Interest

In this section we will be looking at Interest, we will look at it in three parts:
1) Simple Interest,
2) Simple Interest over Multiple Years and finally
3) Simple Interest over a Fraction of a Year.

We shall start by defining Interest:

What is Interest?
Interest is money you gain over time after putting your money in a bank account – it’s the banks way of thanking you for staying with them.
Interest is also the fee you pay for delaying the repayment of a debt – it’s your way of thanking the bank for lending you that money.
Simple Interest
Simple Interest depends on the Interest Rate and is calculated yearly.
Nb: ‘Yearly’, ‘annually’ and ‘p.a’ (‘per annum’) all mean ‘once a year’
Definition 1.1 – Interest
Given that an account pays x% Interest Rate p.a and a person deposits £Y, you can work out how much Interest the person receives p.a:
Interest = x x 0.01 x Y

Definition 1.2
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time

Example 1.1: If Tom’s bank account pays him 2% Interest Rate per annum and he deposits £5,000 into the account then:
i) Calculate the interest Tom receives after one year
ii) Find out how much money Tom will have in the account after one year.
Answer
i) Interest = 2% of £5,000
= 0.02 x 5,000
= 100
Thus Tom receives £100 in interest
ii) 5,000 + 100 = 5,100
Thus Tom will have £5,100 in his account after one year

Explanation:
i) Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 2
Y = Amount deposited = 5,000
Thus
Interest = 2 x 0.01 x 5000
= 0.02 x 5000
= 100
ii) Using Definition 1.2 we know
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
Amount in the account after one year = Amount Originally Deposited + Interest over one year
= 5000 + 100
= 5,100.

Example 1.2: Cody puts £123 into an an account which has an Interest Rate of 4.2% per year.
How much will Cody have in her account after one year?
Answer
Interest = 0.042 x 123
= 5.166
Amount in the bank after one year = Amount originally deposited + interest over one year
= 123 + 5.166
= 128.166
Thus Cody will have £128.17 in her account after one year.
Explanation:
Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 4.2
Y = Amount deposited = 123
Thus
Interest = 4.2 x 0.01 x 123
= 0.042 x 123
= 5.166
In Cody’s bank she after a year she will have the amount she originally deposited and the interest:
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
= 123 + 5.166
= £128.17

Nb we round to 2.d.p as this answer concerns money.

Simple Interest over Multiple Years
Simple Interest over Multiple Years addresses what happens if you have your bank account for more than one year.
Definition 2.1
Given that an account pays x% Interest Rate p.a, a person deposits £Y and the person leaves it in there for Z years, you can work out how much interest they gain after Z years:

Interest after Z years = Z x Interest [ or Z x x x 0.01 x Y ]

Example 2.1: If Kit leaves £240 in his bank account for 4 years and the account pays interest at a rate of 7% p.a. How much does he have in his account after 4 years?
Answer
Interest = 0.07 x 240
= 16.8

Interest after 4 years = 4 x 16.8
= 67.2
Amount in his bank after 4 years = Amount originally deposited + Interest over 4 years
= 240 + 67.2
= 307.2
Thus Kit has £307.2 in his account after 4 years.
Explanation:
i) Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 7
Y = Amount deposited = 240
Thus
Interest = 7 x 0.01 x 240
= 0.07 x 240
= 16.8
But you want the interest gained after 4 years, from Definition 2.1 we know:
Interest after Z years = Z x Interest
Here Z = 4
Thus
Interest after 4 years = 4 x 16.8
= 67.2
So after 4 years Kit has £67.2 in interest, but you want the amount he has in his bank after 4 years :
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
Amount in the account after 4 years = Amount Originally Deposited + Interest over 4 years
= 240 + 67.2
= 307.2

Example 2.2: Lily earns 4.5% interest per year in her bank account. If she started off with £200, how much will she have in her account after 5 years?

Answer
Interest = 0.045 x 200
= 9

Interest after 5 years = 5 x 9
= 45
Amount in her bank after 5 years = Amount originally deposited + Interest over 5 years = 200 + 45
= 245
Thus Lily has £245 in her bank account after 5 years
Explanation:
i) Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 4.5
Y = Amount deposited = 200
Thus
Interest = 4.5 x 0.01 x 200
= 0.045 x 200
= 9
But you want the interest gained after 5 years, from Definition 2.1 we know:
Interest after Z years = Z x Interest
Now Z = 5
Thus
Interest after 5 years = 5 x 9
= 45
Thus after 5 years Lily has £45 in interest, but you want the amount she has in her bank:
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
Amount in the account after 5 years = Amount Originally Deposited + Interest over 5 years
= 200 + 45
= 245.

Simple Interest over a Fraction of a Year
If the money is taken out of the bank before the year is over then only a fraction of the interest gets paid. The fraction is directly proportional to what point in the year the moneys taken out.
Definition 3.1
Given that an account pays x% Interest Rate p.a, a person deposits £Y and the person leaves it in there for M months, you can work out how much interest they gain after M months:

Interest after M months = M/12 x Interest [ or M/12 x x x 0.01 x Y ]

Example 3.1: Donna earns 5% interest per year in her bank account. If she started off with £200, how much will she have in her account after 6 months?

Answer
Interest = 0.05 x 200
= 10

Interest after 6 Months = 6/12 x 10
= 1/2 x 10
= 5
Amount in her bank after 6 Months = Amount originally deposited + Interest over 6 months
= 200 + 5
= 205
Thus Lily has £205 in her bank account after 6 Months
Explanation:
i) Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 5
Y = Amount deposited = 200
Thus
Interest = 5 x 0.01 x 200
= 0.05 x 200
= 10
But you want the interest gained after 6 months, from Definition 3.1 we know:
Interest after M months = M/12 x Interest
Now M = 6
Thus
Interest after 6 Months = 6/12 x 10
= 5
Thus after 5 years Donna has £5 in interest, but you want the amount she has in her bank:
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
Amount in the account after 6 months = Amount Originally Deposited + Interest over 6 months
= 200 + 5
= 205.

Example 3.2: Cate earns 3.7% interest per year in her bank account. If she started off with £560, how much will she have in her account after 2 months?

Answer
Interest = 0.037 x 560
= 20.72

Interest after 2 Months = 2/12 x 20.72
= 1/6 x 20.72
= 3.45333
= £3.45
Amount in his bank after 2 Months = Amount originally deposited + Interest over 2 months
= 560 + 3.45
= 563.45
Thus Cate has £563.45 in her bank account after 2 Months
Explanation:
i) Using Definition 1.1 we know
Interest = x x 0.01 x Y
Now x = Interest Rate p.a. = 3.7
Y = Amount deposited = 560
Thus
Interest = 3.7 x 0.01 x 560
= 0.037 x 560
= 20.72
But you want the interest gained after 2 months, from Definition 3.1 we know:
Interest after M months = M/12 x Interest
Now M = 2
Thus
Interest after 2 Months = 2/12 x 20.72
= 3.45
Thus after 2 months Cate has £3.45 in interest, but you want the amount she has in her bank:
Amount in the account after specified time = Amount Originally Deposited + Interest over specified time
Amount in the account after 2 months = Amount Originally Deposited + Interest over 2 months
= 560 + 3.45
= 563.45