Calculating percentages mentally Some percentages are easy to work out mentally: To find 1% Divide by 100 To find 10% Divide by 10 To find 25% Divide by 4 Teacher notes Discuss how each of the percentages on the board can be calculated mentally. 1% means 1/100, so to find 1% of something we divide by 100. For example, 1% of 623 is 6.23 (moving the digits two places to the right). 10% is equivalent to 1/10, so to find 10% of something we divide by 10. For example, 10% of 27 is 2.7 (moving the digits one place to the left). 25% is equivalent to 1/4, so to find 25% of something we divide by 4. For example, 25% of 48 is 12. We can divide by 4 mentally by halving and halving again. 50% is equivalent to 1/2, so to find 50% of something we divide by 2. For example, 50% of 68 is 34. To find 50% Divide by 2

Spider diagram Teacher notes Start by working out the percentage at the top, 1% of the value in the centre of the spider diagram. Work around the spider diagram in a clockwise direction to find 10%, 25% and 50% of the central value. The remaining four percentages can be calculated mentally by adding, subtracting or multiplying the amounts already found. Discuss any mental strategies used each time. Make the activity more challenging by changing the order or revealing the proportion and asking what percentage of the central amount it represents.

Calculating percentages using fractions Remember, a percentage is a fraction out of 100. 16% of 90, means “16 hundredths of 90”, or 4 18 16 100 × 90 = 16 × 90 100 25 Teacher notes When a calculation is too difficult to work out mentally we need to use an appropriate written method. One way is to use a fractional operator. We know that 16% means 16/100. Remind pupils again that, in maths, ‘of’ means ‘times’. We can therefore multiply 16/100 by 90. 16% of 90 means 16 hundredths times 90. We can write 16 × 90 ÷ 100 like this. Indicate the second stage of the calculation. Both 16 and 100 are divisible by 4 and so we can cancel. 90 and 25 are both divisible by 5, and so we can cancel again. Point out that it does not matter whether we cancel the 16 and 100 first or 90 and 100 first. If we do not cancel at this point then we will have a more difficult multiplication to do. We would also have to cancel at the end. How can we calculate 4 × 18 mentally? To multiply by 4 we can double and double again. Pupils may also suggest working out 4 × 20, 80, and then subtract 8. Or, using partitioning, 10 × 4 is 40, plus 8 × 4, 32, is 72. 72 divided by 5 is 14 remainder 2. Reveal 142/5. 5 = 72 5 = 14 2 5 .

Calculating percentages using fractions What is 23% of 57? We can use fractions: Working 23 100 × 57 × 20 3 23% of 57 = 50 1000 150 1150 = 23 × 57 100 7 140 21 + 161 Teacher notes Again, we know that 23% means 23/100 and that ‘of’ means ‘times’. We can therefore multiply 23/100 by 57. 23% of 57 means 23 hundredths times 57. Go through the method of finding 23/100 × 57 including the grid method for multiplying 23 × 57. Using long multiplication would do equally well and pupils may wish to use this method if they are more confident with it. We have a fraction over a hundred and so we can easily write the answer as a decimal if necessary. What is 1311/100 as a decimal? (13.11) 1 3 1 1 = 1311 100 1 = 13 11 100

Calculating percentages using decimals What is 4% of 9? We can also calculate percentages using an equivalent decimal operator. 4% of 9 = 0.04 × 9 = 4 × 9 ÷ 100 Teacher notes We can also work out percentages by writing the percentage as a decimal. What is 4% (or 4 hundredths) as a decimal? (0.04) Explain that ‘of’ means ‘times’ and so 4% of 9 can be written as 0.04 × 9. How can we work out 0.04 × 9? 0.04 is equal to 4 ÷ 100, so we can work out 4 × 9 and then divide by 100. Ask pupils to work this out mentally before revealing the next step in the calculation. You can see how using a decimal is equivalent to using a fraction. We still multiply by the percentage and divide by 100. Another way we could work this out is to find 1% of 9 which is 0.09 and then multiply by 4 to find 4%. 4 x 0.09 = 0.36. This is called a unitary method. = 36 ÷ 100 = 0.36

Using a calculator By writing a percentage as a decimal, we can work out a percentage using a calculator. Suppose we want to work out 38% of $65. 38% = 0.38 So we key in: Teacher notes Explain that by converting the percentage to a decimal we can also work out percentages on a calculator. It would be quite difficult to work out 38% of $65 without a calculator. We can estimate the answer by working out 40% of $65. 10% of $65 is $6.50 so 40% is 4 × $6.50 = $26. What is 38% as a decimal? Ask pupils how we would write 24.7 in pounds before revealing the answer. And get an answer of 24.7. We write the answer as $24.70.

Using a calculator We can also work out a percentage using a calculator by converting the percentage to a fraction. Suppose we want to work out 57% of $80. 57 100 57% = = 57 ÷ 100 So we key in: Teacher notes Ask pupils to estimate 57% of $80. For example, 60% of $80 = 6 x $8 = $48. Then ask pupils how they think we could use 57% as a fraction to work out 57% of $80 using a calculator. Reveal the correct sequence of key presses on the board. This method requires more key presses (it is less efficient). However, many pupils find it easier to remember. And get an answer of 45.6. We write the answer as $45.60.

Using a calculator We can also work out percentage on a calculator by finding 1% first and multiplying by the required percentage. Suppose we want to work out 37.5% of $59. 1% of $59 is $0.59 so, 37.5% of $59 is $0.59 × 37.5. We key in: Teacher notes Ask pupils if they know what 37.5% is as a fraction (3/8). Ask pupils to estimate what 37.5% of $59 is. Discuss the unitary method for calculating percentages. Discuss how we could write 22.125 in pounds. And get an answer of 22.125. We write the answer as $22.13 (to the nearest penny).