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This icon indicates the slide contains activities created in Flash. These activities are not editable. This icon indicates an accompanying worksheet. This icon indicates teacher’s notes in the Notes field. For more detailed instructions, see the Getting Started presentation. 1 of 40 © Boardworks Ltd 2010 2 of 40 © Boardworks Ltd 2010 Adding and subtracting fractions When fractions have the same denominator it is quite easy to add them together. For example, 3 5 + 1 5 = 3+1 5 = 4 5 We can show this calculation in a diagram: + 3 of 40 = © Boardworks Ltd 2010 Adding and subtracting fractions Similarly, when fractions have the same denominator it is quite easy to subtract them. ÷4 7 8 For example, – 3 8 7–3 8 = = 4 8 = 1 2 ÷4 Fractions should always be cancelled to their lowest terms. We can show this calculation in a diagram: – 4 of 40 = = © Boardworks Ltd 2010 Adding and subtracting fractions Adding fractions with the same denominator is quite easy. 1 7 4 + + 9 9 9 1+7+4 = = 9 12 9 Improper fractions should be written as mixed numbers. ÷3 12 9 = 9 9 3 + 9 = 1 3 9 = 1 1 3 ÷3 + 5 of 40 + = © Boardworks Ltd 2010 Fractions with common denominators Fractions are said to have a common denominator if they have the same denominator. 11 4 5 , and For example, have a common denominator 12 12 12 of 12. Adding these together gives us a total of: 11 4 5 11 + 4 + 5 20 = + + = = 12 12 12 12 12 1 8 = 12 = 1 6 1 2 3 Subtracting these gives us a total of: 11 4 5 11 – 4 – 5 – – = 12 12 12 12 6 of 40 = 2 12 © Boardworks Ltd 2010 Fractions with common denominators 7 of 40 © Boardworks Ltd 2010 Fractions with different denominators Fractions with different denominators are more difficult to add and subtract. If fractions have different denominators we need to do something to make the denominators the same. 5 2 How would you calculate – ? 6 9 – 5 6 8 of 40 ×3 15 = 18 ×3 = 2 9 ×2 4 = 18 ×2 15 – 4 11 = 18 18 © Boardworks Ltd 2010 Using diagrams When fractions that are being added have different denominators, we need to find a common denominator. 3 3 How would you calculate + ? 5 4 Find a common denominator and make equivalent fractions. + 12 20 + = 15 20 = = 9 of 40 12 + 15 20 20 7 + 20 20 = = 27 20 1 7 20 © Boardworks Ltd 2010 Using diagrams When fractions that are being added have different denominators, we need to find a common denominator. How would you calculate 1 1 7 – ? 4 10 Find a common denominator and make equivalent fractions. – 25 20 10 of 40 – = 14 20 = 25 – 14 11 = 20 20 © Boardworks Ltd 2010 Using a common denominator 11 of 40 © Boardworks Ltd 2010 Equivalent fractions 12 of 40 © Boardworks Ltd 2010 Adding and subtracting fractions 13 of 40 © Boardworks Ltd 2010 Fraction cards 14 of 40 © Boardworks Ltd 2010 15 of 40 © Boardworks Ltd 2010 Multiplying integers by fractions Multiplying an integer by a fraction can be done by following a specific set of instructions. Multiply by the numerator and divide by the denominator. It doesn’t matter which order you perform the tasks in, so long as you multiply and divide the correct part of the fraction. 4 54 × 9 = 54 ÷ 9 × 4 or 4 54 × 9 = 54 × 4 ÷ 9 =6×4 = 216 ÷ 9 = 24 = 24 4 This is equivalent to of 54. 9 16 of 40 © Boardworks Ltd 2010 Multiplying integers by fractions In some cases, the question gives an idea about the most logical order to perform the multiplication and division. 5 What is 12 × ? 7 5 12 × = 12 × 5 ÷ 7 7 = 60 ÷ 7 60 = 7 = 17 of 40 8 It is difficult to divide 12 by 7, so in this example, it is better to perform the multiplication first. 4 7 © Boardworks Ltd 2010 Using cancellation to simplify calculations To make calculations simpler, it can sometimes be a good idea to cancel down before performing the multiplication. 7 What is 16 × ? 12 7 We can write 16 × as: 12 4 16 7 28 × = 3 1 123 = 18 of 40 9 1 3 If the number we are multiplying by and the denominator of the fraction share a common factor, cancel the common factor before multiplying. © Boardworks Ltd 2010 Using cancellation to simplify calculations Cancellation is particularly useful in multiplications that involve big numbers that are difficult to work out mentally. 8 What is × 40? 25 8 We can write × 40 as: 25 8 8 40 64 The number we are × = 25 5 1 5 multiplying by and the denominator of the fraction 4 share a common factor, = 5 allowing us to cancel the common factor. 12 19 of 40 © Boardworks Ltd 2010 Multiplying integers by fractions 20 of 40 © Boardworks Ltd 2010 Multiplying a fraction by a fraction When multiplying two fractions together, multiply the numerators together and multiply the denominators together. 3 2 What is × ? 8 5 ÷4 3 4 × 8 5 = 12 40 = 3 10 ÷4 Alternatively, cancel the numerator of the fraction we are multiplying by and the denominator of the first fraction. 21 of 40 3 4 1 3 = × 8 5 10 2 © Boardworks Ltd 2010 Multiplying a fraction by a fraction Dealing with mixed numbers requires an extra step to make the calculation slightly easier. What is 5 5 12 × ? 6 25 Start by writing the calculation with any mixed numbers as improper fractions. To make the calculation easier, cancel any numerators with any denominators. Convert the improper fraction back to a mixed number. 7 2 35 14 12 × = = 6 1 25 5 5 22 of 40 2 4 5 © Boardworks Ltd 2010 Multiplying fractions 23 of 40 © Boardworks Ltd 2010 24 of 40 © Boardworks Ltd 2010 Dividing an integer by a fraction Dividing an integer by a fraction with a value less than 1 will always result in a larger number. 1 What is 4 ÷ ? 3 How many thirds are there in 4? Here are 4 rectangles: If they are divided into thirds, you can see that there are 12 thirds in the four rectangles. 1 4÷ = 12 3 25 of 40 © Boardworks Ltd 2010 Dividing an integer by a fraction Calculations where an integer is divided by a fraction can be thought of as ‘how many of the fraction are in the integer?’ 2 What is 4 ÷ ? 5 How many two fifths are there in 4? Here are 4 rectangles: Divide them into fifths. Count the number of two fifths. 2 4÷ = 10 5 26 of 40 © Boardworks Ltd 2010 Dividing an integer by a fraction It’s not always possible to represent fractions using diagrams. We need another way to perform the calculations. 3 What is 6 ÷ ? 4 How many three quarters are there in six? 1 6÷ = 6 × 4 = 24 4 There are 4 quarters in each whole. 3 = 24 ÷ 3 =8 4 We can check this by multiplying. 6÷ 3 8× =8÷4×3 =6 4 27 of 40 © Boardworks Ltd 2010 Dividing integers by fractions 28 of 40 © Boardworks Ltd 2010 Dividing a fraction by a fraction Dividing a fraction by a fraction works in much a similar way. 1 1 What is ÷ ? 8 2 How many eighths are in one half? This diagram shows half of a rectangle. If the shape is now divided into eighths, you can see how many eights there are in one half. 1 1 ÷ =4 8 2 29 of 40 © Boardworks Ltd 2010 Dividing a fraction by a fraction With more complex fraction divisions, we can write an equivalent calculation to help perform the division. 4 2 What is ÷ ? 5 3 4 2 How many are in ? 5 3 An equivalent calculation involves multiplying by the denominator and dividing by the numerator. 4 5 2 2 can be written as: ÷ × 5 4 3 3 ÷2 5 2 × = 4 3 10 12 = 5 6 We have swapped the numerator and denominator and multiplied by this. Remember to simplify. ÷2 30 of 40 © Boardworks Ltd 2010 Dividing a fraction by a fraction With mixed numbers, convert the mixed number to an improper fraction before continuing with the calculation. What is 3 3 6 3 ÷ ? 7 5 3 18 = 5 5 18 6 ÷ = 5 7 21 = 5 31 of 40 = 4 3 18 7 × 5 61 1 5 Cancelling the fractions at this stage makes the next stages of the calculation slightly easier. © Boardworks Ltd 2010 Dividing fractions 32 of 40 © Boardworks Ltd 2010 Multiplying and dividing by fractions 33 of 40 © Boardworks Ltd 2010 34 of 40 © Boardworks Ltd 2010 How much was spent? 35 of 40 © Boardworks Ltd 2010 Bills to pay Mary takes home £1,200 a month in her job working at a bank. Of the money she takes home, she spends: 4 paying the rent on her flat 7 2 of the remainder paying for food. 5 How much does she pay in rent? How much does she spend on food? What fraction of her wages is she left with after paying her rent and spending money on food? 36 of 40 © Boardworks Ltd 2010 Fenced in John is painting the fence in his back garden. 5 His 10 litre tin of paint is full. 6 How much paint is in his tin? 8 Each fence panel requires 9 of a litre of paint. How many fence panels can he paint? John has 18 panels to paint in total, so he buys another 8 litre tin of paint. Does he have enough paint to complete all the panels in the fence? Show your working. 37 of 40 © Boardworks Ltd 2010 Healthy bones Joe likes to drink a lot of milk. Every day during the week, he drinks a set amount. 2 In the morning, he has of a pint. 3 In the afternoon, he has another 2 3 of a pint. At the weekend, he drinks 3 of a 4 pint at breakfast. How much milk does he drink each week? A 1 pint carton costs £0.43, a 2 pint carton costs £0.76 and a 4 pint carton costs £1.12. Investigate the cheapest way for Joe to get as much milk as he needs each week. 38 of 40 © Boardworks Ltd 2010 Fill in the gaps 39 of 40 © Boardworks Ltd 2010 Mixed up information Some students have some information about their year group. What fraction of the students were boys? What fraction had neither a blazer nor a jumper? What fraction had Science as their favourite subject? What fraction voted for Jamie as student rep? What is the smallest possible number of students in the year group? 40 of 40 © Boardworks Ltd 2010