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Simple interest is calculated as a percentage of the principal and stays the same over time.

Example

Saoirse puts \(£250\) into a savings account which gives simple interest at a rate of \(7.5\%\) per annum (per year).

How much will Saoirse have saved after \(3\) years?

Every year, \(7.5\%\) of \(£250\) will be added as interest to Saoirse’s account.

\({7.5}\% {~of~} {£250} = {£18.75}\)

Each year \(£18.75\) interest will be added.

After \(3\) years interest to be added \(= {3}\times £18.75 = £56.25\).

Saoirse will have saved the principal + interest \(= £250 + £56.25 = £306.25\)

After \(3\) years Saoirse will have saved \(£306.25\).

It can be helpful to use a formula to calculate simple interest, provided you give the variables the correct values.

The formula is:

Simple Interest = \(\frac{(P ×T×R)}{100}\)

Where
P = Principal (in £s)
T = Time (in years)
R = Interest rate (\(\%\) p.a.)

Example

To show how the formula works, we can recalculate the last example:

Rory borrows \(£300\) from his bank.

The bank charges simple interest at a rate of \(9\%\) p.a. (per annum).

How much will Rory owe after \(4\) years?

P = \(£300\)

T = \(4\) years

R = \(9\%\) p.a.

Put these values into the formula.

Simple Interest = \(\frac{(300 × 4 × 9)}{100}\)

\(= £108\)

Rory owes \(£108\) interest + the principal of \(£300\)

\(= £408\)

If you are using the formula to calculate simple interest, don’t forget to add the principal if you want to know the total amount owed/saved.

Compound interest is interest that is calculated on the principal plus the amount of interest already earned.

Therefore, the amount of money that earns interest increases every year.

Example

Daniel invests \(£400\) at a compound interest rate of \(6\%\).

How much interest will he have earned after \(3\) years?

Interest earned in first year

\(= 6\% ~of~ £400\)

\(= £24\)

Principal for second year
\(= £400 + £24\)
\(= £424\)

Interest earned in second year
\(= 6\%~ of~ £424\)
\(= £25.44\)

Principal for second year
\(= £424 + £25.44 = £449.44\)

Interest earned in third year
\(= 6\%~ of ~£449.44\)
\(= £26.97\)

Total amount of interest earned
\(= £24 + £25.44 + £26.97\)\(= \boldsymbol{£76.41}\)

If compound interest is to be added over a large number of years, the calculation becomes very long and complex. In this case, it is convenient to use a formula.

Total amount \(= {P}\times{(1 +}\frac{R}{100})^t\)

P = Principal (original amount)
R = compound interest rate (%)
T = time (years)

Example

Daniel invests \(£400\) at a compound interest rate of \(6\%\).
How much interest will he have earned after \(8\) years?

P = \(£400\)
R = \(6\%\) per annum (p.a.).T = \(8\) years

Total amount after \(3\) years \(= {P}\times{(1 +}\frac{R}{100})^t\)\(= 400 (1 + 0.06)^8\)
\(= 400(1.06)^8\)
\(= £637.54\)

Interest earned \(= £637.54 - £400 = £237.54\)