Quantitative: Simple and Compound Interest

How to Prepare for Less Common Quant Topics:

With limited time and a lot to cover, a difficult choice many GRE students face is how much time to devote to the “errata” of the GRE Quant section. Among the more difficult problems, the GRE may throw students a curve ball: perhaps a difficult probability question, a complex permutation, standard deviation, a challenging quadratic, or the graph of an unfamiliar function.  Sometimes, ETS will combine concepts to increase the level of difficulty. For instance, counting problems can integrate probability to add complexity and the likelihood students will make errors in reasoning or calculations.

For this post, we will focus on simple and compound interest. Since I teach math for a living and still can’t determine whether I’m better off investing in an IRA, paying into the principal of my house, or sticking my money in a mattress, I can empathize with students’ dismay at anything that hints at finance on the GRE. Fortunately, as with all Quant problems on the GRE, you:

• Don’t need to do any complicated math to get to the right answer.
• Can expect a “shortcut” to the solution that elides lengthy calculations.

Follow the jump for a discussion of the concepts and an example problem.

The Eighth Wonder of the World 

To begin, we must distinguish between simple and compound interest:

• Simple interest is interest paid on the principal alone.
  • The formula for simple interest is i = pRt, where i is interest, p is principal, R is the rate, and t is time.
  • For example, given a principal of $500, a rate of 5%, and a time of 3 years, the interest would be:
    • i = 500 x 5% x 3
    • i = 500 x 1/20 x 3
    • i = 25 x 3
    • i = 75
• Compound interest is interest calculated on both the principal and the accrued interest.
  • The formula for compound interest is i = p × (1+R)^t – p, where i is interest, p is principal, R is rate, and t is time (here assuming compounded annually).
  • For example, given the same principal of $500, a rate of 5%, and a time of 3 years, the interest would be:
    • i = 500 × (1+5%)³ – 500
    • i = 500 x (1.05)³ – 500
    • i = 500 x ~1.158 – 500
    • i = 578.81 – 500
    • i = 78.81

As you can see, compound interest is significantly more complicated than simple interest, so what should you do if you encounter a problem that tests one of these concepts?

First, don’t panic! Remain confident! As mentioned earlier in this post, there will be a simpler, more direct route to finding the solution to these problems than by using these formulas.

Second, follow the same process as you would for any other GRE Quant question. Let’s take a look at this process through the use of an example question.

We Can Do This the Easy Way or the Hard Way 

Consider the following example:

If money is invested at R percent interest, compounded annually, the amount of the investment will double in approximately 70/R years. Nichelle, who just turned 37, placed $3,000 in a retirement account that pays 5% interest. She plans to retire when she turns 65. Approximately how much will her account be worth when she turns 65?

A. $18,000

B. $15,750

C. $12,000

D. $9,000

E. $4,500

Now let’s step through the process to answer this question:

1. Read through the problem and assess what the question is asking. We are asked a problem involving interest, compound interest to be precise. Given an original amount, an interest rate, and a length of time, we are asked to find the final value of an investment.
2. Assess what information is given. We know Nichelle’s initial age, 37. We know her final age, 65. We know the initial amount, $3,000. We are given an interest rate, 5%, and we are given a useful formula: The approximate length of time it takes for an investment to double if compounded annually can be expressed as 70/R years.
3. Now put this information together into a step-by-step plan to get from the initial information to your target solution.
   1. What is the first piece of information we might want to know? Here I might want to know exactly how many years we’re talking about. In this case, we can find that information by taking Nichelle’s final age and subtracting her initial age. 65 – 37 = 28.
   2. Now consider different possible ways of doing this problem. Are we expected to plug numbers into the compound interest formula? No way! In fact, the problem itself gives us another formula. Let’s think about how we might be able to use this. The approximate length of time it takes for an investment to double if compounded annually can be expressed as 70/R years. Do we know what R is? Yes! It’s 5%. So what happens when we put 5 in for R? We get 70/5 years. Let’s do this calculation. We get 14 years.
   3. What do we know now? It would take this investment 14 years to double in value. Does this number look familiar? It might strike you as half of 28. How could we use this information to our advantage? We could say that now we know that the $3,000 investment will double in 14 years.
   4. How much will it be worth in 14 years? $6,000. Then if we have 14 more years to go, how much will it be worth at the end of the 28 years? $12,000. Is that one of the answers? Yes!
   5. Pick C and we’re done.

More and Less than Meets the Eye

Notice in our example, there was no outside knowledge of interest required to reach the solution. To succeed at the GRE, students are not expected to become walking encyclopedias of formulas and equations. Instead, students must assess the information given and come up with a game plan to put the pieces together to reach a solution.

You may have noticed that none of the math steps we did in our example problem was particularly impossible. With some practice (and the use of the provided calculator, if necessary), students can perform with ease the arithmetic necessary to solve a problem such as this. Then one might ask why this is a difficult problem.

The difficulty arises from (1) not getting overwhelmed with unfamiliar terms or concepts, (2) coolly assessing what information is at our disposal, (3) putting together a plan of attack to find the solution, (4) executing this plan confidently and accurately.

Not to overstate the point, but what we have done on this problem is in some respects a microcosm of the process on many GRE Quant Problem Solving questions.

We hope you have found this post helpful. If you have questions, please comment below. I would also like to encourage you to visit our GRE Forums, where you can engage with instructors and other students to discuss GRE preparation. If you wish to pursue your GRE preparation with PowerScore, please consider one of our outstanding GRE prep options. Finally, we are pleased to announce new sessions in the continuation of our Free GRE Webinar Series. Register today!