First, a few practice problems. Remember: no calculator!

**Q1**. If

(A) compounding annually

(B) compounding quarterly

(C) compounding monthly

(D) compounding daily

(E) All four of these would produce the same total

**Q2**. If

(A)

(B)

(C)

(D)

(E)

**Q3**. At the beginning of January 2003, Elizabeth invested money in an account that collected interest, compounding more frequently than a year. Assume the annual percentage rate of interest remained constant. What is the total amount she has invested after seven years?

Statement #1: her initial investment was

Statement #2: the account accrued

**Q4**. Sarah invested

(A)

(B)

(C)

(D)

(E)

Solutions to these will be given at the end of the article.

In grade school, you learn about simple interest, largely because we want to teach little kids something about the idea of interest, and that's the only kind of interest that children can understand. No one anywhere in the real world actually uses simple interest: it's a pure mathematical fiction.

Here's how it works. There's an initial amount

Again, this is a fiction we teach children, the mathematical equivalent of Santa Claus. This never takes place in the real world.

With compound interest, in each successive year or period, you collect more interest not merely on the principle but on all the interest you have accrued up to that point in time. Interest on interest: that's the big idea of compound interest.

Here's how it plays out. Again, there's an initial amount

Here, I was demonstrating everything step-by-step for clarity, but if we wanted to calculate the total amount after a large number of years, we would just use a formula. We know that each year, the amount increased by

We don't have to do it step-by-step: we can just jump to the answer we need, using multipliers. Of course, for this exact value, we would need a calculator, and you don't get a calculator on the QUANT Quantitative section. Sometimes, though, the QUANT lists some answers in “formula form”, and you would just have to recognize this particular expression,

If we graph compound interest against time, we get an upward curving graph (purple), which curves away from the simple interest straight line (green):

The curve of the graph, that is to say, the multiplying effect of the interest, gets more pronounces as time goes on.

**BIG IDEA #1: as long as there is more than one compounding period, then compound interest always earns more than simple interest.**

A year is a long time to wait to get any interest. Historically, some banks have compounded over shortened compounding period. Here is a table of common compounding periods:

Technically, the fraction for “compounding daily” would be

Now, how does this work? Let's say the bank gives

For compounding quarterly, we divide the annual rate by four and compound four times each year. For compounding monthly, we divide the annual rate by twelve and compound twelve times a year. Similarly, for daily or any other conceivable compounding period.

How do the amounts of interest accrued compare for different compounding periods? To compare this, let's pick a larger initial value,

simple \space interest = \${2,000,000} compounding \space annually = \${2,653,297.71} compounding \space quarterly = \${2,701484.94} compounding \space monthly = \${2,712,640.29} compounding \space daily = \${2,718,095.80} compounding \space hourly = \${2,718,274.07}

As we go down that list, notice the values keep increasing as we decrease the size of the compounding period (and, hence, increase the total number of compound periods). This leads to:

**BIG IDEA #2: We always get more interest, and larger account value overall, when the compounding period decreases; the more compounding periods we have, the more interest we earn.**

Admittedly, the difference between “compounding daily” and “compounding hourly” only turn out to be a measly

The formula for calculating continuously compound interest involves e, and is more complicated than anything you need to understand for the QUANT. I will simply point out, in the example above, with one million dollars invested at

Most banks use monthly compounding interest, for accounts and for mortgages: this make sense for accounts with monthly statement or payments. Credit cards tend to use continually compounding interest, because charges or payments could occur at any point, at any time of any day of the month. In addition to understanding the mathematics of compound interest, it's good to have a general idea of how it works in the real world: after all, the history or logic of compound interest would be a very apt topic for a Reading Comprehension passage or a Critical Reasoning prompt on the QUANT!

If you had any “aha's” while reading this article, you may want to go back a take another look at the four practice problems above. If you would like to express anything on these themes, or if you have a question about anything I said in this article, please let us know in the comments section.

**Q1**. The smaller the compounding period is, the greater the number of times the interest will be compounded. Of course, if we compound monthly instead of quarterly, then we are compounding by

**Answer = (D)**

**Q2**. Notice that, since R is the annual percent as a ** decimal**, we can form a multiplier simply by adding one:

**Method One: Step-by-step**

After one year, we multiply by the multiplier once

That's the total amount at the end of the first year. The

At the end of the second year, that entire amount is multiplied by the multiplier. We need to FOIL.

That's the total amount at the end of the second year. The amount

At the end of the third year, this entire amount is again multiplied by the multiplier.

That's the

**Answer = (C)**

**Method Two: some fancy algebra**

Over the course of three years, the initial amount A is multiplied by the multiplier

Now, if you happen to know it offhand, we can use the cube of a sum formula:

Thus,

and,

**Answer = (C)**

**Q3**. In order to determine the total amount at the end of an investment, we would need to know three things: (a) the initial deposit; (b) the annual percentage rate; and (c) the compounding period.

Statement #1 tells us the initial deposit but not the annual percentage rate. **Insufficient**.

Statement #2 tells us the annual percentage rate but not the initial deposit. **Insufficient**.

Together, we know both the initial deposit and the annual percentage rate, but we still don't know the compounding period. All we know is that it's less than a year, but quarterly compounding vs. monthly compounding vs. daily compounding would produce different total amounts at the end. Without knowing the exact compounding period, we cannot calculate a precise answer. Even together, the statements are **insufficient**.

**Answer = (E)**

**Q4**. Without a calculator available, this is a problem screaming for estimation. The problem even uses the magic word “approximately” to indicate that estimation is a good idea, and the answer choices are spread far apart, making it easier to estimate an individual answer.

Let's round the deposit up to **(A)**.