**There are a few common maths hurdles that budding accountants can trip up on. First Intuition tutor Gareth John provides some useful tips to avoid these mistakes. **

## Calculating averages

When businesses are dealing with large volumes of production involving large amounts of resource and costs, it is useful to be able to break these down into averages. This allows us to consider things on a smaller scale.

For example, if we produced 100,000 units using 200,000 kg of material then the average amount of material per unit would be 200,000 kg/100,000 units = 2 kg per unit. This can be useful for:

- Helping to plan resource requirements. If we are going to produce 2,000 units next week then we need to purchase around 4,000 kg of materials.
- Identifying unit production costs. If material costs £5 per kg then the total material cost per unit would be £10.
- Setting prices. Once we have worked out the average costs for making a unit we can add a margin to set a profitable price.

The main thing to be careful about when calculating averages is that you divide the two figures ‘the right way around’. A common mistake is to divide them ‘the wrong way around’ i.e. working out 100,000/200,000 = 0.5 kg per unit rather than the correct 2 kg per unit.

Have a go at these examples:

- Gohere Railways provide meals for their passengers. They made a total of 1,000,000 meals using a total quantity of 250,000 kg of ingredients. What is the average usage of materials per meal?
- Runup Ltd spent £210,000 on 700,000 kg of materials. What is the average cost per kg of their purchases?
- Walkdown Ltd made 200,000 units in 40,000 labour hours. What was the average time taken to make each unit?
- Strollaway plc paid staff £428,750 for 35,000 hours of work. What was the average wage rate per hour?

## Percentage changes

Percentages are a useful way to look at how figures are changing, perhaps over time. If I tell you that the price of a car has changed from £13,500 in year 1 to £14,850 in year 2, then we can see that the price has risen between the two years, but it is not immediately clear by how much. Using percentages (‘per cent’ means ‘parts of a hundred’) can make this much clearer.

To work out the increase as a percentage we can first work out the increase in pounds. £14,850 – £13,500 = £1,350. We normally work out percentage changes based on the starting figure, so in this case we would work it out as £1,350/£13,500 x 100 = 10%.

Another way to work this out is to calculate £14,850/ £13,500 which gives 1.1 which is ‘1 + the percentage increase as a decimal’ or ‘1 + 0.1’ or ‘1 + 10%’. As ever make sure you divide the two figures ‘the right way around’ with the starting figure at the bottom.

Have a go at these examples:

- A cost has increased from £80 to £100. What is the percentage increase in the cost?
- A cost had fallen from £150 to £120. What is the percentage decrease in the cost?
- A price has risen from £18 to £20. What is the percentage increase in the price?
- The interest rate on our bank loan is increased from 10% to 13%. What is the percentage increase in the interest rate?

## Using margin and mark up information

The relationship between costs and prices is a critical one in any business as it determines the profit (or loss) that is made.

- Margin is short for ‘margin on sales’ so the profit element is shown as a percentage of the sales value.
- Mark-up is short for ‘mark-up on cost’ so the profit element is shown as a percentage of the cost.

Here are some examples:

- If the sales price is set at £190 and the company has a target profit margin of 40% what is the maximum target cost that they can afford to incur?
- If a company has a target profit margin of 25% and spends £150 producing a unit what must the selling price be set at?
- If the sales price is set at £190 and the company uses a mark-up of 40% what is the production cost of the product?
- If a company spends £150 producing a unit and sets the price to give a mark-up of 25% what will the selling price be?

Once you have had a go at the examples in this article you can watch me talk through my answers**. **

You may encounter some of these calculations when using the **incomplete records** technique.

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**
Gareth John is a qualified chartered accountant and tutor at First Intuition.
**