Interest and Compound Interest

There are two types of interest problems on the GMAT, and they include simple interest and compound interest. Simple interest is the most basic and is a function of P, the principle amount of money invested, the interest rate earned on the principle, i, and the amount of time the money is invested, t (this is usually stated in periods, such as years or months). The resulting equation is:       I = iPt In basic terms, the above equation tells us the amount of interest that would be earned on a principle amount invested (P), for a given time (t) at a given interest rate (i). Example: If you invested $1,000 (P = your principle) for one year (t = one year) at 6% simple interest (i = given interest rate), you would get $60 in interest at the end of the year and would have a total of $1,060. For compound interest, you would earn slightly more. Let’s look at similar type problem, though this one involves compound interest. Mr. Riley deposits $500 into an account that pays 10% interest, compounded semiannually. How much money will be in Mr. Riley’s account at the end of one year? For compound interest, first you need to divide the interest rate by how many compound periods there are. So for in the above question, because we are compounding semiannually, we need to divide 10% by 2 (because of 2 compounding periods), and if we were compounding quarterly, we would need to divide 10% by 4. In the above question, Mr. Riley deposited $500 into his account at a rate of 10% compounded semiannually and the bank will divide his interest into two equal parts. They will pay 5% interest (10%/2) at the end of six months, and then will pay another 5% at the end of the year. Compound interest can essentially be translated into “interest paid on interest”, meaning that after one period, you are paid interest on the interest that was paid in prior periods, hence the phrase “compounding”. So at the end of the six months, Mr. Riley has $525 because the bank paid $25 in interest ($500*5%) into his account. For the second half of the year, Mr. Riley is then paid 5% on the $525 balance that was in his account at the end of the first six months. This interest is equal to $525*5% = $26.25. Therefore, at the end of the year, Mr. Riley has $551.25, which is equal to his balance of $500, plus the $25 interest paid at the end of 6 months, plus $26.25 paid at the end of the year. Mr. Riley earns $1.25 more with this compound interest than he would have been paid if he were paid only 10% simple interest (would have been only $550). The lesson? Compound interest always pays more! Let’s look at another similar type of problem that involves interest. Money invested at x%, compounded annually, triples in value in approximately every 112/x years. If $2500 is invested at a rate of 8%, compounded annually, what will be its approximate worth in 28 years? A. $3,750 B. $5,600 C. $8,100 D. $15,000 E. $22,500 At first glance, this one seems pretty tricky because you are given x% as the interest rate and it asks you about compounding and it might seem difficult where to find a starting point for this. For this one, it might be a bit easier to think about this without the use of compound interest, which might unnecessarily confuse you. Here, we are given x% as 8%, so all we need to do is take 112/8 = 14. Thus, we know that the money triples in value every 14 years. Further, we know that the money will triple exactly twice in 28 years, once in 14 years and one more time at the 28^th year. So first we need to multiply the original $2500 invested by 3 to get the balance at the end of year 14 (because it triples), to get $7,500 (or $2,500*3). Now, we know that this balance of $7,500 will triple again, so the final balance at the end of the next 14 year period will be $22,500 (or $7,500*3). The correct answer choice is E. Overall, the three types of interest problems you will most likely encounter come test day will be simple interest, compound interest, and word problems involving the mention of interest, but that can be solved without the application of interest or compound interest methods. The key to deciphering between compound interest and simple interest is to see how many periods the interest is paid….interest paid in one period is simple interest and interest “paid on interest” in multiple periods is compound interest. Finally, remember that some questions can be solved intuitively. Check out Grockit for more GMAT quantitative practice!

Good luck!

Three types of pencils, J,K, and L, cost $0.05, $0.10, and $0.25 each, respectively. If a box of 32 of these pencils costs a total of $3.40 and if there are twice as many K pencils as L pencils in the box, how many J pencils are in the box?

(A) 6
(B) 12
(C) 14
(D) 18
(E) 20

Highlight to see answer: C

Please post your explanations in the comments below!

For me, one of the most exciting parts of Kaplan’s new GMAT revision for 2010 is its emphasis on Identifying the Task in Quantitative Problem Solving questions.  Often, Problem Solving questions give a test-taker a large amount of information–sometimes as much as five or six equations’ worth–and when the pressure is on and the clock is ticking, it can be tough to get this information organized enough to answer the question, not to mention the amount of time that can be lost in doing so.

One of the most efficient ways of zeroing in on the answer in complicated questions is to use what I call a “master equation.”  For example, if I see the phrase “average (arithmetic mean)” in the first line of a Quantitative question, I immediately shoot down to my noteboard and jot down the average formula: “Average = (Sum of terms)/(Number of terms),” or my preferred abbreviated version, “Avg = sum/#.”

The logic behind it?  Well, if a question stem is mentioning an average in its first few words, I’m going to have to use the formula at some point, whether the question ends up needing the value of the average, the sum of the terms, or some other relationship.  With the formula already written before I even finish reading the question stem, I save myself the time and effort of recalling it later on, when I’ll be more concerned about the tricky, GMAT-ty aspects of solving the problem.  If I’m lucky–say, for example, the question from above gives us an average and a number of terms, and asks for the sum of the terms–I can even plug values directly into my master equation, solve algebraically, and get to the answer that much faster.  This can be very effective even complicated algebra questions where you’re not sure what the first step is–at least you can use the equations you know to do SOMETHING, instead of sitting there without knowing what to do.  When you keep your eventual goal (your “master equation”) prominent and proud at the very top of your scratch work, you’ll find it very difficult to lose focus!

In 1980 the government spent $12 billion for direct cash payments to single parents with dependent children. If this was 2,000 percent of the amount spent in 1956, what was the amount spent in 1956? (1 billion = 1,000,000,000)

(A) $6 million
(B) $24 million
(C) $60 million
(D) $240 million
(E) $600 million

Highlight to see answer: E

Please post your explanations in the comments below!