 +50 + 12 * 50
=+38 6
=+32% increase overall
Hope this helps!
$200 * 9% is not $182.. it’s 180? I do not understand your reasoning, can someone please let me know what I did wrong.
Hi Shanell,
Happy to help! 🙂
$180 would be a 10% decrease ($200 * 0.9 = $180). Instead, we are talking about a 9% decrease ($200 * 0.91 = $182). If you want to figure out where your method led you astray, can you please show me your process? 🙂
Hope this helps!
Hey Mike, can number 6 be solved using a table or a ven diagram? I was able to using a table, but wanted to make sure if that is the right approach? Appreciate your work btw! 🙂
Samson,
Yes, my friend, you certainly could use a table or venn diagram for that. It’s always good if you can discover more than one way to solve a problem. I’m glad you have been finding this blog helpful. Best of luck to you!
Mike 🙂
Hi, Mike,
I read the article at http://magoosh.com/gmat/2013/gmatquantitativeratioandproportions/ and I still have a question regarding the methodology used to obtain the ratio in question 8.
When R = 50%, how do you get the ratio of 5:3 for balcony to balcony to orchestra? I can sort of look and see that sales will be 50/50 after each ticket has sold $150 and then obtain the ratio from there, but I am having problems mathematically since were dealing with the revenue ratio and ticket sales ratio at the same time. This poses a problem for me when I play with the problem and attempt to use different revenue ratios.
For example, how would you determine the ticket ratio if you wanted to test R = 25% or R = 10%?
Thank you for your time,
Nick
Nick,
I rewrote the text of the QE a little.
Think about it. Balcony tickets cost $30 and orchestra tickets cost $50, a 3/5 ratio. To undo a 3/5 ratio, we multiply by 5/3, and indeed, if we buy either 5 balcony tickets or 3 orchestra tickets, we pay the same price. That’s the ratio that equalizes the revenue.
Suppose we wanted R = 25%, which means (revenue from balcony tickets) = 3*(revenue from orchestra tickets). Well, go back to the 5 B and 3 O example, and just triple the latter — 5 B and 9 O will produce the desire ratio of revenue. That means 5/14 of ticket sold were balcony tickets.
Does all this make sense?
Mike 🙂
Thank you, your reply helped a lot!
Dear Nick,
You are quite welcome, my friend. 🙂 Best of luck to you!
Mike 🙂
I have a question about #8 problem.
in order for the revenue from balcony tickets to equal the revenue from orchestra tickets, they would have to be sold in a ratio of 5:3, which means that 5/8 of the tickets sold would be balcony tickets.
How do we make this conclusion? “which means that 5/8 of the tickets sold would be balcony tickets.”
I guess this is a ration problem but I’m having a hard time understand this. Thank you.
Dear Bylitta,
That’s a great question. Please see this blog:
http://magoosh.com/gmat/2013/gmatquantitativeratioandproportions/
the section on “portioning”, for the answer.
Mike 🙂
I completely understand now! This is amazing thank you so much!
Dear bylitta,
You are more than welcome, my friend. 🙂 Best of luck to you!
Mike 🙂
Why isn’t the answer to question (3) 3:2? B:R is ratio 180%:120% to what they both have not 120%:180%.
Oh never mind! Got it!
Hello Mike,
Can you please provide some more medium to hard questions (question sources or links) similar to the last question above in which backsolving method can be used. Although I have understood the method, I need some more questions for practice.
Thank You…
Dear Shuvabrata,
Check out this blog article
http://magoosh.com/gmat/2013/backsolvingongmatmath/
Mike 🙂
Can you please let me know what difficulty level do the above problems fall in?? I got all of them correct…. I just to see how much I should expect in the Math Section.
Dear Tango,
Well, it’s always hard to estimate difficulty, but none of them is especially easy, and the last in particular is challenging. I would say at least a few of them are 700 level. If solving all eight of these was no problem for you, then you certainly are in good shape for the GMAT Q section, at least on the topics of percents.
Good job!
Mike 🙂
Hey Mike. How can number 4 be done without the plug in approach? I am trying to make sense of the concepts of the percent change formulas conceptually.
Dear Barb,
Without plugin, this problem becomes a sea of algebra. Here’s how I would do it algebraically. Let P be the price of KW, let A = A’s assets, and B = B’s assets. We know P = 1.5*A and P = 2B, which tells us that 1.5A = 2B or 0.75A = B. Therefore, combined assets A + B = A + 0,75A = 1.75*A, and so now the question is: P is what percent of 1.75*A? In other words, 1.5A is what percent of 1.75*A, or 1.5 is what percent of 1.75? We can double both — 3 is what percent of 3.5? Double again — 6 is what percent of 7? Well, that’s close to 100%, just less than that. We know 1/7 is about 14%, so 6/7 would be about 100 – 14 = 86 percent.
There’s no way to get to the answer without somehow wrestling a bit with the number. Personally, I think the plug in approach is easier than this algebraic approach, but you need to something like this — set the numbers in motion and follow their consequences. Without doing that in some form, I don’t think you can intuit the answer.
Does this make sense?
Mike 🙂
Hi Mike! Thank you so much for posting these. They are very helpful. Why is it that the answer for number 3 is 2:3. When I worked it out I got 3:2 since it is B:R. I am not sure if I am missing a concept here which is causing me to do this backwards?
Dear Tiffany,
Hmmm. I would have to see your steps to know where things went awry. I will say: just think about this conceptually. The price is 80% more than Bert has — in other words, it’s almost double what Bert has, and Bert has only a little more than half the price. By contrast, the same price is 20% more than Rebecca has — in other words, she has almost the whole price all by herself. Bert has barely more than 1/2 the price, and Rebecca has almost the whole price by herself, so Rebecca must have MORE than Bert. That means, in a B:R ratio, the R number must be bigger than the B number. Check all your work to make sure that, at each step, it makes sense that R > B. In fact, since we know the answer is 2:3, plug in B = 2 and R = 3 in each link of your work, and that will help you find the mistake.
Does all this make sense?
Mike 🙂
hello,
I don’t understand why in the #7 the result is not the same as 1.8*1.2*0.6, just multiplying the multipliers?
Thank you
Dear Titouprince,
We would simply multiply the multipliers if a *single item* went up 80%, then up 20%, and then down 40%. Here, we are talking about different items — in fact, unequal batches of books. The 150 books that were sold at 80% more than W were sold and gone by the time they sold the next batch of 100 at 40% more than W. Multiplying together the multipliers is good when we want the cumulative effect of all the percent changes on a single thing. Here, the different percents are happening to different batches of books — nothing is accumulating in any one place. Does this make sense?
Mike 🙂
Thanks a lot!
You’re quite welcome! Best of luck to you!
Mike 🙂
Hi Chris, I think there is a spelling error in question 6:
” If R% of the revenue for the concert was from the sale of balcony tickets, and B% of the tickets sold were balcony tickets”
Dennis,
This is Mike, the author of this blog. A spelling error in #8? I’m sorry, but I don’t follow. Which word do you believe is misspelled?
Mike 🙂
Hi Mike,
I am under the impression that solution given for problem 6 is right.Could please explain why it is wrong .This is based on the comment by Musa “P((not R) and (not S)) = 0.1, however this must be 0,20. So the right answer should be 0,60 nat 0,50 “
Dear Satish,
The solution for #6 right now is indeed correct. When I first published this blog, at the end of September, problem #6 had a typo in it — one number in the question was different from what I intended, and consequently, the problem and solution did not match. A couple weeks ago, Musa made a comment pointing this out, and at that time, I corrected the typo, to make the problem & solution are perfectly consistent. This is the version you now see. Does this make sense?
Mike 🙂
Hi Mike,
Can you show algebraic approach to Prob 8.
Dear GMAT4o,
Well, I added the algebraic solution to #8 as part of the solutions listed above. I certainly could not do all that in under 5 minutes, and I doubt anyone short of Will Hunting could. I hope it helps to see it for algebra practice, but solving with numerical plugins & backsolving is definitely more efficient.
Mike 🙂
Thanks Mike,
I used the back solving method but was somehow getting numbers and constants mixed up with algebraic approach. this as just to understand where was i going wrong.
Regards
You’re welcome. I’m glad you found it helpful. That 8th problem is a particularly tricky one. Best of luck to you.
Mike 🙂
Hi, I think the answer for the question 6 is wrong. In the solution iti is stated that: P((not R) and (not S)) = 0.1, however this must be 0,20. So the right answer should be 0,60 nat 0,50
Dear Musa,
You’re perfectly correct. Thank you very much for pointing this out. 🙂 In the process of writing this blog, I decided to change the problem slightly, and thought I had consistently changed everything. I meant for the doublydeprived fox to be 10%, not 20% of the population, so I just changed that in the question, to make it consistent with the solution. Good eye for detail — that skill will serve you well on GMAT Quant! 🙂 Thanks again!
Mike 🙂
Hey Mike for Q 6 whats the right answer I am getting 0.6 too
Is it right?
0.7 + 0.5 +0.1 – intersection= 1
Intersection equals = 0.3
We want A or B event that give 0.7 0.3 =0.4 and 0.50.3 =0.2
Hence 0.6
Dear Son,
Notice that all the answers and explanations are given on the page. Please read the OE of #6, and let me know if you have any further questions.
Mike 🙂