Simple Interest

Q: How do I find the number of years if I know the interest?

Rearrange the formula by dividing both sides by $P r$: $t = \frac{I}{P r}$. In the example, 150 dollars divided by 1000 dollars times 0.05 gives 3 years.

In one sentence: Simple interest says the interest is the starting amount times the yearly rate times the number of years, and interest never earns interest of its own.

$I = P r t$

Read it out loud: “I equals P times r times t. I is the interest, P is the principal, the amount you start with, r is the rate per year as a decimal, and t is the number of years. So interest equals principal times rate times time.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Percent and decimal: A percent is a number out of 100, from the Latin per centum, meaning per hundred. The sign % means divide by 100, so 5% is $\frac{5}{100} = 0.05$ . The formula needs the decimal form, so every rate is converted this way before you use it. More practice is on the math basics page.
Principal: In money, the principal is the starting amount: the sum you deposit or borrow before any interest is added. It is measured in dollars. The interest is always worked out from the principal, which is the main sum that the rest of the calculation is built on.
Per year and units: The word per means divide, and per year means for each year. A rate of 0.05 per year means 0.05 dollars of interest for each dollar of principal, for one year. Units help you check a formula: dollars times per year times years leaves dollars, so the answer is an amount of money.

What each symbol means

Symbol	Meaning
$I$	Interest, the extra money paid for using money, measured in dollars. It is the number the formula gives you. The total owed or saved is the principal plus I.
$P$	Principal, the amount you start with, measured in dollars. It is the sum deposited or borrowed before any interest is added, and in simple interest it is the only amount that earns interest.
$r$	Annual interest rate, the price of money for one year, written as a decimal. A rate of 5% is written 0.05, because 5% means 5 out of every 100. Its unit is per year, so 0.05 means 0.05 dollars of interest for each dollar of principal in one year.
$t$	Time, the number of years the money is borrowed or saved, measured in years so that it matches the yearly rate. Six months is t = 0.5, and three years and six months is t = 3.5.

The formula piece by piece

$r$: The yearly rate as a decimal, for example 0.05 for 5%. It is the share of the principal paid as interest in one year. It is always per year, so it must be paired with a time in years.
$P r$: The principal times the rate: the interest for one year. For 1000 dollars at 0.05, that is 50 dollars. Only the principal is used, which is why the yearly interest stays the same from one year to the next.
$P r t$: The one-year interest multiplied by the number of years, which gives the interest for the whole period. For 3 years, the 50 dollars is counted 3 times, giving 150 dollars.
$I = P r t$: The whole formula in one sentence: the interest equals the principal times the yearly rate times the number of years.

What the formula is doing, and why

Simple interest is the most direct way to price money over time. You pick a starting amount, the principal $P$ , and a rate $r$ for each year. The formula $I = P r t$ tells you how much interest builds up over $t$ years. Each year adds the same amount, because the same starting amount earns the same rate every year.

Why multiply? Interest is a share of the principal, so it adds up in steps. Multiplying $P$ by $r$ gives the interest for one year: 1000 dollars at 0.05 per year earns 50 dollars in one year. Multiplying by $t$ stacks that same yearly amount once for each year. Three years of 50 dollars is 150 dollars, which is why the three letters are multiplied together.

Why the rate is a decimal? Because 5% means five hundredths. If you keep the rate as 5 instead of 0.05, the answer comes out 100 times too big. Writing the rate as a fraction keeps the units honest: dollars times per year times years leaves dollars.

What the formula leaves out. Simple interest never earns interest on earlier interest. The 50 dollars earned in year one stays in the side pile and does not grow. This is why simple interest makes a straight line on a graph: the total rises by the same step each year. Many savings accounts and credit card balances use compounding instead, which is covered on the compound interest page.

An everyday picture

Think of a yearly cash-back reward on a sofa that cost you 1000 dollars. If the store gives back 5% of the price you paid each year, the reward is 50 dollars every year. The reward for year two is still 50 dollars, not more, because it is always worked out on the price you paid and never on the rewards you already received. Simple interest works the same way: the base never grows, so the yearly amount never grows.

Worked example: A loan between friends

Sam lends Alex 1000 dollars. They agree on simple interest at 5% per year, and Alex will repay the whole loan after 3 years. We want the interest Alex pays and the total Alex repays. In the formula, $P = 1000\ \text{dollars}$ , $r = 0.05$ per year and $t = 3$ years.

Write the formula and put in the numbers. We replace each letter with its value: the principal is 1000 dollars, the rate is 0.05 per year, and the time is 3 years. So $I = 1000 \times 0.05 \times 3$ , and the units are dollars times per year times years, which leaves dollars because the years cancel.
Find the interest for one year. Multiplying the principal by the rate tells us how much interest one year adds. Here $1000 \times 0.05 = 50$ , so the loan earns 50 dollars of interest in each year, measured on the 1000 dollars Sam first handed over.
Multiply by the number of years. Each year adds the same 50 dollars, so three years add three copies of it. We compute $50 \times 3 = 150$ , which means the interest for the whole loan is 150 dollars.
Add the interest to the principal to get the total. The total Alex repays is the 1000 dollars borrowed plus the 150 dollars of interest. That gives $1000 + 150 = 1150$ dollars in total, which is the amount Alex hands back after three years.
Check the year-by-year picture. After year one, the amount owed is 1050 dollars. The interest for year two is still 50 dollars, not more, because the formula works only on the original 1000 dollars. After year three the total is $1000 + 3 \times 50 = 1150$ dollars, the same answer as before.

Answer: Alex pays 150 dollars of interest over 3 years and repays 1150 dollars in total. Simple interest adds the same 50 dollars each year, because it is always worked out on the original 1000 dollars.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the rate as a whole number. Write 5% as 0.05, not 5. With 5 in the formula, 1000 dollars for 3 years would earn 15,000 dollars of interest instead of 150 dollars.
Using months as if they were years. The rate is per year, so the time must be in years too. Six months is 0.5 years, not 6.
Letting interest earn interest. Simple interest keeps the base fixed, so each year adds the same amount. If you add the rate to the growing balance each year, you have switched to compound interest, which uses a different formula.
Confusing the interest with the total. The formula gives I, the interest only. The total owed or saved is P + I, which is 1150 dollars in the example.

Where you will meet it

Short personal loans between friends or family, where the agreed charge is a fixed rate per year on the amount borrowed.
Some car loans and personal loans, which are quoted with simple interest worked out on the unpaid principal.
Bonds that pay a fixed coupon, where each payment is the face value times the coupon rate for the period.
Textbook problems and quick bank examples, where the rule is easy to check by hand.

Questions people ask

Is simple interest better than compound interest?

Neither is better in general. For savings, compound interest grows more with the same rate and time, because interest earns interest. For a debt, the same fact means compound interest costs more. The compound interest page compares the two side by side.

How do I find the number of years if I know the interest?

Rearrange the formula by dividing both sides by $P r$ : $t = \frac{I}{P r}$ . In the example, 150 dollars divided by 1000 dollars times 0.05 gives 3 years.

What if the time is in months?

Convert it to years first by dividing the number of months by 12. Four months is $\frac{4}{12}$ years, about 0.33 years, so 1000 dollars at 5% earns about 16.67 dollars in four months.

Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
Rule of 72: The rule of 72 says that dividing 72 by a yearly percentage rate gives a quick estimate of the number of years it takes money to double.
Effective Annual Rate: The effective annual rate says what a stated yearly rate really adds up to over a year, once interest is added more than once and earns interest of its own.