Reorient Your View, Redux
A couple of months ago, we talked about Reorienting Your View on quant problems. (If you havent yet read the first two articles, feel free to do so right now.)
The test writers will often try to imply (or outright set you up for) a certain solution path. I probably dont need to tell you that this solution path is rarely the best way to solve the problem! It makes sense that they would set us up this way: business schools are interested in finding people who think for themselves and dont just follow the path that theyre given.
How have you been doing with this? Have you gotten better at reorienting your view?
Try this GMATPrep problem to see.
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
What did you get? How confident are you in your answer? How much time did you spend?
Alert: its fine not to be 100% confident in your answer. This is quite common. Its NOT fine to have spent extra time on a problem when youre not fairly confident in the answer. Lets face it: you already knew before you started using that extra time that you didnt really know what you were doing. So cut yourself off faster! Dont waste time.
Okay, back to our regularly-scheduled lesson. How did you set this one up? The problem is a classic double ratio problem: they start us off with one ratio, then change the scenario and give us the second, new ratio.
People who know how to do this problem will typically set it up in the following way. (Notice my language choice I didnt say heres how to do it!)
The initial ratio for K : A is 5 : 3. Call the unknown multiplier x. Kaye, then, had 5x stamps and Alberto had 3x stamps.
Next, Kaye gives Alberto 10 stamps, so now she has only [pmath]5x-10[/pmath] stamps. Alberto, on the other hand, gained stamps, so he now has [pmath]3x + 10[/pmath] stamps. These two expressions represent the new numbers, so they correspond to the new ratio of 7 : 5. Make an equation and solve for x:
[pmath]{5x-10}/{3x+10}=7/5[/pmath]
[pmath]25x-50 = 21x + 70[/pmath]
[pmath]4x = 120[/pmath]
[pmath]x = 30[/pmath]
The unknown multiplier is 30, so initially Kaye had 5(30) = 150 stamps and Alberto had 3(30) = 90 stamps. After Kaye gives Alberto 10 stamps, she has 140 and he has 100, for a difference of 40 stamps.
The correct answer is (C).
This method certainly workswe got the right answer. But we fell into the standard mindset, so we did more work (and used up more time and mental energy) than necessary. Take a look at that problem again: how could you save time and energy?
You solved for the first ratio first but you dont actually care what the first set of numbers is because they asked for the difference from the second set of numbers. Can you solve directly for that?
Sure! Set up the initial equation to solve for the unknown multiplier of the second ratio, not the first. Nothing says that you have to solve for the first ratio!
Heres how that math would work:
Kaye has 7x stamps (after the exchange) and Alberto has 5x. Reverse the exchange: Kaye started with [pmath]7x + 10[/pmath] stamps and Alberto started with [pmath]5x-10[/pmath]. Set up the equation with the first ratio:
[pmath]{7x+10}/{5x-10}=5/3[/pmath]
[pmath]21x + 30 = 25x - 50[/pmath]
[pmath]80 = 4x[/pmath]
[pmath]x = 20[/pmath]
The unknown multiplier for the second ratio is 20. Kaye, then, had 20(7) = 140 stamps after the exchange and Alberto had 20(5) = 100, for a difference of 40 stamps.
But wait! Theres another shortcut here. Go back to the fact that the unknown multiplier is 20. The problem asks for the difference between Kaye and Alberto. Instead of solving for Kaye and Alberto individually (and then subtracting), solve directly for that difference. The difference in a 7 : 5 ratio is 2 (because 7 5 = 2). The unknown multiplier is 20. So the difference between K and A is 20(2) = 40.
The correct answer isstill!(C).
How do you learn how to do this right from the start? In the first place, a direct solution method usually exists, so look for it. Try to solve as directly as possible for whatever the problem asks; be suspicious if the first path that jumps out involves solving for something else first.
If, after 10 or 20 seconds of brainstorming, you cant think of that more direct path, then go ahead with the longer method of solving for something else first. Remember: while the clock is ticking, react in the way that youd want to react if you were taking the real test.
Afterwards, take all the time you want to try to figure out a more direct solution method. If you cant find anything, look online (in our OG Archer program, for example, or on the forums) for alternate solution methods. (Do try to figure it out for yourself first, thoughyoull be much better able to recall the more efficient solution method if you are able to figure it out yourself.)
Take-aways for Reorienting Your View:
(1) If a problem seems to imply a certain path, be skeptical. After all, the test writers arent in the business of helping you get a better score on the test! Take a step back and choose an approach based on your own knowledge and strengths.
(2) Whenever possible in multi-variable problems, try to solve directly for whatever the problem asks. You cant always do this; sometimes, you do have to solve for y before you can find x. But if you can avoid that extra step, youll save time and youll also save yourself some opportunities to make mistakes. For instance, Kaye and Alberto were initially 60 stamps apartand that answer is in the answer choices.
(3) If the problem asks for a combination of variables (combo), see whether you can solve directly for that combo. In this case, the problem asked for K A, so we saved a little time and effort in the last step by solving directly for that difference.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.