2 parts of the circle are shaded is called the numerator. the circle is divided into 7 equal parts

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1 Fractions CHAPTER. Fractions revision of this circle is shaded. is a fraction. The top number shows that The top number of the fraction parts of the circle are shaded is called the numerator The bottom number shows that The bottom number of the fraction the circle is divided into equal parts is called the denominator Equivalent fractions are fractions that are equal. If both the numerator and the denominator of a fraction are multiplied by the same number then an equivalent fraction is obtained. 6 and are all equivalent fractions. A fraction can be simplified if the numerator and denominator can both be divided by the same number. This process is called cancelling. 6 6 When a fraction cannot be simplified, it is in its simplest form or in its lowest terms. Example Find the simplest form of 8 Solution 6 is the highest common factor of 8 and Divide both 8 and by 6 The simplest form of 8 is Fractions sometimes have to be put in order of size. To do this when fractions have the same denominator, compare the numerators.

2 CHAPTER Fractions When fractions have different denominators: find the lowest common denominator (lowest common multiple of the denominators) change each fraction to its equivalent fraction with this denominator compare the numerators to put the fractions in order. Example Write the fractions and in order of size. Start with the smallest fraction. Solution The lowest common denominator for the three fractions is Equivalent fractions for each of and with a denominator of are Starting with the smallest fraction, the order is 8 that is An improper fraction is one in which the numerator is greater than the denominator. For example, and 8 are improper fractions. The improper fraction can be thought of as over or as quarters. Similarly, the improper fraction can be thought of as over or as fifths. A mixed number consists of a whole number and a fraction. For example, and 6 8 are mixed numbers. Mixed numbers can be changed to improper fractions and vice versa. For example, to change to an improper fraction, work out how many quarters there are in There are quarters in and so there are 8 ( ) quarters in Add the extra quarters to get quarters. 8 So ( ) To change the improper fraction 6 to a mixed number, firstly work out how many whole ones there are. 6 sixths is ; sixths is and so sixths is whole ones and sixths remainder is the whole number 6 is the fraction Exercise A Copy the fractions and fill in the missing number to make the fractions equivalent. a b c d e 8 f g 6 h 8 8 Copy the fractions and fill in the missing number to make the fractions equivalent. a 8 b 8 8 c d e 6 f g 6 h

3 . Addition and subtraction of fractions CHAPTER Write each fraction in its simplest form. a 6 b 6 8 c Write each fraction in its simplest form. d e f 6 g 8 h 6 a b c d e 8 6 f g 8 h Write each set of fractions in order. Start with the smallest fraction. a b 6 c 6 d e f 6 8 g h 6 Julie and Susan have identical chocolate bars. Julie eats of her chocolate bar. Susan eats 8 of her chocolate bar. Who eats more chocolate? Give a reason for your answer. Ahmid says that is bigger than 6 because is bigger than Is Ahmid correct? You must give a reason for your answer. 8 Change these mixed numbers to improper fractions. a b c 6 d 8 e f 6 Change these improper fractions to mixed numbers. a b c. Addition and subtraction of fractions of the rectangle is shaded red and of the rectangle is shaded green. of the rectangle is shaded. So d e f To add fractions with the same denominator, add the numerators but do not change the denominator. For example, 6 which in its simplest form is To add fractions with different denominators, firstly find the lowest common denominator and then change each fraction to its equivalent fraction with this denominator. Example Work out Solution and The lowest common denominator is Change each fraction to its equivalent fraction with a denominator of So Add the numerators but do not change the denominator.

4 CHAPTER Fractions Example Work out 8 Give your answer as a mixed number. Solution The lowest common denominator is 8 Change to the equivalent fraction with a denominator of Add the numerators but do not change the denominator. Change the improper fraction to a mixed number. 8 8 Fractions can be subtracted in a similar way. Example Work out Solution 6 and 6 The lowest common denominator is Change each fraction to its equivalent fraction with a denominator of Subtract the numerators but do not change the denominator. Exercise B In questions give each answer as a mixed number or a fraction in its simplest form. Work out a b c d 8 e f g h Work out a 8 b c 6 d e f 6 8 g h 8 Work out a b c d e f g h 6 Work out a b c 8 d 8 e 6 f g h 6 Work out a b c 6 d e 8 f 6 g h 6 6

5 . Addition and subtraction of mixed numbers CHAPTER. Addition and subtraction of mixed numbers When adding mixed numbers, add the whole numbers and the fractions separately. Example 6 Work out Solution Add the whole numbers. Add the fractions. Add the two results. Sometimes adding the fractions gives an improper fraction. For example, adding the fraction parts of and gives is an improper fraction. As a mixed number, 6 6 So Mixed numbers can be subtracted in a similar way. Example Work out Solution Subtract the whole numbers. Subtract the fractions. Add the two results. 6 Example 8 Work out Solution 8 Method and Change mixed numbers to improper fractions. 8 and The lowest common denominator is Change each fraction to its equivalent fraction with a denominator of 8 Subtract the numerators but do not change the denominator. Give the answer as a mixed number.

6 CHAPTER Fractions Method Subtract the whole numbers. Subtract the fractions. Add the two results. Exercise C Work out a b 8 c 6 d e f g h Work out a b 8 c d Work out a b c 6 d 8 Work out a b c d 6 e f 6 g h Work out a b 8 8 c 6 8 d 6 e 8 f 6 g 8 h 6. Multiplication of fractions and mixed numbers Multiplication by an integer is the same as repeated addition. So is the same as 8 To multiply a fraction by an integer, multiply the numerator of the fraction by the integer. Do not change the denominator of the fraction. Example Work out 6 Solution 6 6 Multiply the numerator of the fraction by the integer. Do not change the denominator. Simplify the fraction. The answer is an integer in this case. To multiply 6 by Multiply the numerators 6 Multiply the denominators Simplify the fraction. 6 ( ) of the area of the square is shaded. To multiply two fractions, multiply the numerators and then multiply the denominators. 8

7 . Multiplication of fractions and mixed numbers CHAPTER Example Work out Solution Multiply the numerators Multiply the denominators is in its simplest form. When multiplying fractions, it is sometimes possible to simplify the multiplication by cancelling. Example Work out Solution Cancel the and the When multiplying mixed numbers, first write the mixed numbers as improper fractions. Example Cancel the and the Work out Solution 8 8 Change each mixed number into an improper fraction. Cancel the and the Change the improper fraction into a mixed number. Exercise D Give each answer in its simplest form. Work out a b c d e 8 f g 8 h

8 CHAPTER Fractions Work out a b c d e f g h Work out a b 8 c d 6 8 f 6 h 6 e g 8 Work out a b c d e f g h Work out a b c d e 8 f 6 g 6 h 8 6 Work out a 6 6 b 8 c ( 6 ) 8 d ( ). Division of fractions and mixed numbers of this rectangle is shaded red. Divide the red area by Now 8 of the rectangle is shaded. So 8 also 8 So dividing by is the same as multiplying by is the reciprocal of To work out consider how many times goes into There are in whole squares, this is lots of So also So dividing by is the same as multiplying by is the reciprocal of Similarly, the reciprocal of (or ) is and the reciprocal of is To divide by a fraction change the division sign into a multiplication sign write down the reciprocal of the second fraction. 6

9 . Division of fractions and mixed numbers CHAPTER Work out Solution Example Example Work out 6 Give your fraction in its simplest form. The reciprocal of is Multiply by Solution The reciprocal of 6 6 So multiply 6 by is 6 Write the improper fraction as a mixed number. When dividing mixed numbers, first write the mixed numbers as improper fractions. Example Work out Solution Write the mixed numbers as improper fractions. The reciprocal of is Write the improper fraction as a mixed number. Exercise E Give each answer in its simplest form. Work out a 6 b 8 c 6 d 8 e f g h 6 Work out a b c e 8 f g 8 d 6 8 h 6 Work out a b c d e f g h Work out a b c d 6 8 e f 6 g h Work out a ( ) b c 8 ( 8 ) d 6

10 CHAPTER Fractions.6 Fractions of quantities A unit fraction has a numerator of and the denominator is a non-zero positive integer. Examples of unit fractions are and To find a unit fraction of an amount, think of that amount divided into equal parts. Example 6 Find of Solution 6 6 of 6 Finding of an amount is the same as dividing the amount into equal parts. To find a fraction of an amount where the numerator is more than, think of the calculation in two stages. Firstly, divide the amount by the denominator. Then multiply the result by the numerator. Example Find of Solution Divide by The result is 8 Multiply 8 by of 6 Another way to find a fraction of a quantity is to multiply the quantity by the fraction. For example 8 6 In mathematics, the word of means the same as Example 8 Find of Solution 8 of of means the same as of Change the improper fraction into a mixed number. Exercise F Find a of b of 8 c of d 6 of e of f 8 of g of 8 h of 6 Find a of b of c of d of e 6 of f of g of h 8 of 8

11 . Fraction problems CHAPTER Find a of b of grams c of 8 cm d of e 6 of m f of cm g of 8 kg h of 6 km Find a of b of c of 6 d of 6 e 8 of f of 8 g of 8 h of Find a of b 8 of c 6 of d 8 of 6 Find a of 8 b 8 of 6 c 6 of 8 d of 6 e 6 of 6 m f of g of 66 kg h of km Find a of b 8 of m c 6 of 6 km d of 8 cm e of kg f of grams g of h of 68.. Fraction problems Problems can involve fractions. In a cinema Example of the audience are women. 8 of the audience are men. All the rest of the audience are children. What fraction of the audience are children? Solution 8 6 Add and 8 to find the fraction of the audience who are women or men. Subtract from to find the fraction of the audience who are children. of the audience are children. Example A school has 8 pupils. 86 of these pupils are girls. of the girls like swimming. of the boys like swimming. Work out the total number of pupils in the school who like swimming. Solution 86 6 Work out the number of girls who like swimming Work out the number of boys in the school. Work out the number of boys who like swimming. 6 6 pupils like swimming. Work out the total number of pupils who like swimming. 6

12 CHAPTER Fractions Exercise G Simon spends of his money on rent and of his money on transport. a What fraction of his money does he spend on rent and transport altogether? b What fraction of his money is left? Dawn drives for of a journey. The journey lasts for 8 minutes. For how many minutes does Dawn drive? There are 8 students in a school. of the students are boys. Work out the number of boys in the school. Last season, Pearson Athletic won of its matches, drew and lost the rest. What fraction of its matches did it lose? of a garden is lawn. of the garden is a vegetable patch. The rest of the garden is a flower bed. What fraction of the garden is a flower bed? 6 8 of an iceberg lies below the surface of the water. The total volume of an iceberg is m. What volume of this iceberg is below the surface? There are 6 students in a class. Javed says that 8 of these students are boys. Explain why Javed cannot be right. 8 John walks miles to the next village. He then walks a further miles to the river. How far has he walked altogether? Tammy watches films. The first film is hours long and the second one is hours long. Work out the total length of the two films. Two sticks are metres and metres long. Work out the difference between the lengths of the two sticks. of a square is shaded. of the shaded part is shaded blue. What fraction of the whole square is shaded blue? In a crowd, of the people are female. of the females are girls. What fraction of the crowd is girls? DVDs are sold for each. of the goes to the DVD company. How much of the goes to the DVD company? A school buys some textbooks. The total price of the textbooks is The school gets a discount of 8 off the price of the textbooks. Work out how much the school pays for the textbooks. Alison, Becky and Carol take part in a charity relay race. The race is over a total distance of 8 kilometres. Each girl runs an equal distance. Work out how far each girl runs. 6

13 Chapter review questions CHAPTER Chapter summary You should now know: that equivalent fractions are fractions that are equal how to find an equivalent fraction by multiplying both the numerator and denominator by the same number how to cancel a fraction to obtain its simplest form how to order fractions by writing each fraction with the same denominator that an improper fraction is one in which the numerator is greater than the denominator that a mixed number consists of a whole number and a fraction how to convert between mixed numbers and improper fractions that to add (or subtract) fractions with the same denominator, add (or subtract) the numerators but do not change the denominator that to add or subtract fractions with different denominators, firstly find the lowest common denominator and then change each fraction to its equivalent fraction with this denominator how to add (or subtract) mixed numbers by adding (or subtracting) the whole numbers and the fractions separately how to multiply fractions by multiplying the numerators and then multiplying the denominators how to multiply or divide mixed numbers by firstly writing the mixed numbers as improper fractions how to divide by a fraction by changing the division sign into a multiplication sign writing down the reciprocal of the second fraction how to find a fraction of an amount by dividing the amount by the denominator and then multiplying the result by the numerator. Chapter review questions Work out a of 8 b of c of 6 d of e 6 of Work out a of b of 8 c of d of e of Change to improper fractions a b 8 Work out a b c d e Simon spent of his pocket money on a computer game. He spent of his pocket money on a ticket for a football match. Work out the fraction of his pocket money that he had left. (8 June ) 6

14 CHAPTER Fractions 6 Asif, Curtly and Barbara share some money. Asif receives 8 of the money. Barbara receives of the money. What fraction of the money does Curtly receive? (88 March ) Work out ( 6 ) (8 November ) 8 Work out, giving your answers as mixed numbers a b Work out and simplify where possible a b c 8 d e 8 Work out a b c 6 d 8 e Work out and simplify where possible a b c d e 6 Work out and simplify where possible a of b of c of 6 d of 8 e of Work out, giving each answer in its simplest form a 8 b 6 c 6 d 6 e 6 Work out, giving each answer in its simplest form a b c d 8 6 e 8 Work out a 6 b 8 (88 March ) 6 Work out 6 Give your fraction in its simplest form. (88 January ) Work out, giving your answers as mixed numbers a b 8 a Work out the value of Give your answer as a fraction in its simplest form. b Work out the value of + Give your answer as a fraction in its simplest form. (8 June ) A school has pupils. of these pupils are girls. of the girls like sport. of the boys like sport. Work out the total number of pupils in the school who like sport. (8 November ) Work out (88 March ) Work out Give your answer as a mixed number in its simplest form. 66