Interest
When you keep money in a savings account, the bank pays you for the use of it. That payment is called interest. The money you put in is called the principal, and the bank states the interest as a percentage of the principal for each year, called the interest rate.
Save $200 at a rate of 5% a year, and each year the bank adds 5% of $200 to the account. With simple interest, the interest is always worked out on the principal alone.
The same amount every year
5% of $200 is , so one year of interest is $10. The next year the interest is 5% of the principal again, which is $10 again. The $10 already paid earns nothing, because simple interest is never worked out on earlier interest. So simple interest pays the same amount every year.
Year after year
The total in the account is called the balance. Each year adds another $10 to it: after one year the balance is $210, after two years it is $220, and after three years it is $230.
Three years of interest is three equal payments of $10, so the interest comes to 3 × $10 = $30, and the balance is $200 + $30 = $230.
The interest is $10 in every year, so the balance rises by the same $10 each year: $200, $210, $220, $230, $240.
The balance in dollars against the number of years. Each year adds the same $10, so the points lie on a straight line, from $200 at the start to $250 after 5 years. Growth by the same amount at every step is called linear growth.
One multiplication
One year of interest is the principal times the rate, divided by 100: 200 × 5 ÷ 100 = 10. For several years, multiply by the number of years as well.
Write P for the principal, R for the rate as a percentage and T for the time in years. The simple interest I is then . For $200 at 5% for 3 years, , the $30 found year by year. The balance at the end is the principal plus the interest: 200 + 30 = 230 dollars.
Using the formula
$1500 saved at 4% a year for 6 years earns dollars of interest, so the balance at the end is $1860.
T counts years, so a time given in months is turned into years first. 18 months is years, and $800 at 3% a year for 18 months earns dollars.
The formula also works backwards. If $600 earns $54 of simple interest in 3 years, then , which is 54 = 18R. Divide both sides by 18: R = 3, so the rate is 3% a year.
The usual mistakes
Stopping at one year. $10 is the interest for one year. Over 3 years simple interest pays it three times, so the interest is $30.
Giving the balance as the interest. $230 is everything in the account after 3 years. The interest is only the $30 that was added to the $200.
Working a later year out on the balance. 5% of $210 would include interest on the first $10 of interest, and simple interest never pays that. Interest worked out on a balance that includes earlier interest is called compound interest.
When money is taken out
The second problem below takes money out of the account partway through. The interest is still a percentage of the money in the account, so after a withdrawal the next year of interest is worked out on the smaller amount that is left.
Worked example: A Savings Account at Simple Interest: the Interest Year by Year and the Year a Target Is Reached
Question Priya puts $2500 into an account that pays simple interest at 4% a year. Simple interest is worked out on the money she put in, and never on the interest already paid. (a) Find the interest after 6 years, and her balance then. (b) Priya wants a balance of at least $3150. Find the first whole year in which she has it, and her balance in that year.
1.One year's interest is 4% of $2500, which is 4100 × 2500 = $100.
Simple interest is worked out on the $2500 alone, so one year pays 4100 × 2500 = $100. 2.Simple interest pays that same $100 every year, so after n years the interest is 100n dollars and the balance is 2500 + 100n dollars.
Every column carries the same $2500 with equal slices of interest on top: after n years the balance is 2500 + 100n dollars. 3.(a) After 6 years the interest is 6 × 100 = $600, so the balance is 2500 + 600 = $3100.
(a) After 6 years the interest is 6 × 100 = $600 and the balance is $3100. 4.For a balance of at least $3150, solve 2500 + 100n ≥ 3150. Subtract 2500 from both sides to get 100n ≥ 650, and then divide both sides by 100 to get n ≥ 6.5.
The dashed line is the target of $3150. The first six columns are all below it. 5.The interest is paid at the end of each year, so n has to be a whole number. The first whole number that is at least 6.5 is 7.
2500 + 100n ≥ 3150 gives 100n ≥ 650 and n ≥ 6.5, and the interest is only paid at the end of a year. 6.(b) In year 7 the balance is 2500 + 7 × 100 = $3200. Check: after 6 years she had $3100, which is below the target, and after 7 years she has $3200, which is above it.
(b) Year 7 is the first column above the line, with a balance of $3200.
Answer: (a) $600 of interest, so the balance is $3100; (b) year 7, when the balance is $3200
Common mistakes
- Working the second year's interest out as 4% of $2600. That is what compound interest does. Simple interest is always a percentage of the $2500 that was put in at the start, so every year pays exactly $100.
- Rounding n ≥ 6.5 down to 6 years. At 6 years the balance is only $3100, which is short of the target, so the number of whole years has to be rounded up to 7.
More simple and compound interest problems, worked step by step →
Worked example: Annual Simple Interest with Staggered Withdrawals
Question Mr. Lim deposited $12000 into a savings account that paid a simple interest rate of 3.5% per annum. At the end of Year 1, the bank credited the first year's interest into his account, after which Mr. Lim withdrew $4420 from the account. The remaining balance stayed in the account for Year 2 at the same interest rate of 3.5% per annum. (a) What was the total interest earned by Mr. Lim across the two years? (b) What was the total amount in Mr. Lim's account at the end of Year 2?
1.Year 1: Draw a bar of $12000.
Year 1 starts with $12000. 2.Interest credited (3.5%): 3.5100 × 12000 = $420.
3.5% of $12000 is $420 of interest. 3.Bar expands to $12420.
The account holds $12420. 4.Cut off $4420: remaining principal bar is 12420 − 4420 = $8000.
After the $4420 withdrawal, $8000 stays in. 5.Year 2 interest (3.5% of $8000): 3.5100 × 8000 = $280.
Year 2: 3.5% of $8000 is $280. 6.(a) Total interest earned: $420 + $280 = $700.
(a) Interest: $420 + $280 = $700. 7.(b) Final balance: $8000 + $280 = $8280.
(b) At the end of Year 2: 8000 + 280 = $8280.
Answer: (a) $700; (b) $8280
Common mistakes
- Calculating Year 2 interest on the original $12000 principal rather than on the reduced $8000 balance.
- Deducting the $4420 withdrawal from the initial $12000 before adding the Year 1 interest.