In “GMAT Interest Rate Problems – Part 1” we looked at simple interest and the way it’s commonly tested on the GMAT. What makes it “simple” is the fact that interest is accrued over one (1) time period, generally one year. If you invest $5,000 and earn an interest rate of 5%, it’s assumed that the income earned is for one year. But what if the question tells you that the interest is compounded multiple times throughout the year? That’s called a compound interest problem. Not only is compound interest important to know because Albert Einstein called it the 9th Wonder of the World, but it’s also fair game on the GMAT, so let’s take a closer look.
Compound Interest Rate Problems
For whatever reason, students often freak out when they see things like “quarterly” or “semi-annually” on GMAT interest rate problems. Fear not! With one small bit of explanation, you’ll be good to go.
The only real difference between simple interest and compound interest problems on the GMAT is that compound interest problems assume multiple time frames. It may seem like a small distinction, but it’s actually quite important — especially from an income-earned standpoint. Let’s take a look at three variations of the initial question we looked at in Part 1:
Variation 1: If Sarah invests $5,000 at an interest rate of 5%, how much money will she have at the end of a year?
Answer: $5,000 * 1.05 = $5,250.
(Note, a quick way to find the new principle balance is to multiply the current principle times 1.05; if you just multiply it by 0.05, that will give you the amount of interest earned, but you’ll need to add that back on to the original principle of $5,000 to get the new balance; you can skip a step by multiplying by 1.05 straight away).
Variation 2: If Sarah invests $5,000 at an interest rate of 5%, compounded semi-annually, how much money will she have at the end of a year?
Answer: The word semiannually means “twice per year.” (Biannual means the same thing, but the GMAT generally prefers to use the word semiannual). In other words, the bank will pay Sarah half of her interest after 6 months, and the other half at the end of the year. What’s cool about that is that Sarah will actually be earning interest on her interest, which means that she actually earns more money than under a simple interest situation.
The important part is to understand that when you’re dealing with compound interest, you simply need to divide the interest rate by the total number of compounding periods before doing the math. In this case, semiannually means two (2) compounding periods. So, we need to divide 5% by 2, which is 2.5%. Here’s what that means for Sarah:
Step 1: In the first time period (6 months), Sarah’s principle becomes $5,000 * 1.025 = $5,125.
Step 2: In the second period (next 6 months), that new principle balance of $5,125 earns an additional 2.5% interest, or $5,125 * 1.025 = $5,253.13.
Notice that with semiannual compounding, Sarah earns $3.13 more than with simple interest — not enough to retire on, but free money is free money!
Yep, you guessed it — quarterly means four (4) time periods. So, start by dividing 5% by 4 periods, which is 1.25% per quarter. Here’s how that plays out:
Step 1: In quarter 1, $5,000 * 1.0125 = $5,062.5.
Step 2: Start with $5,062.5 and compound that at 1.25%: $5,062.5 * 1.0125 = $5,125.78.
Step 3: Repeat the process for quarter 3: $5,125.78 * 1.0125 = $5,189.85.
Step 4: Finally, for quarter 4: $5,189.85 * 1.0125 = $5,254.73
There you have it. With quarterly compounding, Sarah will have earned $4.73 more than with Simple Interest.
Usually common sense and applying this step-by-step approach are enough on these types of questions, but if you’re the type of person who needs a formula, here it is:
A = P (1 + r/n)ᶯᵗ where
A = Amount in the account (Principle plus interest)
P = Original principle amount
r = interest rate (yearly)
n = number of times per year interest is compounded
t = number of years
In conclusion, let’s try one summary example.
Tom plans to invest x dollars in a savings account that pays interest at an annual rate of 8%, compounded quarterly. Approximately what amount is the minimum that Tom will need to invest to earn over $100 in interest within 6 months?
So, how did you do? If you tried to apply the formula I just gave you, well…I think you’ll find that Working Backwards is, once again, the fastest and easiest way to solve these types of problems. Let’s assume the answer is “C” and see if it works.
First, we need to divide 8% by 2 because the interest is compounded quarterly. So, Tom will receive 2% each quarter on his principle. Since the problem asks us to help Tom earn $100 in just 6 months, that’s only two (2) time periods. So, let’s work backwards and apply what we learned above:
Step 1: Assume Tom invests $2,250 (answer choice C)
Step 2: Calculate new principle balance after quarter 1: $2,250 * 1.02 = $2,295
Step 3: Calculate new principle balance after quarter 2: $2,295 * 1.02 = $2340.90
Step 4: Total interest earned is $2340.90 – $2250 = $90.90
That’s less than $100, so “C” is not the answer. That means we can also eliminate answer choices A and B because they must be too small as well.
Next, let’s try answer choice D.
Step 1: Assume Tom invests $2,500 (answer choice D)
Step 2: Quarter 1 Principle: $2,500 * 1.02 = $2,550
Step 3: Quarter 2 Principle: $2,550 * 1.02 = $2,601
Step 4: Total interest earned is $101
The answer is therefore D.
Great job if you got that. Hopefully now the terminology around simple interest, compound interest, semiannually, and quarterly makes sense so that you’re not rattled or confused when you seem them on test day. Practice these strategies and go out and Dominate the GMAT!