What is interest?
When you put money into a savings account, the bank will use your money, for example by lending it to other people.
They will pay you a certain amount for allowing this.
The money they pay you is known as 鈥榠苍迟别谤别蝉迟鈥.
The rate of interest is calculated on an annual basis or per annumEach year. (% p.a.).
When you borrow money, you will have to pay interest as well as paying back the original amount.
The original amount of money borrowed or loaned is called the 鈥榩谤颈苍肠颈辫补濒鈥.
The 鈥榠nterest rate鈥 is the % of the principal that is added on over the course of one year as interest.
The interest rate charged or earned depends on a lot of factors, including the financial conditions in the country at the time.
The interest rate, including and fees charged over one year, to borrow money is known as Annual Percentage Rate, APR.
The interest rate, including and fees charged over one year, to lend money is known as Annual Equivalent Rate, AER.
APR and AER make it easier to compare savings accounts and loans.
How to work out interest
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Simple interest
Simple interest is calculated as a percentage of the principal and stays the same over time.
Example
Saoirse puts \(拢250\) 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 \(拢250\) will be added as interest to Saoirse鈥檚 account.
\({7.5}\% {~of~} {拢250} = {拢18.75}\)
Each year \(拢18.75\) interest will be added.
After \(3\) years interest to be added \(= {3}\times 拢18.75 = 拢56.25\).
Saoirse will have saved the principal + interest \(= 拢250 + 拢56.25 = 拢306.25\)
After \(3\) years Saoirse will have saved \(拢306.25\).
Question
Rory borrows \(拢300\) from his bank.
The bank charges \(9\%\) simple interest per annum.
How much will Rory owe after \(4\) years?
Answer
Each year, \(9\%\) of \(拢300\) will be added to the amount that Rory owes.
\(9\%\) of \(拢300 = 拢27\)
Each year \(拢27\) interest will be added.
After \(4\) years \({4} \times 拢27 = 拢108\) will be added.
Rory will owe the principal + interest
\(= 拢300 + 拢108 = 拢408\)
After \(4\) years Rory will owe \(拢408\).
Simple interest formula
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 = \(\frac{(P 脳T脳R)}{100}\)
Where
P = Principal (in 拢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 \(拢300\) 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 = \(拢300\)
T = \(4\) years
R = \(9\%\) p.a.
Put these values into the formula.
Simple Interest = \(\frac{(300 脳 4 脳 9)}{100}\)
\(= 拢108\)
Rory owes \(拢108\) interest + the principal of \(拢300\)
\(= 拢408\)
If you are using the formula to calculate simple interest, don鈥檛 forget to add the principal if you want to know the total amount owed/saved.
Question
Use the simple interest formula to calculate the interest gained on \(拢2500\) over \(4\) years at a rate of \(6\%\) per annum.
Answer
Simple Interest = \(\frac{(P 脳 T 脳 R)}{100}\)
P = \(拢2500\)
T = \(4\) years
R = \(6\%\) p.a.
Interest = \(\frac{({2500}\times{4}\times{6})}{100}\)
\(= 拢600\)
Compound Interest
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 \(拢400\) at a compound interest rate of \(6\%\).
How much interest will he have earned after \(3\) years?
Interest earned in first year
\(= 6\% ~of~ 拢400\)
\(= 拢24\)
Principal for second year
\(= 拢400 + 拢24\)
\(= 拢424\)
Interest earned in second year
\(= 6\%~ of~ 拢424\)
\(= 拢25.44\)
Principal for second year
\(= 拢424 + 拢25.44 = 拢449.44\)
Interest earned in third year
\(= 6\%~ of ~拢449.44\)
\(= 拢26.97\)
Total amount of interest earned
\(= 拢24 + 拢25.44 + 拢26.97\)\(= \boldsymbol{拢76.41}\)
Question
Amelia borrows \(拢1500\) at a compound interest rate of \(8\%\) per annum (p.a.).
How much does she owe after \(2\) years?
Answer
Interest to be added in first year
\(= 8 \percent~of~拢1500\)
\(= 拢120\)
Principal for second year
\(= 拢1500 + 拢120\)
\(= 拢1620\)
Interest to be added in second year
\(= 8 \percent~of~拢1620\)
\(= 拢129.60\)
Amelia now owes \(= 拢1620 + 拢129.60\)
\(= \boldsymbol{拢1749.60}\)
Compound Interest Formula
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 \(拢400\) at a compound interest rate of \(6\%\).
How much interest will he have earned after \(8\) years?
P = \(拢400\)
R = \(6\%\) per annum (p.a.).T = \(8\) years
Total amount after \(3\) years \(= {P}\times{(1 +}\frac{R}{100})^t\)\(= 400 (1 + 0.06)^8\)
\(= 400(1.06)^8\)
\(= 拢637.54\)
Interest earned \(= 拢637.54 - 拢400 = 拢237.54\)
Question
Ryan borrows \(拢850\) at a compound interest rate of \(9 \percent\) per annum (p.a.).
Use the compound interest formula to calculate how much will he owe after \(4\) years?
Answer
P = \(拢850\)
R = \(9 \percent\) per annum (p.a.).
T = \(4\) years
Total amount after \(4\) years \(= {P}\times{(1 +}\frac{R}{100})^t\)
\(= 850\times(1 + 0.09)^4\)
\(= 850\times(1.09)^4\)
\(= 拢1199.84\) (nearest penny)
Ryan owes \(拢1199.84\) after \(4\) years.
Test section
Question 1
Calculate the simple interest on \(拢7000\) borrowed for \(5\) years at an interest rate of \(5.5\%\) per annum (p.a.).
Answer
袄(拢1925袄)
Question 2
Jamie puts \(拢450\) into a savings account.
How much will he have in his account after \(4\) years at a simple interest rate of \(12\%\) per annum (p.a.)?
Answer
袄(拢666袄)
Question 3
Emily invests \(拢1200\).
After four years, how much will her investment be worth if she is paid \(12\%\) compound interest per annum?
Answer
袄(拢1888.22袄)
Question 4
Mo borrows \(拢2400\) at 1\(5\%\) per annum (p.a.) compound interest for \(4\) years.
How much compound interest will he have to pay ?
Answer
袄(拢1797.62袄)
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