Know the GMAT Code: Interest Rates on the GMAT
Im excited about the problem I have to share with you today in the latest installment of our Know the Code series. :) Interest rate problems can be extremely annoying on the GMATyou might find yourself spending 4 minutes and still having to guess in the end. So your first decision is whether you even want to tackle these kinds of problems in the first place.
But there are some things you can learn that could make answering interest rate questions a lot less irritating. Try out this Integrated Reasoning (IR) Two-Part problem from the GMATPrep free practice exams. (Note: This one is an IR question, but I could absolutely see them testing the same principle on a Quant problem.)
If youre planning to guess on 3 questions in the IR section, then you can give yourself 3 minutes and 20 seconds to do this problem. If youre planning to guess on 2 questions, then give yourself 3 minutes.
*Loan X has a principal of $10,000x and a yearly simple interest rate of 4%. Loan Y has a principal of $10,000y and a yearly simple interest rate of 8%. Loans X and Y will be consolidated to form Loan Z with a principal of $(10,000x + 10,000y) and a yearly simple interest rate of r%, where. In the table, select a value for x and a value for y corresponding to a yearly simple interest rate of 5% for the consolidated loan. Make only two selections, one in each column.
Ready?
1-second Glance. Two-Part. Wall of words! Storywill need to translate.
Read and Jot. This ones so complex that I needed to read it twice, and I didnt start writing anything till the second time. Sometimes you need to do that with story problems.
There are two loans with some details and then a third one thats a combination of the first two. Interesting. Lets start with the first two.
Reflect. Im also going to do a loop on my first two steps. Im going to reflect a bit here, then continue with the rest of my second read-through and jot down the rest.
Hmm. If both of these were exactly $10k in principle, then combining them would give me a combined interest rate of 6%the exact or "straight," average of the two interest rates. But the principles have these extra variables, x and y, and they probably dont represent the same valuethat would be too easy.
I have noticed one important thing, though: this problem is really a weighted average problem in disguise, with the x and the y representing the relative weights of the two original loans. The combined loan will depend on how much each of the original loans is weighted.
Read and Jot some more.
The first part is okay, but what is up with that weird formula for r? (I don't know what it means, so I haven't jotted it down yet.) And then that last bittheyre telling me to calculate based on a simple interest rate of 5%...for the consolidated loan.
Hey! Thats Loan Z. They actually just told us that r% = 5%. :) Nice!
And heres the even nicer thing: go back to that weird formula. Plug in r = 5.
[pmath]r={4x+8y}/{x+y}[/pmath]
[pmath]5={4x+8y}/{x+y}[/pmath]
Thats ugly. So make it less ugly. Simplify!
[pmath]5(x+y)=4x+8y[/pmath]
[pmath]5x+5y=4x+8y[/pmath]
[pmath]x=3y[/pmath]
Thats certainly a much nicer equation. But whats the significance? What is that telling us?
The question asks us to find a value for x and a value for y that correspond with all of the given information. This equation gives a relationship between those two variables. Whatever y is, multiply it by 3 to get x.
Go take a look at the possible answer choices. If y were 21, what would x have to be?
If y = 21, then x = (3)21 = 63. However, that value, 63, isnt in the answers, so y doesnt equal 21.
Try the next one. If y = 32, then x = (3)32 = 96. Bingo! That value is in the answers! The value for y is 32 and the value for x is 96. Done!
Now, wait a sec. What just happened here? How did that really work?
If youre comfortable with the idea that the problem asked you for relative values of x and y, and all you really had to do was find that relative relationship and then find the two answers that fit that relationship, youre good to go.
If, on the other hand, you want to understand the underlying principles hereand, by the way, if you're interested in learning an even faster way to solvethen read on.
Remember, at the beginning, when I mentioned that this was a weighted average problem? We never followed up on that. Now were going to.
Loan X is 4% and Loan Y is 8%. And then they tell us the rate for the combined loan: its 5%. Thats really key!
If the combined loan rate is 5%, then we can figure out the relative proportion of Loan X to Loan Y using the teeter-totter method (we discuss this in the Weighted Averages chapter of our Word Problems Strategy Guide). And remember that Loan X = 10,000x and Loan Y = 10,000y. In other words, the relative values of x and y equal the relative weighting that each loan is given in the overall calculation.
Heres how it works:
If the teeter totter were perfectly balanced, then the combined rate would be exactly halfway between Loan X and Loan Y, at 6%. Its not perfectly balanced, though; its tilted over towards Loan X.
That leads to our first important conclusion: Loan X is more heavily represented, so the value of x is larger than the value of y. Keep that in mind if you get stuck and have to guess later.
Next, we can actually figure out the exact proportion of x to y. Heres how:
There are two sub-distances: 5 - 4 = 1 and 8 - 5 = 3. The shorter one goes with the smaller loan, Y. The longer one goes with the larger loan, X. The values themselves represent the ratio of the two loans: x : y = 3 : 1. In other words, x is 3 times as large as Y.
Thats the same info that the earlier equation told us, and you can follow the same logic to get to the answer pairing 32 and 96. In other words, if you recognize that this is a weighted average, you can find the 3 : 1 ratio just by drawing a number line and doing some pretty basic subtraction. No algebra needed.
As I mentioned earlier, I can definitely see them using this same principle on a regular quant question. The only major difference would be that IR questions do tend to provide more information than you need to answer a question, while quant questions do not. So, in quant-question form, the question stem would be streamlined: Youd be given only what you need in order to answer the question.
Key Takeaways for Knowing the Code:
(1) Take long story problems in parts. You may need to read the whole thing first to understand the basic story, then read it a second time in order to jot down information and reflect on how to move forward.
(2) Dont skip that Reflect step! In this case, there were two important keys to notice: first, that this is a weighted average problem in disguise, and second, that r = 5.
(3) Turn any knowledge you gain into Know the Code flash cards:
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.