= $100(1.03)12Interest rates are something that you might be acutely aware of in real life, especially as they apply to mortgages and credit cards, but unless you work directly in a field that touches on loans or mortgages, you probably don’t think about the formulas used to calculate them.
Fortunately, you don’t have to puzzle over calculating out the interest year-by-year or step-by-step. You can memorize and apply formulas to help you manage the information from the question stem much more efficiently and accurately than by simply applying brute force to the arithmetic.
Interest rate questions rely on three important pieces of information: (1) the principal, or the amount initially invested; (2) the interest rate, or the rate at which the investment grows; and (3) the time period during which the investment accrues interest.
Attention to the Right Detail is necessary even when applying the formulas—and no detail is more important to interest rate questions than whether the interest is simple or compound. Simple and compound interest use different formulas:
Simple interest is interest applied only to the principal, not to the interest that has already accrued. Use this formula:
(Total of principal and interest) = Principal × (1 + rt), where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods
Example: If $100 were invested at 12 percent simple annual interest, what would be the total value of the investment after 3 years?
Total = $100 × (1 + 0.12 × 3) = $100(1.36) = $136
Compound interest is interest applied to the principal and any previously accrued interest. Use this formula:
(Total of principal and interest) = Principal × (1 + r)t, where r equals the interest rate per time period expressed as a decimal and t equals the number of time periods
Example: If $100 were invested at 12 percent interest compounded annually, what would be the total value of the investment after 3 years?
Total = $100 × (1 + 0.12)3 = $100(1.12)3
Despite the GMAT’s general preference for simplified values in the answer choices, you will often see answer choices for a compound interest rate problem written as expressions similar to the result above. You can be thankful that you won’t have to calculate values such as 1.123, since you don’t have the use of a calculator on the Quant section.
On Test Day, you might also encounter a more difficult question that requires dealing with annual interest but payments not on an annual basis. In this case, r equals the annual rate divided by the number of times per year it is applied, and t equals the number of years multiplied by the number of times per year interest is applied.
Example: If $100 were invested at 12 percent annual interest, compounded quarterly, what would be the total value of the investment after 3 years?
Total = $100 ×
= $100(1.03)12
Now let’s use the Kaplan Method on a Problem Solving question dealing with interest rates:
This may not look like an interest rate question at first glance, but interest rates are just percent increases applied multiple times. The number of bacteria increases 50% every 2 hours. You could think of this as 50% compounded interest applied once every 2 hours.
Instead of calculating forward, you have to calculate back—if there are 108 million in the dish at 2:00 p.m., when were there 32 million?
This problem could be solved as a straightforward percent change, but let’s look at it through the lens of interest rates:
108 = 32 × (1.5)t
Clearly, then t = 3. That’s three 2-hour increases for a total of 6 hours. Since there were 108 million at 2:00 p.m., there were 32 million 6 hours earlier at 8:00 a.m.
The answer is (D).
This is also an excellent backsolving question, as you could easily just try out a time for 32 million and see whether that’s consistent with 108 million at 2:00 p.m.
Let’s say you try (B) first:
|
Time |
Millions of bacteria |
|
8:00 p.m. |
32 |
|
10:00 p.m. |
48 |
|
Midnight |
72 |
|
2:00 a.m. |
108 |
That’s either far too soon (12 hours too soon) or far too late (12 hours too late), depending on how you envision the day. Either way, (B) is wrong by a lot, and just shifting 2 hours to 6:00 p.m. won’t help matters. Eliminate (A) as well.
Now let’s try (D):
|
Time |
Millions of bacteria |
|
8:00 a.m. |
32 |
|
10:00 a.m. |
48 |
|
Noon |
72 |
|
2:00 p.m. |
108 |
Exactly what it should be (you might also have seen that (D) would be correct when you noticed that (B), 8:00 p.m., was off by exactly 12 hours). The answer is (D).
Reread the question stem, making sure that you didn’t accidentally skip over any important aspects of the situation.
years, where p is the percent interest, compounded annually. If Thelma invests $40,000 in a long-term
CD that pays 5 percent interest, compounded annually, what will be the approximate
total value of the investment when Thelma is ready to retire 42 years later?