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Fractions Mastery: Understanding, Comparing, and Applied Skills

Develop expertise in fractions: Equivalence, Comparison, Operations, Decimals Conversion, Mid-points. Practice using diagrams and examples to approach fractions from basics to advanced concepts.

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Fractions Mastery: Understanding, Comparing, and Applied Skills

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  1. KS4 Mathematics N3 Fractions

  2. N3 Fractions Contents • A N3.2 Finding fractions of quantities • A N3.1 Equivalent fractions N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  3. Equivalent fractions ×2 ×3 3 6 18 = 4 8 24 ×2 ×3 Look at this diagram: =

  4. Equivalent fractions ×3 ×4 2 6 24 3 9 36 ×3 ×4 Look at this diagram: = =

  5. Equivalent fractions ÷3 ÷2 18 6 3 30 10 5 ÷3 ÷2 Look at this diagram: = =

  6. Equivalent fractions spider diagram

  7. Cancelling fractions to their lowest terms Which of these fractions are expressed in their lowest terms? 14 20 3 15 14 32 16 27 13 21 35 15 A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors. 7 5 2 8 7 5 Fractions which are not shown in their lowest terms can be simplified by cancelling.

  8. Mixed numbers and improper fractions 15 15 15 is an improper fraction. 4 4 4 3 3 4 When the numerator of a fraction is larger than the denominator it is called an improper fraction. For example, We can write improper fractions as mixed numbers. can be shown as =

  9. Improper fraction to mixed numbers + + + + = 1 1 1 1 8 8 8 5 8 5 = + + + + 8 8 8 8 8 8 37 37 37 = 8 8 8 4 4 4 4 5 5 5 = 8 8 Convert to a mixed number. This number is the remainder. 4 37 ÷ 8 = 4 remainder 5 This is the number of times 8 divides into 37.

  10. Mixed numbers to improper fractions 3 3 1 1 1 = + + + 7 7 7 2 2 2 2 7 7 7 7 7 7 7 = + + + 23 = 7 7 Convert to a mixed number. … and add this number … To do this in one step, 3 3 … to get the numerator. 2 2 23 = 7 7 Multiply these numbers together …

  11. Find the missing number

  12. N3 Fractions Contents N3.1 Equivalent fractions • A • A N3.2 Finding fractions of quantities N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  13. Finding a fraction of an amount 2 2 What is of £18? 3 3 of £18 We can see this in a diagram: = £18 ÷ 3 × 2 = £12

  14. Finding a fraction of an amount What is of £20? 7 10 7 of £20 10 Let’s look at this in a diagram again: = £20 ÷ 10 × 7 = £14

  15. Finding a fraction of an amount 5 1 5 What is of £24? 6 6 6 of £24 = of £24 × 5 = £24 ÷ 6 × 5 = £4 × 5 = £20

  16. Finding a fraction of an amount To find of an amount we can multiply by 4 and divide by 7. 4 4 1 7 7 7 36 5 kg = kg 7 What is of 9 kg? We could also divide by 7 and then multiply by 4. 4 × 9 kg = 36 kg 36 kg ÷ 7 =

  17. Finding a fraction of an amount 2 3 multiply by the numerator and divide by the denominator of 18 litres When we work out a fraction of an amount we For example, = 18 litres ÷ 3 × 2 = 6 litres × 2 = 12 litres

  18. Finding a fraction of an amount 2 5 To find of an amount we need to add 1 times the amount to two fifths of the amount. 1 2 2 2 7 5 5 5 5 of 3.5 m = 1 so, of 3.5 m = We could also multiply by 1 What is of 3.5m? 1 × 3.5 m = 3.5 m and 1.4 m 3.5 m + 1.4 m = 4.9 m

  19. MathsBlox

  20. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A N3.5 Multiplying and dividing fractions • A

  21. Using decimals to compare fractions Which is bigger or ? 7 7 20 20 7 20 3 3 3 8 8 8 > We can compare two fractions by converting them to decimals. = 3 ÷ 8 = 0.375 = 7 ÷ 20 = 0.35 0.375 > 0.35 so

  22. Using equivalent fractions 5 5 5 5 12 24 12 12 24 12 Now, write and as equivalent fractions over 24. 3 3 3 3 ×2 ×3 8 8 8 8 = < = ×2 ×3 Which is bigger or ? Another way to compare two fractions is to convert them to equivalent fractions. First we need to find the lowest common multiple of 8 and 12. The lowest common multiple of 8 and 12 is 24. 9 10 and so,

  23. Ordering fractions

  24. Mid-points

  25. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.4 Adding and subtracting fractions N3.3 Comparing and ordering fractions • A • A N3.5 Multiplying and dividing fractions • A

  26. Adding and subtracting fractions 3 + 1 3 1 4 5 5 5 5 When fractions have the same denominator it is quite easy to add them together and to subtract them. For example, + = = We can show this calculation in a diagram: + =

  27. Adding and subtracting fractions 7 – 3 4 – 8 8 3 1 7 2 8 8 1 = = = 2 Fractions should always be cancelled down to their lowest terms. We can show this calculation in a diagram: = –

  28. Adding and subtracting fractions + + 1 3 4 1 7 9 3 9 9 9 1 + 7 + 4 12 1 1 9 9 1 = = = = 3 Top-heavy or improper fractions should be written as mixed numbers. Again, we can show this calculation in a diagram: = + +

  29. Fractions with common denominators 11 11 4 4 5 , and 12 12 12 12 12 5 11 + 4 + 5 8 2 1 1 20 12 12 12 3 12 Fractions are said to have a common denominator if they have the same denominator. For example, all have a common denominator of 12. We can add them together: = + + = = =

  30. Fractions with different denominators 2 9 What is – ? 15 11 4 15 – 4 18 18 18 18 5 6 Fractions with different denominators are more difficult to add and subtract. For example, We can show this calculation using diagrams: – = – = =

  31. Using diagrams 3 4 What is + ? 12 15 27 7 1 12 + 15 20 20 20 20 20 3 5 + = + = = =

  32. Using diagrams 1 4 7 1 10 25 14 25 – 14 11 20 20 20 20 What is – ? – = – = =

  33. Using a common denominator 3 5 7 1 3 1 + What is + ? 4 12 9 4 1 4 1) Write any mixed numbers as improper fractions. = 2) Find the lowest common multiple of 4, 9 and 12. The multiples of 12 are: 12, 24, 36 . . . 36 is the lowest common denominator.

  34. Using a common denominator 5 7 1 1 3 1 + What is + ? 12 4 9 9 4 ×4 ×9 ×3 5 63 82 10 4 15 12 36 18 36 36 36 36 36 36 36 ×9 ×3 ×4 5 2 2 63 + 4 + 15 = + + = = = 36 3) Write each fraction over the lowest common denominator. 63 4 15 = = = 4) Add the fractions together.

  35. Adding and subtracting fractions

  36. Using a calculator It is also possible to add and subtract fractions using the key on a calculator. a a b b c c we can key in For example, to enter 4 8 = Pressing the key converts this to: 4 8 The calculator displays this as:

  37. Using a calculator 2 4 3 5 a a b b + c c 2 3 + = 5 4 15 7 1 We write this as To calculate: using a calculator, we key in: The calculator will display the answer as:

  38. Fraction cards

  39. N3 Fractions Contents N3.1 Equivalent fractions • A N3.2 Finding fractions of quantities • A N3.5 Multiplying and dividing fractions N3.3 Comparing and ordering fractions • A N3.4 Adding and subtracting fractions • A • A

  40. Multiplying fractions by integers 4 9 multiply by the numerator and 4 9 divide by the denominator This is equivalent to of 54. 54 × When we multiply a fraction by an integer we: For example, = 54 ÷ 9 × 4 = 6 × 4 = 24

  41. Multiplying fractions by integers 4 5 5 7 7 7 What is 12 × ? 12 × 60 = 7 8 = = 12 × 5 ÷ 7 = 60 ÷ 7

  42. Using cancellation to simplify calculations 7 7 7 What is 16 × ? 12 12 12 We can write 16 × as: 16 × 1 3 9 1 = 3 4 28 = 3

  43. Using cancellation to simplify calculations 8 8 What is × 40? 25 25 We can write × 40 as: 8 40 25 1 64 × 5 12 4 = 5 8 = 5

  44. Multiplying a fraction by a fraction 3 3 4 2 8 8 5 5 = × 40 3 = 10 What is × ? To multiply two fractions together, multiply the numerators together and multiply the denominators together: 3 12 10 We could also cancel at this step.

  45. Multiplying a fraction by a fraction 12 5 5 4 What is × ? 25 6 5 35 12 × = 6 25 5 2 = Start by writing the calculation with any mixed numbers as improper fractions. To make the calculation easier, cancel any numerators with any denominators. 7 2 14 1 5

  46. Multiplying fractions

  47. Dividing an integer by a fraction 1 1 1 3 3 3 What is 4 ÷ ? 4 ÷ means, “How many thirds are there in 4?” 4 ÷ = 12 Here are 4 rectangles: Let’s divide them into thirds.

  48. Dividing an integer by a fraction 2 2 2 What is 4 ÷ ? 5 5 5 4 ÷ means, “How many two fifths are there in 4?” 4 ÷ = 10 Here are 4 rectangles: Let’s divide them into fifths, and count the number of two fifths.

  49. Dividing an integer by a fraction 3 3 1 3 3 What is 6 ÷ ? 4 4 4 4 4 6 ÷ means, ‘How many three quarters are there in six?’ 6 ÷ = 6 × 4 6 ÷ = 24 ÷ 3 8 × = 8 ÷ 4 × 3 There are 4 quarters in each whole. = 24 So, = 8 We can check this by multiplying. = 6

  50. Dividing a fraction by a fraction 1 1 1 1 1 1 1 What is ÷ ? 2 2 8 2 8 8 2 ÷ ÷ means, ‘How many eighths are there in one half?’ Here is of a rectangle: Now, let’s divide the shape into eighths. = 4

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