Breaking Down A GMATPrep Fractions Word Problem
Recently, we took a look at how to translate various word problems into math. Todays problem expands on that lesson, using a particular GMATPrep problem that also tests us in the area of fractions.
Set your timer for 2 minutes. and GO!
*Alices take-home pay last year was the same each month, and she saved the same fraction of her take-home pay each month. The total amount of money that she had saved at the end of the year was 3 times the amount of that portion of her monthly take-home pay that she did not save. If all the money that she saved last year was from her take-home pay, what fraction of her take-home pay did she save each month?(A) [pmath]1/2[/pmath]
(B) [pmath]1/3[/pmath]
(C) [pmath]1/4[/pmath]
(D) [pmath]1/5[/pmath]
(E) [pmath]1/6[/pmath]
Hmm. So, last year, Alice made a certain amount of money and she also saved a certain amount. Every month, she made the same amount, and every month, she saved the same amount. So far, so good. The second sentence is a bit complicated; its giving us a relationship between the amount of money she saved for the entire year versus the amount of her monthly income that she did NOT save. Ugh. Okay.
Finally, what does the question ask? They ask us for a fraction, not an actual amount of money that makes sense, because they never tell us anything about the actual amounts of money shes making or saving. A quick glance at the answers confirms this theyre asking us for a fractional amount, or a proportion, of something.
What implications does that have for how I can solve this? I can do it the regular mathematical / algebraic way. but I could also pick my own numbers and just do the problem arithmetically. Either way is fine the choice is based on personal preference but the general rule is that, the harder you find the problem in general, the easier its probably going to be to pick your own numbers and use arithmetic rather than algebra.
Back to the question. More specifically, the question is asking for the fraction of take-home pay that she saved each month. In other words, they want to know monthly amount saved / monthly take-home pay.
Okay, lets try a real number (in an actual testing situation, either youd try this or youd try algebra well try algebra next in this article). Are there any clues in the problem that would indicate a good starting point? The initial inclination is just to start with the very first thing mentioned in the problem but this may or may not be the best place to start. Generally we want to start with the most specific thing / number, or the number that is used as the starting point to generate other numbers.
Lets see theres something in the second sentence about one thing being 3 times the amount of another thing. Theres a relationship there, so lets start with that. Which half is the smaller half? The second half of the sentence. (The first half was 3 times the amount of the second half. Which half is bigger? Try a real number if youre not sure. Then start with the smaller one.)
Okay, the second half of the sentence is the amount of that portion of her monthly take-home pay that she did not save. Hmm. How much do we want her to have not saved each month? Lets say she didnt save $2 per month. Now, the total amount of money that she had saved at the end of the year was 3 times that $2. 3 times 2? Thats $6. Lets see thats the total that she saved for the year so monthly that would be not an integer. Grr.
Okay, go back for a second. What do I want in order to get an integer? Something divisible by 12, because Im trying to take a yearly amount and calculate the monthly amount. 3 times what equals a multiple of 12? Oh, how about 3 times 4! Excellent, now she did not save $4 per month, and she saved $4 * 3 = $12 for the entire year. That means she saved $12 / 12 = $1 each month.
Now we know she did save $1 each month and she did not save $4 per month! That means her take-home pay was $1 + $4 = $5 per month. Great. Whats the question were trying to answer again? Right, the amount she saved per month as a proportion of her take-home pay each month. Hey, weve got what we need! $1 / $5 or 1/5.
The correct answer is D.
Thats one way to do it. In general, if you are struggling with translation problems long, wordy problems that require you to translate all the words and concepts into math then you are probably going to do better with the above method. Note also that struggling can refer to accuracy and timing. If youre getting these right but consistently taking too long, start working on doing these arithmetically, as above. That may sound funny once you look at the algebra below you might say, wait a minute, that looks a lot faster than the arithmetic method!
Youre right, it is as long as you are already good at these kinds of problems. If youre taking way too long, chances are very good that your inefficiency is due to difficulties with setting the problem up abstractly (or algebraically) and / or making mistakes with the algebra (which is much less intuitive than arithmetic). So even though the straight algebra itself looks faster, thats only true if you are already totally fine with translating into algebra and doing the manipulation. If youre not use the above method.
Okay, lets take a look at the algebra. Our first task: assigning a bunch of variables.
monthly take-home pay = t
monthly saved pay = s
What else do we know from those two variables?
yearly take-home pay = 12t
yearly saved pay = 12s
monthly pay not saved = t s
Lets start translating. Sentence 2 becomes:
12s = 3(t s)
The second half of the question becomes:
[pmath]s/t[/pmath]=?
Rearrange the first equation to get [pmath]s/t[/pmath] by itself on one side:
12s = 3(t s)
12s = 3t 3s
15s = 3t
[pmath]s/t=3/15=1/5[/pmath]
And D is the right answer again. (Phew!)
Key Takeaways for Wordy Fraction Problems:
- You may have a choice between using algebra or arithmetic. Know how to choose the best path for yourself. In general, if you are efficient and accurate when translating, and youre fine with the actual algebraic manipulation that needs to happen after that, go ahead with the algebra. If you tend to make mistakes with the translation, or you get tangled up in the algebra, or you spend way too much time, then develop the arithmetic method itll take some practice, but youll learn to do many of these accurately within 2 minutes. Dont feel like you should be able to do it algebraically. Stubborn = lower score on the GMAT. They dont check your work they dont care how you do it and neither should you!
- Make the situation as real and concrete as you can. You are Alice. This is your pay were talking about. Draw your wallet and a couple of arrows going to one box representing your bank and another representing your favorite restaurant? Your coffee habit? Your kids? Wheres the money going? Label clearly. Make it real. Then "step" through the problem as though it's really happening to you.
- Practice makes perfect. Do enough wordy problems in both ways (algebraically and arithmetically) that you feel comfortable (a) executing on each method in certain circumstances, (b) knowing when it's better for you to do algebra and when it's better to do arithmetic, and (c) being able to make an accurate assessment 15 to 20 seconds into a new problem as to whether this one should be an algebra problem or an arithmetic problem.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.