3.2 Percentage change: using multiplying factors
Consider the following problem.
A customer goes into a restaurant and pays for a meal. Because they had the ‘early bird’ menu, their bill is discounted by 25%. There is also a 10% service charge added to their bill. What difference does it make to the customer's bill if the waiter deducts the 25% and then adds the 10% service charge, or alternatively, adds the service charge and then makes the deduction?
Activity 10 Does the order matter?
Work out the solution to the above problem in both ways and see what the difference is. Don’t forget that the second percentage change is enacted on the result of the first percentage change, i.e. having deducted 25% from the bill, the 10% service charge is worked out on the resulting discounted bill.
You could choose to calculate the solution of the problem using different amounts for the bill. Or you could prove it once and for all using a general case, which requires you to use algebra, by using a letter to represent the amount of the bill.
One form of the solution uses algebra to find the general case:
Let the cost of the meal be c.
For 25% deduction followed by 10% increase:
For 10% increase followed by 25% deduction:
Were you surprised at the result? It does not matter which way round the final bill is calculated because a 25% deduction followed by adding on 10% works out the same as adding 10% first followed by deducting 25%.
To see why this is, consider the use of multiplying factors for each of the percentage changes.
A 10% increase means that, having started with 100% of the amount and adding an extra 10%, you end up with 110% of the amount.
A 25% decrease means that, having started with 100% of the amount and subtracting 25%, you end up with 75% of the amount.
Using decimals for the multiplying factors (although you can also use fractions as the multiplying factors), an elegant solution to the problem can be found as follows:
Subtracting 25% then adding 10% gives c
Adding 10% then subtracting 25% gives c
Both multiplications come to 0.825c because it does not matter which way round you multiply (this is known as commutativity).
In short, a percentage change can be calculated by using the appropriate multiplying factor, which is calculated by considering what has happened to the original 100%.
The use of multiplying factors for percentage change does make the calculations much simpler. However, it may be a difficult concept for younger learners to grasp. If this is the case, it is better to allow learners to work the long way round.
In mathematics it is always best to work with the conceptual understanding that learners have demonstrated and to build on this carefully. Learners can do, understand and succeed when they have grasped the conceptual underpinning for the mathematics they are doing. This was summarised by Richard Skemp (1976)as relational understanding. Otherwise, if there is no understanding, learners will resort to trying to learn rules which are easily forgotten. (Skemp referred to the learning of rules and procedures without a conceptual underpinning as instrumental understanding.)