# Algebra: Interest Problems

## Interest Problems

There are three good reasons to deposit your life savings in a bank account, rather than hide it in your closet or mattress:

- A bank is safer, and if your money is stolen, there are usually federal laws that insure your investment.
- A bank affords you the unique opportunity to write with pens chained to desks. Even though your money allows banks, themselves, to rake in the dough, for some reason they are very adamant that you not accidentally take their pens, each worth only pennies.
- You earn interest on your money without having to exert any effort at all.

##### Talk the Talk

The amount of money you initially deposit in an interest problem is called the **principal**.

Interest is a great thing. It's free money that you earn just by keeping your money in a safe place. In algebra, you may be asked to solve problems in which you calculate the interest earned by some initial investment (which is called the *principal*) over some length of time. Specifically, there are two major types of interest problems you may be asked to solve: simple interest and compound interest.

### Simple Interest

If your money grows according to simple interest, you're basically just earning a small percentage of your initial investment each year as interest. For instance, if the principal of an account is $100 and your annual interest rate is 6.75%, at the conclusion of every year you will have earned an additional $6.75 (since $6.75 is 6.75% of $100).

Here's the bad news: Even though your account will grow slightly every year, you won't earn more interest! In simple interest problems, you only earn interest on the initial investment, no matter how long you've had an active bank account or how much interest that money has accrued.

The formula for calculating simple interest is:

*i = prt*

where *p* is your principal, *r* is the annual interest rate expressed as a decimal, and *i* is the interest you have earned after the money has been invested for *t* years.

**Example 1**: You were a very thrifty and money-savvy child. Instead of spending the money the tooth fairy gave you for your baby teeth, you invested that cash in one lump sum of $32.00 as a teenager, in a bank account with a fixed annual interest rate of 7.75%. What is the balance of the account exactly 30 years later?

##### Critical Point

To convert a percent into a decimal, drop the percent sign and multiply by .01. For instance, the decimal equivalent of 6.75% is (6.75)(.01) = .0675. (Conversely, to change a decimal into a percent, multiply by 100 and stick a percent sign on the end. Therefore, the percent equivalent of .45 is (.45)(100) = 45%.)

**Solution**: To calculate the balance of an account, just add the interest you earned to the principal. Of course, you still need to figure what that interest is. Use the formula *i = prt*, where *p* = 32, *r* = .0775 (the decimal equivalent of 7.75%), and *t* = 30.

*i = prt**i*= (32)(.0775)(30)*i*= 74.4

You earned $74.40 in interest over that 30-year period, so if you add in the initial investment, your total balance is:

- balance = principal + interest earned
- = $32 + $74.40
- = $106.40

### Compound Interest

Most banks don't use simple interest; the more money you deposit, the more money they can potentially make, so they want to encourage you to deposit as much as possible into your account. One way they do this is via *compound interest*, in which you earn money based on your original principal *and* the interest you've accrued to that point.

##### Talk the Talk

If your bank account accrues **compound interest**, then you earn interest based upon your entire balance, rather than just the initial investment.

Let's say you deposit $100 in an account whose interest is compounded annually at a rate of 6.0%. At the end of the first year, you will have a balance of $106, just like you would with simple interest. However, at the end of the second year, you'll earn 6.0% interest on the new balance of $106, not just the original balance of $100.

Even better, most banks don't just compound their interest once a year. Whether they compound weekly (52 times a year), monthly (12 times a year), or quarterly (4 times a year) can make a big difference in your bottom line.

The formula for calculating compound interest is slightly more complicated than simple interest; it looks like this:

##### Critical Point

The more times the interest in your account is compounded, the more money you'll earn. The best possible scenario would be continuously compounding interest, which compounds an infinite number of times. That sort of thing is possible; in fact, you'll learn to do it in precalculus.

##### Kelley's Cautions

Notice that the compound interest formula gives you the total *balance*, whereas the simple interest formula gives you the *interest* only—you had to add the principal to the interest in Example 1 in order to calculate the simple interest balance.

In this formula, *p* is the principal investment, *r* is once again the annual interest rate in decimal form, *n* is the number of times interest is compounded in one year, and *b* is the balance in your account after exactly *t* years have elapsed.

**Example 2**: How much more money would you make if you invested $3,000 in a savings account whose 6.25% annual interest rate was compounded monthly rather than quarterly, if you planned on leaving the money alone for 18 months? (To keep our answers consistent, round all decimals to seven decimal places as you calculate.)

**Solution**: You'll have to calculate two separate balances, one with *n* = 12 for monthly compounding, and one with *n* = 4 for quarterly interest compounding. The other variables will match for both problems: *p* = 3,000, *r* = .0625, and *t* = 1.5. Be careful! The variable *t* must measure years, not months; since 18 months is exactly a year and a half, *t* = 1.5, not 18.

##### Kelley's Cautions

In both simple and com-pound interest problems, *t* must be measured in years. Therefore, if your investment collects interest for 24 months, *t* = 2, not 24, since 24 months equals two years.

Calculate the balance if you compound monthly.

- = 3,000(1 + .0052083)
^{18} - = 3,000(1.0980173)
- = 3294.0519

Since banks don't award fractions of a penny, your final answer should only contain 2 decimal places: $3,294.05. Now calculate the balance if interest is only compounded quarterly.

- 3,000 (1 + 0.015625)
^{6} - 3,000(1.0974893)
- = 3,292.4679

This time, your balance is $3,292.47. Subtract the two balances to find the total difference: $3,294.05 - $3,292.47 = $1.58. Sure, $1.58 isn't a huge difference, but the larger the principal and the longer you leave the money in, the larger that difference will grow.

##### You've Got Problems

Problem 1: Calculate the *balance* of an account if its $5,000 principal earns:

(a) Simple interest at an annual interest rate of 8.25% for 20 years.

(b) Interest compounded weekly (*n* = 52) at an annual interest rate of 8.25% for 20 years.

If necessary, round decimals to 7 places during your calculations.

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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