What is the simple interest formula?
Short answer. Simple interest is I = P × r × t: principal times the annual rate as a decimal times the years. $5,000 at 4% for 3 years earns 5,000 × 0.04 × 3 = $600, a total of A = P × (1 + r × t) = $5,600. Interest is charged only on the original principal.
Three letters, one multiplication. The formula is short enough to fit on a receipt, which is probably why it gets taught first and then quietly ignored by every bank you will ever use.
I = P × r × t I interest, in money P principal, the amount borrowed or deposited r annual rate, as a decimal (4% = 0.04) t time, in years A = P + I = P × (1 + r × t)
"Simple" means the interest is worked out on P and nothing else. Interest that has already built up does not earn more interest. Every year adds exactly the same amount.
Worked examples
The only real work is getting r and t into the right shape. Once they are a decimal and a number of years, it is one line on the keypad.
$5,000 at 4% for 3 years I = 5,000 × 0.04 × 3 = $600 A = 5,000 + 600 = $5,600 $2,400 at 6% for 9 months t = 9 ÷ 12 = 0.75 I = 2,400 × 0.06 × 0.75 = $108 A = 2,400 + 108 = $2,508 $10,000 at 5% for 90 days t = 90 ÷ 365 I = 10,000 × 0.05 × 90 ÷ 365 = 123.287… ≈ $123.29
The percent-to-decimal step is the same one behind every percentage calculation: divide by 100. 4% is 0.04, 6.5% is 0.065, and 0.5% is 0.005, which is where people most often slip a zero.
The units trap
The rate and the time have to be in the same unit. An annual rate needs years. Feed it months and you get an answer twelve times too big.
$5,000 at 4% a year for 36 months 5,000 × 0.04 × 36 = 7,200 wrong 5,000 × 0.04 × (36÷12) = 600 right
The trap also runs the other way. Some short-term lenders quote a monthly rate. 1.5% a month for 8 months on $1,000 is 1,000 × 0.015 × 8 = $120, and here t stays in months because r is per month. The same loan written as 18% a year gives 1,000 × 0.18 × 8 ÷ 12 = $120. Same answer, as long as the units agree.
Days: 365 or 360?
For periods in days, you have to pick how long a year is. Using 365 is called exact interest. Using 360 is called ordinary interest, or the banker's rule, and it survives because 360 divides so neatly. It also charges more, because each day is a bigger slice of a shorter year.
$10,000 at 5% for 90 days 365-day year 10,000 × 0.05 × 90 ÷ 365 = $123.29 360-day year 10,000 × 0.05 × 90 ÷ 360 = $125.00 difference = $1.71
Small here, not small on a large commercial loan. If a contract says "actual/360", that is the second line.
Solving for the rate, the time, or the principal
Four quantities, one equation. Know three and you can get the fourth by dividing.
r = I ÷ (P × t) $150 earned on $2,500 over 2 years
150 ÷ (2,500 × 2) = 0.03 = 3% a year
t = I ÷ (P × r) how long for $1,200 at 5% to earn $300?
300 ÷ (1,200 × 0.05) = 5 years
P = I ÷ (r × t) what earns $450 at 3% over 5 years?
450 ÷ (0.03 × 5) = $3,000
One neat consequence: under simple interest, money doubles when r × t = 1, so the doubling time is 1 ÷ r. At 8%, that is 1 ÷ 0.08 = 12.5 years.
Why this is not what your bank does
Savings accounts and credit cards compound. Each period, the interest is added to the balance, and the next period's interest is charged on the bigger number. Over a year or two the difference is pennies. Over ten years it is not.
| $10,000 at 5% | Simple | Compounded yearly |
|---|---|---|
| After 1 year | $10,500.00 | $10,500.00 |
| After 2 years | $11,000.00 | $11,025.00 |
| After 5 years | $12,500.00 | $12,762.82 |
| After 10 years | $15,000.00 | $16,288.95 |
The simple column grows by $500 a year, forever. The compound column pulls ahead by $1,288.95 at ten years because the interest is earning interest. Push the compounding to its limit, splitting the year into ever smaller periods, and you arrive at the constant e.
Simple interest still shows up in real life. A bond that pays its coupon out in cash pays simple interest on its face value, since the coupons are not reinvested for you. Loans described as "simple interest" charge interest on the outstanding balance only, never on unpaid interest.
What a car loan actually charges
Most car loans and mortgages are amortized: a fixed monthly payment, where each month's interest is simple interest on whatever is still owed. Take $20,000 at 6.5% over 5 years. The first month's interest is 20,000 × 0.065 ÷ 12 = $108.33, and the rest of the payment chips away at the balance.
$20,000 at 6.5% over 5 years, paid monthly monthly payment ≈ $391.32 total interest over 60 payments ≈ $3,479 naive P × r × t on the full $20,000 = 20,000 × 0.065 × 5 = $6,500
The naive number almost doubles the real cost, because you do not owe the full $20,000 for five years. You owe less every month. P × r × t is exact for money that sits untouched. For money you are paying back, it is only the first month's story.
Why $1,500 at 7% for 3 years comes out as 315.00000000000006
Type 1500 * 0.07 * 3 into JavaScript or Python and both print 315.00000000000006. The answer is 315. The problem is that 0.07 has no exact binary representation, so the computer multiplies something a hair above 0.07, the same fault behind 0.1 + 0.2 = 0.30000000000000004. A spreadsheet will hide the tail until you compare the result with 315 or add up a few hundred rows of it. Prism's keypad works in exact decimal arithmetic, so 1,500 × 0.07 × 3 is 315 and the interest on your statement matches the interest on your screen.
P × r × t on the keypad, amortized loans in the loan tool
Simple interest is one line on Prism's exact keypad, and History keeps the working if you need to check it later. For a car loan or mortgage, the loan tool takes an amount, an APR and a term and shows the monthly payment and total interest. No ads, no subscription. See the app.