Picture this: You go to the bank and open a savings account. They offer you a CD with an interest rate of 5% (not likely in today’s market, but stick with me here). You deposit $5,000 into that CD. Question: How much money will you have at the end of the year?
Pretty simple, right? Five percent of $5,000 is $250, so at the end of the year, you’ll have $5,250 in the account. We deal with this type of math every day. Why, then, do interest rate problems seem so difficult on the GMAT? They shouldn’t be, so let’s break them down and make them easier.
Simple Interest Rate Problems
As we just discussed, simple interest is the most basic type of interest rate question tested on the GMAT. It’s simply a function of the principle amount invested and the interest rate. If you’re the type of person who likes formulas, here it is:
Interest Earned = (Principle Invested) * (Interest Rate)
Remember that the interest rate needs to be expressed as a decimal. In our example above, where the interest rate is 5%, the math would therefore be:
Interest Earned = $5,000 * 0.05 = $250
With that basic understanding in mind, give this more difficult GMAT problem solving question a try.
Sarah invests $2,400 at 5% interest. How much additional money must she invest at 8% so that the total annual income will be equal to 6% of her entire investment?
If you tried to solve this question algebraically, hopefully you recognized that it’s essentially just a giant weighted average problem. We could solve it algebraically the same way we’d solve a GMAT Mixture problem (to review, check out “Problem Solving – Common GMAT Word Problems”).
The good news, especially if you hate algebra, is that there’s an easier and more efficient way to solve this question. We simply need to Work Backwards.
Answer Explanation: Let’s assume that the additional principle Sarah invests is one of those five answer choices. Which answer choice should we start with to test? (Hint: Use a little common sense). The “rules” for Working Backwards tell us to start with answer choice C. However, do we even need to test $2,400? No — we know that can’t possibly be the correct answer. How?
Think about it this way. If Sarah invests the same amount ($2,400) at 5% as she does at 8%, what is the average percentage return she’ll receive? Answer: Simply the average, or midpoint, of those two interest rates, or 6.5%. Does that make sense?
And yet the problem tells us that Sarah must receive an average return of 6.0%, so the answer cannot be C. In fact, because her average return must be LESS than what it would be if she invests an additional $2,400, then will see need to invest more than $2,400 or less than $2,400 to receive that lower average interest rate of 6.0%?
LESS, of course. So not only can we eliminate answer choice C straight away, we can also eliminate any answer choices that are greater than $2,400, i.e. answer choices D and E. Now we’re only left to choose between answer choices A and B. Let’s choose one to test. I’ll pick A, just because I think $1,000 will be easier to work with. If it works out, great! Then that’s our answer. If it doesn’t work out, then we’ll know the answer is B. Let’s dive in.
Here’s what we know: Sarah has already invested $2,400 at 5% interest. What, then, is her income on that principle?
Interest Earned at 5% = $2,400 * 0.05 = $120
Now, if we’re assuming the answer is A, that means she’s investing an additional $1,000 at 8% interest. That math is:
Interest Earned at 8% = $1,000 * 0.08 = $80
So, the total amount of interest income she’s earned is $200 ($120 + $80).
Likewise, the total amount of principle she will have invested is $3,400 ($2,400 + an additional $1,000).
The question now becomes, is $200 exactly 6% of $3,400, which is what the question tells us it needs to be? No, it’s not. To determine that, you could either do $200 / $3,400, which is annoying math without a calculator, or, what I think is easier, you can just take 6% of $3,400 and see what that comes out to. That’s easier math to do on your scratch paper:
$3,400 * 0.06 = $204
$204 is NOT the same as what we know she in fact earned ($200) on a $1,000 additional investment at 8%, so the answer is not A. The correct answer must therefore be B! Great job if you got that.
Once you understand the mechanics of simple interest, you should be able to apply it to any variation of problem the GMAT can throw at you. These types of GMAT math questions become more difficult, however, when they start giving you things like semi-annual interest rates and multiple time periods. I’ll cover that in my next blog post (“GMAT Interest Rate Problems – Part 2: Compound Interest”). For now, practice this so that you can Dominate the GMAT!
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