Teaching mathematics
Teaching mathematics

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Teaching mathematics

3.1 Calculating percentages of amounts

When calculating a percentage of an amount, many learners reach for the method of putting the percentage over one hundred and multiplying by the amount over one. As an example, 35% of 60 is calculated below using this method.

multiline equation row 1 35 percent of postfix times 60 equation left hand side equals right hand side 35 divided by 100 multiplication 60 divided by one row 2 equation left hand side equals right hand side 35 multiplication 60 divided by 100 multiplication one row 3 equation sequence 2100 divided by 100 equals 21

This method works if you are good at arithmetic and can remember the procedure! However, it is more intuitive to use some common knowledge: that 10% is equivalent to one-tenth, 1% is equivalent to one-hundredth, 50% is equivalent to one-half, and 25% is equivalent to one-quarter. Other percentages can be worked out from these.

The mathematical cake, which you met in Section 1 on equivalence between fractions, decimals and percentages, also proves useful in helping younger learners to understand the concept of percentages of amounts. It can be used to find multiples of 10% of amounts by imagining the cake cut into ten slices as before. Watch Video 4 to find out about a useful approach to working out percentages of an amount.

Download this video clip.Video player: Video 4 Using the mathematical cake diagram to find percentages of an amount
Skip transcript: Video 4 Using the mathematical cake diagram to find percentages of an amount

Transcript: Video 4 Using the mathematical cake diagram to find percentages of an amount

INSTRUCTOR:
Here is a mathematical cake. Mathematical cakes are split into 10 equal slices. The whole cake is 100%. So each slice is 100% divided by 10, which is 10%. Here is a percentage question. Suppose the whole cake costs 3 pounds. How much would 70% of the cake cost? Well, 70% is seven slices of cake. If we work out how much one slice, or 10%, of the cake costs, then we can find the cost of 70% of the cake.
100% of the cake cost 3 pounds. So one slice costs 3 pounds divided by 10. That's 300 pennies divided by 10, or 30 pence. We can write this as naught pounds 30. One slice cost naught pounds 30, two slices cost naught pounds 60, three slices cost naught pounds 90, four slices costs 1 pound 20, and so on, all the way up the cake. Finally, 10 slices cost 3 pounds.
Now we can read off the answer to 70% of 3 pounds as 2 pounds 10. Work out these other percentages using the mathematical cake-- 40% of 3 pounds. Read across, and that's 1 pound 20. 80% of 3 pounds-- read across, and that's 2 pounds 40. 50% of 3 pounds-- read across, and that's 1 pound 50.
Suppose the cake weighs 500 grammes. We can use the mathematical cake method to work out multiples of 10% of 500 grammes. 10% of 500 grammes is the same as 1/10 of 500 grammes, or 50 grammes. We can label the right-hand side of the cake going up in 50 grammes. No slices weigh naught grammes.
One slice weighs 50 grammes, two slices weigh 100 grammes, and so on all the way up the cake. Finally, 10 slices weigh 500 grammes. Then we can read off the answer to the following questions-- 20% of 500 grammes, 90% of 500 grammes, 30% of 500 grammes. Do these questions for yourself and then click on reveal answers to check your answers.
End transcript: Video 4 Using the mathematical cake diagram to find percentages of an amount
Video 4 Using the mathematical cake diagram to find percentages of an amount
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Answer

multiline equation line 1 20 percent of postfix times equation left hand side 500 times normal g equals right hand side 100 times g line 2 90 percent of postfix times equation left hand side 500 times normal g equals right hand side 450 times g line 3 30 percent of postfix times equation left hand side 500 times normal g equals right hand side 150 times g

Once learners have been introduced to the mathematical cake diagram, they will certainly be familiar with finding 10% of an amount and multiples of 10% of an amount. It is also intuitive to find 25% and 50%.

For example, 35% of 60 can be found by doing the following:

  • 10 percent of postfix times 60 equals six
  • 30 percent of postfix times 60 equals 18 (multiplying 10% of 60 by 3)
  • five percent of postfix times 60 equals three (finding half of 10% of 60)
  • 35 percent of postfix times 60 equals 21 (adding the results of finding 30% and 5%)

Activity 9 Find percentages of 60

Timing: Allow 10 minutes

A useful exercise to do with your learners is to ask them what percentages of 60 they can find, encouraging them to modify the methods they have previously used for finding simple percentages of amounts. Try this one yourself (Figure 4).

Figure 4 Finding percentages of 60

Discussion

There are many possible questions and answers that you could have posed in the diagram.

Examples could be 50% of 60, 30% of 60, or 25% of 60.

Extend learners’ understanding by posing questions such as: 15% of 60, how could you find that? What about 76% of 60? Or 120% of 60?

Ask learners to pose ‘What percentage of 60?’ questions themselves.

So far, if your learners have been using the methods shown above, they will have found percentages of amounts by making reference to the equivalent fractions.

Decimal equivalents can also be used when calculating percentages of amounts and this is a more helpful form when using a calculator. Referring back to the mathematical cake diagram in Section 1.1, remember that 10 percent equals 0.1 comma 20 percent equals 0.2 comma etc.

To convert from percentages to decimals, the percentage has been divided by 100. This follows from an earlier observation about fractions, which stated an alternative meaning of the numerator and denominator, i.e. numerator divided by denominator. In other words:

x percent equation sequence equals x divided by 100 equals x division 100

For example, if calculating 76% of 60, the learner keys into the calculator:

equation left hand side 76 division 100 multiplication 60 equals right hand side

However, it becomes easy to convert 76% straight away to 0.76 and then multiply by 60.

It is worth raising a caution here about using the percentage button on a calculator. Since the percentage button functions in different ways on different makes of calculator , it is best not to use it. Instead, encourage learners to convert the percentage to a decimal and then multiply by the amount.

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