In this chapter, you will learn to:

It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal or present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is often specified for a year, it may be specified for a week, a month, or a quarter, etc. The credit card companies often list their charges as monthly rates, sometimes it is as high as 1.5% a month.

Example 9.2. 

Problem

Ursula borrows $600 for 5 months at a simple interest rate of 15% per year. Find the interest, and the total amount she is obligated to pay?

The interest is computed by multiplying the principal with the interest rate and the time.

(9.1)I=Prt

(9.2)

The total amount is

(9.3)

Incidentally, the total amount can be computed directly as

(9.4)

Example 9.3. 

Problem

Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?

The total amount or the future value is given by .

(9.5)

Example 9.4. 

Problem

Darnel owes a total of $3060 which includes 12% interest for the three years he borrowed the money. How much did he originally borrow?

This time we are asked to compute the principal P.

(9.6)

So Darnel originally borrowed $2250.

Example 9.5. 

Problem

A Visa credit card company charges a 1.5% finance charge each month on the unpaid balance. If Martha owes $2350 and has not paid her bill for three months, how much does she owe?

Before we attempt the problem, the reader should note that in this problem the rate of finance charge is given per month and not per year.

The total amount Martha owes is the previous unpaid balance plus the finance charge.

(9.7)

Once again, we can compute the amount directly by using the formula

(9.8)

Banks often deduct the simple interest from the loan amount at the time that the loan is made. When this happens, we say the loan has been discounted. The interest that is deducted is called the discount, and the actual amount that is given to the borrower is called the proceeds. The amount the borrower is obligated to repay is called the maturity value.

Example 9.7. 

Problem

Francisco borrows $1200 for 10 months at a simple interest rate of 15% per year. Determine the discount and the proceeds.

The discount D is the interest on the loan that the bank deducts from the loan amount.

(9.9)

(9.10)

Therefore, the bank deducts $150 from the maturity value of $1200, and gives Francisco $1050. Francisco is obligated to repay the bank $1200.

In this case, the discount D=$150, and the proceeds P=$1200−$150=$1050.

Example 9.8. 

Problem

If Francisco wants to receive $1200 for 10 months at a simple interest rate of 15% per year, what amount of loan should he apply for?

In this problem, we are given the proceeds P and are being asked to find the maturity value M.

We have P=$1200, r=.15, t=10/12 . We need to find M.

We know

(9.11)P=M−D

but

(9.12)

therefore

(9.13)

(9.14)

(9.15)

Therefore, Francisco should ask for a loan for $1371.43.

The bank will discount $171.43 and Francisco will receive $1200.

In the the section called “Simple Interest and Discount”, we did problems involving simple interest. Simple interest is charged when the lending period is short and often less than a year. When the money is loaned or borrowed for a longer time period, the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded.

Suppose we deposit $200 in an account that pays 8% interest. At the end of one year, we will have $200+$200(.08)=$200(1+.08)=$216.

Now suppose we put this amount, $216, in the same account. After another year, we will have $216+$216(.08)=$216(1+.08)=$233.28.

So an initial deposit of $200 has accumulated to $233.28 in two years. Further note that had it been simple interest, this amount would have accumulated to only $232. The reason the amount is slightly higher is because the interest ($16) we earned the first year, was put back into the account. And this $16 amount itself earned for one year an interest of $16(.08)=$1.28, thus resulting in the increase. So we have earned interest on the principal as well as on the past interest, and that is why we call it compound interest.

Now suppose we leave this amount, $233.28, in the bank for another year, the final amount will be $233.28+$233.28(.08)=$233.28(1+.08)=$251.94.

Now let us look at the mathematical part of this problem so that we can devise an easier way to solve these problems.

After one year, we had

After two years, we had

But $216=$200(1+.08), therefore, the above expression becomes

After three years, we get

(9.18)

Which can be written as

Suppose we are asked to find the total amount at the end of 5 years, we will get

We summarize as follows:

Banks often compound interest more than one time a year. Consider a bank that pays 8% interest but compounds it four times a year, or quarterly. This means that every quarter the bank will pay an interest equal to one-fourth of 8%, or 2%.

Now if we deposit $200 in the bank, after one quarter we will have or $204.

After two quarters, we will have or $208.08.

After one year, we will have or $216.49.

After three years, we will have or $253.65, etc.

(9.21)

Therefore, if we invest a lump-sum amount of P dollars at an interest rate r, compounded n times a year, then after t years the final amount is given by