© 2004  Rasmus ehf

Percentages and interest

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Lesson 6

Interest:  

If you keep money in a bank, the bank pays you for the use of the money. The money they pay is interest. Interest is calculated as a percent of the bank balance.

If you have 1500 euros in a bank account for a whole year and the interest rate is 12% pa. (pa. means per annum = per year), you can find the amount of interest by calculating the the percentage.

      interest rate (% per year) × principal = interest

0.12 × 1500 euros = 180

The amount of interest earned for the year is 180 euros. 

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Interest over time:

If you have 1500 euros in a bank account for half a year or 6 months, the interest earned is:

 is the interest for half a year. 

This can be calculated all at once by including the time factor (t) in the formula. 

  interest rate × principal × time = interest

0.12 × 1500 euros ×0.5 = 90 euros is the interest for half a year. 
half a year

If the amount is kept in a bank account for 3 months, the interest is calculated as follows: 

 interest rate × principal × time = interest        (rPt = I)

0.12 × 1500 euros × 0.25 = 45 euros

The interest earned for 3 months is 45 euros.                        Top of page!     

Compounding period:

A month is calculated as 30 days and 12 months in a year make 360 days. 

If you deposit 1500 euros in a bank account on August 26 at 12% interest pa., how much money will there be in the account on Nov. 8? 

First calculate the number of days in the interest compounding period:
period days
August (30-26) 4 
Sept. 30
Okt. 30
Nov. 8
Total:  72

72 days out of 360 are 0.2 of a year.  The interest earned is: 

0.12 × 1500 euros × 0.2 = 36 euros.

You will have the principal of 1500 euros plus interest 36  = 1536 euros.

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Calculating interest rates

Let's say that you have earned 45 euros in interest on a principal of 1500 euros over the last 3 months. How can you find out what the annual interest rate is?

 Note: three months is one fourth of a year (3/12 = 0.25 ).

interest rate × principal × time = interest Use the formula for calculating interest.
p × 1500 euros × 0.25 = 375 euros Put in the known values.
p × 375 euros = 45 euros Simplify by multiplying the principal by time
p = 45/375 = 0.12 = 12/100 = 12% Divide by the factor of r and convert the decimal into a percentage. 

r · P · t = interest     r = interest rate,      P = principal,     t = time

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Calculating the compounding period by using the interest formula

The interest earned on a principal of 18000 euros at 12% pa. was 72 euros. How long (compounding period) was the money kept in the bank account?

interest rate × principal × time = interest Use the formula for calculating interest.
0.12 × 18000 × t = v Put in the values for the interest rate and the principal.
18000 × t = 72 Simplify by multiplying the principal by the interest rate.
 t = 72/18000 = 0.4 Divide by the factor of t .
0.4 = 4/10 and 360 × 4/100 = 144 days Multiply the number of days per year by the result to get the number of days in the compounding period. 

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Calculating the principal

Principal in an account has earned interest for 288 days at 12 % pa. The amount in the bank account is now 1644 euros. How do we find the principal (the amount of money put in the account)? 

Note: 288 days of 360 is 0.8 of a year.

interest rate × principal × time = interest Use the formula for calculating interest.
0.12 × whole × 0.8 = interest Put in the values for the interest rate and the time.
p + 0.12 × whole × 0.8 = interests + p Add the principal P to both sides of the equation. 
p + 0.096p = 1644 Multiply 0.12 and 0.8 and put in the value for the principal plus interest. 
1.096p = 1644 1P + 0.096P = 1.096P
p = 1644/1.096 = 1500 euros Divide by the factor of P to find the principal. 

The principal was 1500 euros and it earned interest for 288 days.

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Interest compounded over years

Find 12% interest on 1500 euros for one year.

0.12 × 1500 euros = 180 euros  1500 euros + 180 euros = 1680 euros
1500 euros × 1.12 = 1680 euros 1680 euros is 112% of 1500.

Compounded interest can be calculated in one operation by adding 1 to the interest percent rate (1.12) and multiplying. The result is the principal plus the interest earned for one year. To find the new principal for additional years, continue to multiply by 1.12.

1 year 1500 × 1.12 = 1680 euros 1500 euros × 1.12 = 1680 euros
2 years 1680 × 1.12 = 1881.6 euros  1500 euros × 1.12 × 1.12 = 1881.6
3 years 1881.6 × 1.12 = 2107.4 euros 1500 euros × 1.12 × 1.12 × 1.12 = 2107.4 euros

Here we see the compounded principal after 3 and 5 years at 12% interest:

Compounded principal after 3 years    1500 euros × 1.123 = 2107.4 euros
Compounded principal after 5 years    1500 euros × 1.125 = 2643.5 euros

The principal plus interest after 5 years (compounded interest) is:

The formula for compounded principal is     

where:

H = compounded principal
h = initial principal
p = interest rate
x = number of years

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Try Quiz 6 in Percentages and interest.