Finance


Simple Interest


Once you are happy with percentages and formulas, then calculating interest should be a straightforward extension of these concepts.

If we look at Simple Interest first, because that is the simplest (sorry about the pun) to understand, although you are probably unlikely to come across it in real life.

We need to start with a definition - the principal is the original amount invested. Stating the concept of simple interest in a nutshell - interest on money in an account is calculated only on the principal, no matter how many years a sum of money is left in the account. This just means that the actual interest added is exactly the same amount each year (or whatever period is used for calculating interest).

So if you invest £ 1000 at 3% interest per annum, then that will attract £30 interest. Every year, £30 will be added to your account. After 20 years, the money in the account will have grown to £1600, but you will still be receiving £30 a year interest. The interest is just calculated on the principal, i.e. the original £1000.

The concept is fairly straightforward, but for completeness we could state a formula for calculating Simple Interest -

\[ I = \frac{PRT}{100} \]

where

I - interest earned
P - principal, i.e. initial amount invested
R - rate of interest (usually per year)
T - time period (in units used for rate of interest, usually years)

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Compound Interest


Compound interest is the one you are more likely to meet in real life. Here the interest is added after the first year to produce a new total, then for the second year interest is calculated on this new total, not on the principal. The idea carries on in the same way for successive years.

The arithmetic is obviously a bit more complicated. If you were to calculate it year by year, then using the same figures of £1000 principal and 3% per annum interest as used in the example for Simple Interest

First Year

Interest = 3% of 1000

\[ = \frac{3}{100} \times 1000 \]

   = £ 30

So total in account = 1000 + 30 = £ 1030

Second Year

Interest = 3% of 1030

\[ = \frac{3}{100} \times 1030 \]

   = £ 30.90

So total in account = 1030 + 30.90 = £ 1060.90

Third Year

Interest = 3% of 1060.90

\[ = \frac{3}{100} \times 1060.90 \]

   = £ 31.83 (to nearest penny)

So total in account = 1060.90 + 31.83 = £ 1092.73

and so on..

A formula can be stated for Compound Interest which can simplify things

\[ A = P \left( 1 + \frac{R}{100} \right)^T \]

where

A - total amount in account
P - principal, i.e. initial amount invested
R - rate of interest (usually per year)
T - time period (in units used for rate of interest, usually years)


Working from basic principles, you can see how this formula works -

First Year

Total in account = principal + "percentage rate of interest" of principal

or

Total in account = principal + $\frac{\mbox{rate of interest}}{100}$ × principal

Adopting the single-letter representation used above, and denoting the first-year total by A1, this becomes

\[A_1 = P + \left(\frac{R}{100}\right) \times P = P + \left(\frac{R}{100}\right)P = P\left( 1 + \frac{R}{100} \right) \]

Second Year

Total in account = first-year total + "percentage rate of interest" of first-year total

i.e.

Total in account = first-year total + $\frac{\mbox{rate of interest}}{100}$ × first-year total

Adopting the single-letter representation already used, and denoting the second-year total by A2

\[ A_2 = A_1 \left( 1 + \frac{R}{100} \right) \]

which was derived using the same train of logic as for the first year.

Since we know A1 from above, we substitute it into this expression

\[ A_2 = P\left( 1+ \frac{R}{100} \right)\left( 1 + \frac{R}{100} \right) \]

\[ A_2 = P\left( 1+ \frac{R}{100} \right)^2 \]

You can see how the general formula is emerging. If you need to, extend the process for another year to convince yourself of where the formula comes from.

Nowadays there is a tendency to always quote an annual interest rate, but in the past there were situations that could lead to misunderstandings if the annual rate was not considered, e.g.

  • Building societies adding interest every 6 months, making this a better option than bank accounts which only credited interest every year, when compound interest is considered.

  • Credit card companies quoting interest rates per month, which can give a misleading idea of the annual rate given the influence of compound interest.


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Hire of Equipment


The type of questions you will be asked will quite often consist of

  • a fixed basic charge
  • to which will be added a charge per day

Stated as a formula, this would be

P = C + nD

where

P - total price,   C - fixed basic charge;   n - no. of days;   D - charge per day

Example

The hire of a cement mixer involves a £12 basic charge, plus £9 per day. How much will it cost to hire it for 4 days ?

Total cost = 12 + (4 × 9) = 12 + 36 = £48

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Value Added Tax

Do not add VAT to anything in your calculations, unless you are instructed to do so in the questions - especially if you are given no information as to what rate the tax is levied ( you are not expected to have knowledge of the current VAT rate).

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Past Exam Questions


1. Betty plans to hire a minibus for a 4-day scout camp. She estimates that she will do a total of 125 miles. The hire company charges £30 per day plus 15p per mile. How much will the minibus cost for the trip?

  • A    £121.87
  • B    £138.75
  • C    £180.00
  • D    £245.00




The cost of hiring a floor sander is shown below

1.   What is the cost of hiring the floor sander for 4 days ?

  • A    £ 60
  • B    £ 72
  • C    £ 84
  • D    £ 120

2.   A customer has £ 120. What is the longest time she can hire the sander for ?

  • A    3 days
  • B    5 days
  • C    6 days
  • D    7 days

3.   Which formula shows the cost of hiring the sander for n days ?

  • A    cost in £ = 12 + 18 + n
  • B    cost in £ = 12 + 18n
  • C    cost in £ = 12n + 18
  • D    cost in £ = 12




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