Simple Interest

What interest means

Interest is money connected with lending and borrowing. A saver may receive interest from a bank, while a borrower may pay interest to a lender. The starting sum is called the principal, and the interest rate tells you what percentage of that principal is paid or charged during each stated period.

For example, an interest rate of 4% per annum means 4% for each year. The abbreviation p.a. means per annum, or “per year”.

How simple interest grows

With simple interest, the interest for every period is calculated from the original principal. It is not recalculated from a growing balance. Consequently, the same cash amount is added in every equal period.

Suppose £4,800 is borrowed at 4% simple interest per year. One year's interest is

$$\frac{4}{100}\times £4{,}800=£192.$$

The fixed annual addition can be represented like this:

After three years, the simple interest is therefore £576. If the question asks for the total amount owed, add the original £4,800:

$$£4{,}800+£576=£5{,}376.$$

This constant addition is the defining feature of simple interest. By contrast, compound interest is recalculated using an amount that includes earlier interest; that is studied separately.

A formula for simple interest

Let:

• $P$ be the principal;
• $r$ be the percentage interest rate per period;
• $t$ be the number of those periods;
• $$ be the total .

Then

$$I=P\times\frac{r}{100}\times t.$$

Dividing $r$ by 100 converts the percentage rate into a decimal multiplier. The expression $P\times\frac{r}{100}$ gives the interest for one period, and multiplying by gives the interest over all the periods.

For £4,800 at 4% per year for three years:

$$I=4800\times\frac{4}{100}\times3=576.$$

The units of $t$ must match the period used by the rate. If the rate is annual, $t$ must represent a number of years.

Interest only or the total amount

A calculation can ask for two different results:

• total interest: $I$;
• total amount: the principal plus the interest.

If $A$ represents the total amount, then

$$A=P+I.$$

Substituting the simple-interest formula gives

$$A=P+P\times\frac{r}{100}\times t$$

or, after factorising,

$$A=P\left(1+\frac{rt}{100}\right).$$

For a savings account, $A$ is the final balance. For a loan, it is the principal plus the interest charged, assuming the simplified question includes no repayments or extra fees.

Applying the method in a financial context

Consider £5,000 invested for seven years at 3.5% simple interest per annum. First find one year's interest:

$$5000\times0.035=175.$$

The account earns £175 each year. Over seven years, the interest is

$$175\times7=1225.$$

The question may stop there if it asks for the interest. If it asks for the final balance, add the principal:

$$5000+1225=6225.$$

The investment therefore earns £1,225 in interest and has a total value of £6,225 after seven years. Stating what the result represents is important because £1,225 and £6,225 answer different questions.