GMATPrep Problem Solving: Kaye’s Stamps
Some people really like ratio problems while others struggle with these.What do you think?
Lets talk about a go-to solution method when handling a problem of this type. Try this GMATPrep problem:
* The number of stamps that Kaye and Alberto had were in the ratio 5 : 3, respectively. After Kaye gave Alberto 10 of her stamps, the ratio of the number Kaye had to the number Alberto had was 7 : 5. As a result of this gift, Kaye had how many more stamps than Alberto?(A) 20
(B) 30
(C) 40
(D) 60
(E) 90
My very first thought as I read this problem: I have to be very careful with my work here, because it would be really easy to solve for the wrong thing (and, of course, that wrong answer will probably be among the answer choices).
As an aside, Ive found that this attitude is one of the biggest differences between someone who has the potential to hit a top score on quant and someone who wont make it. When you see something and you think, I know how to do this! the top test-taker is going to go in The Zone and pay even more attention to detail, thinking I am going to be really careful not to make a mistake on this one! Someone who isnt going to hit a tip-top score will instead start to coast a little mentally, thinking, Yeah, Ive already got this. Even worse, someone might think, I can speed up on this one since I know how to do it.
No! Dont speed up! You dont necessarily have to take the full 2 minutes, but dont go any faster than youd normally go. Dont increase the chances that you make a careless mistake!
Okay, lets solve this thing.
First, make very clear on your scrap paper what you want: Kaye NEW minus Alberto NEW. Not just Kaye (new or old). Not Kayes original number of stamps minus Albertos original number.
Skip a few lines and write this on the scrap paper and put a big circle around it: [pmath]K_n - A_n[/pmath]. Do the actual work up above this text and, when you're done, youll run into the reminder that you want Kaye NEW minus Alberto NEW.
Also, make sure you organize your work carefully as you go so that you know which portions represent the original numbers versus the new ones.
Lets see.
[pmath]K_o : A_o[/pmath]= 5 : 3
[pmath]K_n : A_n[/pmath]= 7 : 5
And then theres that info about Kaye giving 10 stamps to Alberto. How do we put that all together?
Heres cool trick #1. The ratio can also be written to reflect the real number of stamps using the unknown multiplier. If we call the unknown multiplier x, then Kayes original number of stamps is 5x. Because the unknown multiplier is the same within any ratio, Albertos original number of stamps is 3x.
Okay. What happens next? Kaye gives Alberto 10 stamps. Now, Kaye has 5x 10 stamps, and Alberto has 3x + 10 stamps.
That brings us to cool trick #2. The ratio of these two NEW numbers of stamps equals 7 : 5. We can actually write an equation:
[pmath]{5x-10}/{3x+10}=7/5[/pmath]
Theres only one variablesolve for x!
(5x 10)(5) = (3x + 10)(7)
25x 50 = 21x + 70
4x = 120
x = 30
Glance at the answers. This is NOT what we want! Dont pick answer (B).
Weve found the unknown multiplier, not the new number of stamps for each person. Go figure that out now.
Also, save yourself a little time: remember what your goal is. I dont care what the two people had originally, so Im not going to solve for those numbers. I'm going to solve directly for the two new numbers.
[pmath]K_n[/pmath]= 5x 10 = 5(30) 10 = 140
[pmath]A_n[/pmath]= 3x + 10 = 3(30) + 10 = 100
The difference between those two numbers is 40. The correct answer is (C).
Interestingly, the difference between the two original numbers is 60 (if you figure that out); if you figure that out, you might mistakenly pick it. You actually save yourself from making that mistake by not doing that math. Answer (D) is a trap! Also, answer (E) matches Albertos original number of stamps.
There are certainly many examples of conceptually harder quant problems on this test. The key takeaway on this problem is finding the most efficient, effective path through the math and being systematic so that you don't make a careless mistake. Why use more brain energy than you have to?
Key Takeaways for Changing Ratio Story Problems
(1) You can make connections between the ratios and the real numbers by introducing the unknown multiplier into the problem.
(2) If the problem also gives you a real number (in this problem, they gave us 10), then you can use the unknown multiplier plus that real number plus the new ratio to make an equation. This equation will have just one variable (since the unknown multiplier, by definition, is always the same within any one ratio).
(3) Remember your goal. Try to solve as directly as possible for whatever the problem asks; avoid unnecessary math!
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.