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FRACTIONS A fraction of a whole When you talk about a fraction, you are talking about a part of a whole, namely a fraction of a whole. 1 3 1 4 In the above two figures, the sections shaded-in represent a fraction of the whole. In the first one the shaded area is a quarter of the whole and in the second one it is a third. Numerators other than one Fractions can be made up of more than one piece of the whole. 2 3 The above fraction is two-thirds, where two pieces of the same size have been taken together. When you place the third piece in position you complete the circle and have: 3 3 Three thirds or in other words, one whole circle. 1 Updated March 2013 The number on the bottom tells you the size (name) of the fraction you have, third, quarter etc. and is called the denominator. The number on the top tells you how many of that fraction you have and is called the numerator. NUMERATOR DENOMINATOR 3 - the numerator is 3 and the denominator is 5, telling you that you have three, 5 one fifth fractions. e.g. Equivalent fractions How many different ways can you express the fraction 1 ? 2 1 2 3 4 6 etc. 2 4 6 8 12 To find equivalent fractions you multiply the numerator and the denominator by the same number. 1 1 2 2 3 3 2 6 1 1 4 4 5 5 4 20 You can also do this with fractions that have numerators other than 1. 2 2 2 4 3 3 2 6 3 3 5 15 4 45 20 2 Updated March 2013 Sometimes you need to work the other way round, you have a fraction that needs to be simplified Given 12 you need to express it in its simplest form. 24 You need to find a number that will divide into both the numerator and the denominator. In this case 12 will divide into both. 12 12 1 24 12 2 15 15 5 3 25 25 5 5 12 12 6 18 18 6 2 3 Relative size of fractions It is important to be able to compare fractions and decide which is the larger and which the smaller. Compare 1 1 and 3 4 1 1 and the darker area is . As you can see, the darker area is 3 4 smaller than the shaded area. The total shaded area is 3 Updated March 2013 You represent greater than by using the symbol > and less than by using the symbol < So we can write: 1 1 4 3 The rule is: The larger the number on the bottom, the smaller the actual size of the piece. 1 1 2 8 e.g. 1 1 15 5 Mixed numbers and improper fractions Both Mixed Numbers and Improper Fractions are greater than one. They are just different ways of writing a fractional expression that is greater than one. A Mixed Number consists of a number of whole parts together with a fractional part. e.g. 2 3 4 1 4 5 13 1 2 An Improper Fraction is a fraction where the numerator is greater than the denominator e.g. 11 4 9 5 27 2 It is important to be able to convert from one to the other. Mixed and whole numbers to improper fractions You have 5 wholes and you want to find how many halves there are There are 2 halves in a whole, so there are 2 5 halves = 10 halves 10 Which can be written as 2 How many thirds are there in 4 wholes i.e. 4 ? 12 3 3 4 Updated March 2013 How many thirds are there in 3 1 ? 3 There are 3 thirds in a whole so you have: (3 3) + 1 thirds = 9 + 1 = 10 thirds Which can be written as 10 3 How many quarters are there in 4 (4 4) + 3 = 19 i.e. 3 4 19 4 The rule is: Multiply the whole number by the denominator, add the numerator and write above denominator 5 3 (5 5) 3 25 3 28 5 5 5 5 2 4 (2 7) 4 14 4 18 7 7 7 7 Improper fractions to mixed numbers You need to be able to work the other way. Given an improper fraction, you need to be able to write it as a mixed number. Write 5 as a mixed number. 3 The denominator tells you that the type of fraction is a third. The numerator tells you that you have 5 thirds. If you divide 5 by 3, you get 1 with 2 thirds left over. So you have 1 whole plus 2 thirds: 1 2 3 5 Updated March 2013 Write 19 as a mixed number 4 Divide 19 by 4 to find how many wholes you have: 19 4 = 4 and 3 left over Write 4 3 4 2 4 5 14 as a mixed number 5 14 5 = 2 and 4 left over Fraction multiplication What is 1 1 ? of 2 4 The above diagram shows the quarter shaded in. If you halve the shaded part each 1 of the now two shaded parts will be of the whole. 8 Find 1 1 of 3 2 The shaded part is 1 2 This has been divided into 3 parts. Each of the shaded parts is Each shaded part is 1 1 of 3 2 1 of the whole. 6 6 Updated March 2013 What is 2 3 of ? 3 4 All the shaded area is 3 of the whole. 4 The darker shaded area is 2 of the total shaded 3 area. The darker shaded area is So: 1 of the whole 2 2 3 1 of 3 4 2 Look at the answers: 1 1 1 of 2 4 8 1 1 1 of 3 2 6 2 3 1 of 3 4 2 To find the answer you multiply the numerators together and multiply the denominators together. 1 1 1 2 4 8 1 1 1 3 2 6 23 6 1 3 4 12 2 Since the solution can be arrived at by multiplying the numerators together and the denominators together, the of can be replaced by the x sign. This corresponds with whole numbers situations where 2 lots of 3 is the same as 2 x 3. e.g. 1 1 of 2 4 = 1 1 2 4 = 1 1 2 4 = 1 8 There are 2 other situations where the word of occurs with fractions: 3 lots of 1 4 and 1 of 4 2 7 Updated March 2013 The same rule can be applied here, remember 3 lots of Using the above rule, replace lots of with x : 3 1 3 is 4 4 1 4 The rule calls for numerators and denominators in both numbers What is the denominator for 3? Think of 3 wholes as 3 whole ones then it is logical to use 3 1 Remember the denominator indicates the number of pieces that the whole is cut into, in this case it is only 1. 3 lots of 1 becomes 4 4 lots of 3 = 5 Now look at 3 1 1 4 = 3 1 = 1 4 43 12 2 or 2 1 5 5 5 1 of 4 2 The of can be replaced with x again It becomes 1 of 4 = 2 What is 1 of $10? 5 1 4 = 2 1 4 2 = 2 8 Updated March 2013 3 4 Or If I had to share my $10 amongst 5 people, how much would each get? 1 1 1 0 2 10 2 5 5 1 1 i.e. 1 of $10 is $2 5 3 of a class of 20 passed the exam, how many passed? 4 3 3 2 0 5 20 15 4 4 1 1 i.e. 15 passed the exam Solving fraction problems using pictures Diagrams can be useful when solving problems involving fractions. Three-fifths of a sum of money is $60. What is the sum of money? Draw rectangle A to represent the whole amount TOTAL? A Divide it into 5 equal parts, each part representing one-fifth B Shade in three-fifths and label it $60 $60 The $60 can then be divided into 3 lots of $20 - so $20 is now seen to be onefifth of the total. $60 $20 Updated March 2013 $20 $20 9 $20 $20 The rest of the sections can be filled in with the value $20, demonstrating that the total is $100 3 of a sum of money is $4.50. What is the sum of money? 4 $4.50 $1.50 $1.50 $1.50 $1.50 Each quarter is $1.50 ($4.50 3) There are 4 quarters, so the whole sum of money is: $1.50 4 = $6 So far there have only been two fractions multiplied together, but you can multiply any number of fractions together. 3 7 5 10 9 14 You could do as before and multiply the numerators together and the denominators together 3 7 5 105 10 9 14 1260 Which you now have to simplify by finding numbers that will divide into both the top and the bottom. It is better to simplify before multiplying out. Look for a number on the top and a number on the bottom that have a common factor ( i.e. that will divide by the same number). 3 and 9 will both divide by 3 3 7 5 10 9 3 14 5 and 10 will both divide by 5 3 1 7 5 1 1 0 2 9 3 14 1 10 Updated March 2013 7 and 14 will both divide by 7 Leaving 3 1 7 1 5 1 1 0 2 9 3 1 4 2 1 1 1 1 2 3 2 12 This method is called cancelling. The important part is to cancel or simplify before multiplying out. 10 5 22 11 8 25 10 5 2 2 2 1 11 8 25 11 and 22 both divide by 11 5 and 25 both divide by 5 10 and 8 both divide by 2 10 5 1 2 2 2 1 11 8 2 5 5 1 0 5 5 1 2 2 2 1 11 8 4 2 5 5 5 1 2 There are still numbers that will cancel out 1 4 5 5 and 5 both divide by 5 5 1 1 2 1 4 5 1 5 1 1 2 2 and 4 both divide by 2 1 4 5 1 Finally leaving 1 1 1 1 1 2 1 2 11 Updated March 2013