Simple interest pays the same amount every year, a fixed percent of the starting sum
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.
Simple and Compound Interest · A Savings Account at Simple Interest: the Interest Year by Year and the Year a Target Is Reached
QuestionPriya 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
Watch out for
- 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.
Percentages · A Savings Account Earning Simple Interest, with Money Taken Out After a Year
QuestionMr. Lim put $12000 into a savings account that pays simple interest at 3.5% a year. At the end of the first year, the year's interest was added to the account, and then Mr. Lim took $4420 out. The money left stayed in the account for a second year at the same rate. (a) How much interest did the account earn over the two years? (b) How much money was in the account at the end of the second year?
1.In the first year the account earns 3.5% of $12000: 3.5100 × 12000 = $420.
The first year earns 3.5% of $12000: 3.5100 × 12000 = $420. 2.The interest is added, so the account holds 12000 + 420 = $12420. Mr. Lim takes $4420 out, which leaves 12420 − 4420 = $8000.
With the interest added, the account holds $12420. Taking $4420 out leaves $8000. 3.In the second year the interest is worked out on the $8000 that is in the account: 3.5100 × 8000 = $280.
The second year earns 3.5% of the $8000 in the account: 3.5100 × 8000 = $280. 4.(a) The interest over the two years is 420 + 280 = $700.
(a) The interest over the two years is 420 + 280 = $700. 5.(b) At the end of the second year the account holds 8000 + 280 = $8280. Check: $12000 was put in, $700 of interest was added and $4420 was taken out, and 12000 + 700 − 4420 = 8280.
(b) At the end of the second year the account holds 8000 + 280 = $8280.
Answer: (a) $700; (b) $8280
Watch out for
- Working out the second year's interest on $12000, which gives $420 again and $840 in all. Interest is paid on the money in the account during the year, and after the withdrawal that is only $8000.
- Taking the $4420 out of $12000 before adding the first year's interest, which leaves $7580. The interest was added at the end of the first year, before the money was taken out, so the $4420 comes out of $12420.
Your turn
1.$600 is invested at 4% simple interest per year. How much interest does it earn in 4 years?