Financial Calculations (AQA Level 3 Mathematical Studies (Core Maths))

Order of Operations (BIDMAS/BODMAS)

What is the order of operations (BODMAS/BIDMAS)?

• If there is more than one operation in a calculation then they should be done in the following order

  • Brackets: ( )

    • Perform any calculation(s) inside brackets first

  • pOwers or Indices (sometimes Order):  ², ³, √ and similar

    • These include powers, roots, reciprocals

  • Divisions or Multiplications: × or ÷

    • If there are more than one of these then work them out left to right

    • Division includes fractions

  • Additions or Subtractions: + or -

    • If there are more than one of these then work them out left to right

• The acronym BODMAS or BIDMAS can help you remember the order of operations

• Modern calculators are programmed to process calculations in this order

  • The calculation can then be typed in to the calculator exactly how it is written on paper

  • Ensure you enter long or complicated calculations carefully, checking if your answer seems sensible

    • Type in extra brackets if needed, to help define which calculation to perform first

In what order should fractions and roots be dealt with?

• Fractions mean division in calculations (BODMAS/BIDMAS)

• There may be "invisible brackets" around the numerator and around the denominator
  

  • E.g.  means

  • Instead of using brackets we extend the fraction line to show exactly what is on the top and what is on the bottom

• Roots are pOwers (BODMAS/BIDMAS)

• There may be "invisible brackets" under the root

  • E.g.  means

  • Instead of using brackets we extend the top line on the root symbol to show everything that is to be rooted

Bounds & Limits of Accuracy

What are bounds?

• Bounds are the values that a rounded number can lie between

  • The smallest value that a number can take is the lower bound (LB)

  • The largest value that a number must be less than is the upper bound (UB)

• The bounds for a number, , can be written as 

  • Note that the lower bound is included in the range of values but the upper bound is not

How do we find the upper and lower bounds for a rounded number?

• Identify the degree of accuracy to which the number has been rounded

  • E.g. 1 d.p., nearest 100, 2 s.f. etc.

• Divide the degree of accuracy by 2

• Add this value to the number to find the upper bound

  • E.g. If the number has been rounded to the nearest 10, add on 5

• Subtract this value from the number to find the lower bound

  • E.g. If the number has been rounded to the nearest 2 d.p., subtract 0.005

• The error interval is the range between the upper and lower bounds

  • Error interval:

How do we find the error interval for a truncated number?

• You may be given a question where the number has been truncated

• Truncating is "chopping off" a number rather than considering the value of the digits

  • E.g. The value 2.549 truncated to two decimal places would be 2.54 rather than 2.55

• As an example of finding the error interval for a truncated number,

  • Consider the first 3 digits of an answer, , to a calculation that have been written down as 2.95

    • The smallest value that the answer could have been is 2.95

    • The largest value that the number could have been up to (but not equal to) is 2.96 before it was truncated to 2.95

    • It could not have been 2.94... as that would truncate to 2.94

    • The error interval for the size of the number is 

• The truncated value should be the same as the smallest value in the error interval

Approximations in Finance

• Approximations are used in finance when an exact answer is not needed

• It is often a balance between speed and accuracy of calculation

  • In some scenarios it is more useful to quickly arrive at an answer

    • E.g. When budgeting for a building repair, is it going to cost £300 or £3000 or £30 000?

  • In other scenarios it is more useful to spend a greater amount of time finding an accurate answer

    • E.g. Once the scale of the building repair has been decided, comparing quotes from contractors

• The level of accuracy must be considered when making financial approximations

  • When creating a personal budget for monthly spending, rounding incomes and outgoings to the nearest £10 is probably appropriate

  • However when a global-scale company is estimating project costs, rounding figures to the nearest £1 m may be more appropriate

  • When banks pay interest to individuals or businesses, they may round interest down to the nearest penny

    • For an individual with tens of thousands of pounds, this may not make too much difference

    • For a business with potentially millions of pounds, it can have a larger impact

• The type of rounding should also be considered, taking the context into account

  • For a business, estimated costs can be rounded upwards, whilst income is rounded down

    • This would provide a more conservative estimate of profit (income - costs) and a larger safety margin if something doesn't go to plan

• Fermi estimation is often used to find approximations in financial contexts