Working with Percentages

Writing percentages as fractions ‘Percent’ means ‘out of 100’. To write a percentage as a fraction we write it over a hundred. For example: 23 46 100 46 100 = 23 50 46% = Cancelling: 50 9 180 100 180 100 = 9 5 = 1 4 5 180% = Cancelling: Explain that we can easily write a whole number percentage as a fraction over 100. However, if there is a common factor between the numerator and the denominator, we must cancel the fraction down to its lowest terms. If the percentage is not a whole number, as in the last example, we must find an equivalent fraction with a whole number numerator and denominator. We can then cancel if necessary. 5 3 7.5 100 = 15 200 15 200 = 3 40 7.5% = Cancelling: 40

Writing percentages as decimals We can write percentages as decimals by dividing by 100. The easiest way to do this is to slide the decimal two places to the left That’s because “percent “ means “out of 100” 46% = 46 ÷ 100 = 0.46 7% = 7 ÷ 100 = 0.07 130% = 130 ÷ 100 = 1.3 Explain that to convert a percentage to a decimal we simply divide the percentage by 100. 0.2% = 0.2 ÷ 100 = 0.002

Writing fractions as percentages To write a fraction as a percentage, we can find an equivalent fraction with a denominator of 100. For example: × 5 = 100 85 = 17 20 85 and 85% 100 × 5 If the denominator of the fraction is a factor of 100 we can make an equivalent fraction over a hundred. Remind pupils that as long as we multiply the numerator and the denominator of a fraction by the same number it will have the same value. Talk through both examples. For the second example, we write the fraction as an improper fraction first. An alternative would be to recognise that 1 = 100% and then convert 7/25 to 28/100. Adding 100% and 28% gives 128%. × 4 1 7 25 = = 32 25 128 = 100 128 and 128% 100 × 4

Alternate method 31.25 5 16 = 5 × 100 ÷ 16 = 31.25% 57.142.. 4 7 = We can convert fractions to percentages even if the denominator doesn’t divide into 100. 31.25 5 16 = 5 × 100 ÷ 16 = 31.25% 57.142.. 4 7 = 4 x 100 ÷ 7 = 57.14% Explain to pupils that when the denominator of a fraction is not a factor of 100 (i.e. 2, 4, 5, 10, 20, 25 or 50) it is more difficult to find an equivalent fraction out of 100. Ask pupils if they can think of a way to convert 5/16, for example, to a decimal. 5/16 means 5 ÷ 16 so we can simply enter this into the calculator. Ask pupils to work this out on their calculators. To write the decimal as a percentage we multiply by 100% (multiply by 100 and write % on the end). Emphasize that this does not change the value of the decimal because 100% is equal to one and multiplying by one has no effect. Ask pupils to use their calculators to convert 4/7 to a decimal. We have 0.571428 recurring. Multiplying by 100 gives us 57.14 % (to two decimal places). To convert a mixed number to a percentage we can write it as an improper fraction first. Alternatively, we can convert the fractional part to a percentage and then add it to the whole number. In this example we would have 5/8 = 62.5% plus 100% to give 162.5%. Give pupils further examples to convert using their calculators. Encourage pupils to to check the answer given by the calculator by estimating the given fraction as a fraction of 100. For example, if the fraction is less than 1/2 we would expect the corresponding percentage to be less than 50%. 5 8 = 1 = 13 8 13 × 100 ÷ 8 = 162.5%

Writing decimals as percentages To write a decimal as a percent you can multiply it by 100. You just have to slide the decimal 2 places to the right For example: 0.08 = 0.08 × 100 1.375 = 1.375 × 100 What is 0.47 as a percentage? 0.47 means 47 hundredths, so 0.47 is equivalent to 47%. We can perform this conversion by multiplying by 100%. Remember 100% means the same as ‘1 whole’ or ‘all of it’. So multiplying by 100% is the same as multiplying by 1. The amount remains unchanged. Reveal the examples on the board. = 8% = 137.5%

Calculating percentages using fractions Remember, a percentage is a fraction out of 100. 16% of 90, means “16 hundredths of 90”, or 16 100 × 90 = 4 25 18 × 90 5 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. = 72 5 = 14 2 5 = 14.4

Calculating percentages using decimals Write the percentage as a decimal, and then multiply. Suppose we want to work out 38% of 65. 38% = 0.38 So we calculate: 0.38 x 65 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.

Calculating percentages by finding 10% first What is 34 % of 82? 10% = 82 ÷ 10 = 8.2 So: 30% = 8.2 x 3 = 24.6 Also: 2% = 8.2 ÷ 5 = 1.64 So: 4% = 1.64 x 2 = 3.28

Now try these: Calculate the following by finding 10% first: 22% of 55 2. 34% of 180 3. 17.5% of 400 4. 56% of 60 5. 65% of 300 6. 110% of 94 7. 86% of 78 8. 13% of 9 Calculate the following using fractions: 18% of 45 2. 35% of 90 3. 26% of 35 4. 75% of 2.44 5. 67% of 130 6. 110% of 320 7. 92% of 8 8. 35% of 900 Calculate the following by using a calculator: 12% of 67 2. 56% of 89 3. 15.7% of 12 4. 8% of 43 5. 3.6% of 15 6. 123% of 23

Answers: Calculate the following by finding 10% first: 22% of 55 = 12.1 2. 34% of 180 = 61.2 3. 17.5% of 400 = 70 4. 56% of 60 = 33.6 5. 65% of 300 = 195 6. 110% of 94 = 103.4 7. 86% of 78 = 67.08 8. 13% of 9 = 1.17 Calculate the following using fractions: 18% of 45 = 8.1 2. 35% of 90 = 31.5 3. 26% of 35 = 9.1 4. 75% of 2.44 = 1.83 5. 67% of 130 = 87.1 6. 110% of 320 = 352 7. 92% of 8 = 7.36 8. 35% of 900 = 315 Calculate the following by using a calculator: 12% of 67 = 8.04 2. 56% of 89 = 49.84 3. 15.7% of 12 = 1.884 4. 8% of 43 = 3.44 5. 3.6% of 15 = 0.54 6. 123% of 23 = 28.29