GMAT Prep Story Problem: Make It Real Part 2
How did it go last time with the rate problem? Ive got another story problem for you, but this time were going to cover a different math area.
Just a reminder: heres a link to the first (and long ago) article in this series: making story problems real. When the test gives you a story problem, do what you would do in the real world if your boss asked you a similar question: a back-of-the-envelope calculation to get a close enough answer.
If you havent yet read the earlier articles, go do that first. Learn how to use this method, then come back here and test your new skills on the problem below.
This is a GMATPrep problem from the free exams. Give yourself about 2 minutes. Go!
* Jack and Mark both received hourly wage increases of 6 percent. After the wage increases, Jacks hourly wage was how many dollars per hour more than Marks?(1) Before the wage increases, Jacks hourly wage was $5.00 per hour more than Marks.
(2) Before the wage increases, the ratio of Jacks hourly wage to Marks hourly wage was 4 to 3.
Data sufficiency! On the one hand, awesome: we dont have to do all the math. On the other hand, be careful: DS can get quite tricky.
Okay, you and your (colleague, friend, sisterpick a real person!) work together and you both just got hourly wage increases of 6%. (Youre Jack and your friend is Mark.) Now, the two of you are trying to figure out how much more you make.
Hmm. If you both made the same amount before, then a 6% increase would keep you both at the same level, so youd make $0 more. If you made $100 an hour before, then youd make $106 now, and if your colleague (I'm going to use my co-worker Whit) made $90 an hour before, then shed be makinger, that calculation is annoying.
Actually, 6% is pretty annoying to calculate in general. Is there any way around that?
There are two broad ways; see whether you can figure either one out before you keep reading.
First, you could make sure to choose easy numbers. For example, if you choose $100 for your wage and half of that, $50 an hour, for Whits wage, the calculations become fairly easy. After you calculate the increase for you based on the easier number of $100, you know that her increase is half of yours.
Oh, waitread statement (1). That approach isnt going to work, since this choice limits what you can choose, and thats going to make calculating 6% annoying.
Second, you may be able to substitute in a different percentage. Depending on the details of the problem, the specific percentage may not matter, as long as both hourly wages are increased by the same percentage.
Does that apply in this case? First, the problem asks for a relative amount: the difference in the two wages. Its not always necessary to know the exact numbers in order to figure out a difference.
Second, the two statements continue down this path: they give relative values but not absolute values. (Yes, $5 is a real value, but it represents the difference in wages, not the actual level of wages.) As a result, you can use any percentage you want. How about 50%? Thats much easier to calculate.
Okay, back to the problem. The wages increase by 50%. They want to know the difference between your rate and Whits rate: Y W = ?
(1) Before the wage increases, Jacks hourly wage was $5.00 per hour more than Marks.
Okay, test some real numbers.
Case #1: If your wage was $10, then your new wage would be $10 + $5 = $15. In this case, Whits original wage had to have been $10 - $5 = $5 and so her new wage would be $5 + $2.50 = $7.50. The difference between the two new wages is $7.50.
Case #2: If your wage was $25, then your new wage would be $25 + $12.50 = $37.50. Whits original wage had to have been $25 - $5 = $20, so her new wage would be $20 + $10 = $30. The difference between the two new wages is$7.50!
Wait, seriously? I was expecting the answer to be different. How can they be the same?
At this point, you have two choices: you can try one more set of numbers to see what you get or you can try to figure out whether there really is some rule that would make the difference always $7.50 no matter what.
If you try a third case, you will discover that the difference is once again $7.50. It turns out that this statement is sufficient to answer the question. Can you articulate why it must always work?
The question asks for the difference between their new hourly wages. The statement gives you the difference between their old hourly wages. If you increase the two wages by the same percentage, then you are also increasing the difference between the two wages by that exact same percentage. Since the original difference was $5, the new difference is going to be 50% greater: $5 + $2.50 = $7.50.
(Note: this would work exactly the same way if you used the original 6% given in the problem. It would just be a little more annoying to do the math, thats all.)
Okay, statement (1) is sufficient. Cross off answers BCE and check out statement (2):
(2) Before the wage increases, the ratio of Jacks hourly wage to Marks hourly wage was 4 to 3.
Hmm. A ratio. Maybe this one will work, too, since it also gives us something about the difference? Test a couple of cases to see. (You can still use 50% here instead of 6% in order to make the math easier.)
Case #1: If your initial wage was $4, then your new wage would be $4 + $2 = $6. Whits initial wage would have been $3, so her new wage would be $3 + $1.5 = $4.50. The difference between the new wages is $1.5.
Case #2: If your initial wage was $8, then your new wage would be $8 + $4 = $12. Whits initial wage would have been $6, so her new wage would be $6 + $3 = $9. The difference is now $3!
Statement (2) is not sufficient. The correct answer is (A).
Now, look back over the work for both statements. Are there any takeaways that could get you there faster, without having to test so many cases?
In general, if you have this set-up:
- The starting numbers both increase or decrease by the same percentage, AND
- you know the numerical difference between those two starting numbers
Then you know that the difference will change by that same percentage. If the numbers go up by 5% each, then the difference also goes up by 5%. If youre only asked for the difference, that number can be calculated.
If, on the other hand, the starting difference can change, then the new difference will also change. Notice that in the cases for the second statement, the difference between the old wages went from $1 in the first case to $2 in the second. If that difference is not one consistent number, then the new difference also wont be one consistent number.
Key Takeaways: Make Stories Real
(1) Put yourself in the problem. Plug in some real numbers and test it out. Data Sufficiency problems that dont offer real numbers for some key part of the problem are great candidates for this technique.
(2) In the problem above, the key to knowing you could test cases was the fact that they kept talking about the hourly wages but they never provided real numbers for those hourly wages. The only real number they provided represented a relative difference between the two numbers; that relative difference, however, didnt establish what the actual wages were.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.