Fractions. Chapter Understanding fractions


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1 Chapter Fractions This chapter will show you how to find equivalent fractions and write a fraction in its simplest form put fractions in order of size find a fraction of a quantity use improper fractions and mixed numbers use the four rules for fractions convert between fractions, decimals and percentages recognise rational and irrational numbers handle terminating and recurring decimals write terminating and recurring decimals as fractions. Understanding fractions This cake has been cut into six equal pieces. One of the pieces has been eaten. A fraction has a top number called the numerator and a bottom number called the denominator. You say that 6 (one sixth) of the cake has been eaten. When the numerator is smaller than the denominator the fraction is a proper fraction. A proper fraction is always part of a whole. You can use pictures to show proper fractions. numerator denominator 6 piece eaten The cake is divided into 6 4 equal parts parts shaded equal parts parts shaded 4 is shaded. is shaded. 60 Number M0_CMC_SB_IGCSE_60_U0.indd 60 /6/0 6:44:
2 Fractions EXAMPLE What fraction of this diagram is shaded in? is shaded. equal parts parts are shaded You can use fractions to represent information given in words. EXAMPLE Hassan has 4 DVDs. 0 are sport DVDs, are music DVDs and the rest are film DVDs. Write each of these as fractions of his collection. Numerator number of Sport or Music or Film. Denominator total number of DVDs. Sport 0 4 Music 4 Film 0 4 EXERCISE a For each diagram write (i) the fraction shaded (ii) the fraction unshaded (a) (b) (c) (d) (e) For each of the following, draw a diagram to show the fraction shaded: (a) (b) 4 (c) (d) (e) 7 0 There are 7 animals in a room. There are dogs, 6 cats and rabbits. What fraction of the animals are: (a) dogs (b) cats (c) rabbits? Number 6 M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:44:
3 4 In a vehicle survey there were cars and trucks. (a) How many vehicles were surveyed altogether? (b) What fraction of the vehicles were cars? (c) What fraction were trucks? This is how Joti spent his day. Activity Sleeping School Eating TV Football Homework Hours 0 7 Fraction (a) Copy and complete the table to show the fraction of the day he spent on each activity. (b) Make a similar table for the way you spend your day. 6 A box of sweets contains 6 with green wrappers, 4 with red wrappers and with yellow wrappers. What fraction of the sweets have: (a) green wrappers (b) red wrappers (c) blue wrappers? Equivalent fractions These three diagrams are exactly the same size and have exactly the same amount shaded. 6 4 is shaded is shaded 6 is shaded The diagrams show that They are called equivalent fractions. There is a connection between these three fractions Number M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:44:6
4 Fractions You could also write it like this To find equivalent fractions, multiply or divide the numerator and denominator by the same number. When there is no number that divides exactly into the numerator and denominator, a fraction is in its simplest form. No number divides exactly into both and 4, so 4 is the simplest form of this fraction. EXAMPLE Find three more fractions that are equivalent to. (i) Multiply numerator and denominator by to get 4 (ii) Multiply numerator and denominator by to get 6 7 You can choose any multiplier to give an equivalent fraction, as long as you multiply both the numerator and the denominator. (iii) Multiply numerator and denominator by 0 to get Number 6 M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:44:7
5 EXAMPLE 4 Fill in the missing numbers to make equivalent fractions. % 6 % 0 % 0 4 % Start with the fraction you know, 6 in this example You can use the fraction you were given or any of the answers you have worked out. 0 7 EXAMPLE Write these fractions in their simplest form. (a) 0 (b) 6 4 (a) 0 (b) 6 4 (c) Answer (c) 40 7 Answer 4 Answer You could work out (b) more quickly if you spotted that both 6 and 4 divide exactly by 4 or 6 or. In (c), you might spot that both 40 and 7 divide exactly by 4 or. Even if you do not spot the quickest way, you will get the correct answer if you follow the rules. 64 Number M0_CMC_SB_IGCSE_60_U0.indd 64 /6/0 6:44:
6 Fractions EXERCISE b Find two equivalent fractions for each of these. (a) (b) 4 (c) (d) (e) 7 Find three equivalent fractions for each of these. (a) (b) (c) 4 (d) 7 (e) 0 Write each in its simplest form. Which of these fractions are equivalent to? Copy and complete these equivalent fractions. 4 (a) % 6 % % 0 % (b) 6 % % % 0 6 % (c) % % % % (d) % % 6 % % 6 (e) 0 % % % 4 % 6 Write each of these fractions in its simplest form. (a) 4 (b) 6 (c) (d) 6 (e) 0 (f) 0 (g) 6 0 (h) (i) 6 (j) 4 (k) (l) 7 6 (m) 4 60 (n) 4 4 (o) 6 Which of these fractions are not in their simplest form? Sumala said that 6 60 is equivalent to 4. Is she right? Give a reason for your answer. of the Gobi desert is without water. Write down three fractions equivalent to. Karl ate 0 of an apple pie and Bavna ate. Write these fractions in their simplest form. Who had the most apple pie, Karl or Bavna? Explain your answer Number 6 M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:44:
7 0 In a garden is grass, is flower beds and is a vegetable plot. (a) Write each fraction in its simplest form. (b) Draw a rectangle split into 6 equal parts to represent the garden. Shade in and label the fractions for the grass, flower beds and vegetable plot. Ordering fractions You can see that is bigger than. When fractions have the same denominator, the one with the bigger numerator is the bigger fraction. Which fraction is bigger, 4 or 4? These fractions do not have the same denominator. It is difficult to compare them.. means is greater than You can use equivalent fractions to help you to decide which is bigger Look for a number that is in the timestables of both denominators. 0 is the smallest number in both the 4times and the times tables. It is called the lowest common multiple (LCM). You will meet this again in Chapter is bigger than 0, so 4 is bigger than 4. You can write To order fractions, use equivalent fractions to write them with the same denominator, then compare them. 66 Number M0_CMC_SB_IGCSE_60_U0.indd 66 /6/0 6:44:
8 Fractions EXAMPLE 6 Put these fractions in order, smallest first:, 7 0, is the smallest number you can use. The smallest number that, 0 and 0 all divide into is The order is 60, 0 60, 60 or 0,, 7 0 Multiply both top and bottom by the same number. EXERCISE c In each question, write down the bigger fraction. (a), (b) 6, 6 (c) 4 7, 7 (d) 0, 7 0 (e), 7 (f), 4 (g) 4, 7 0 (h), 6 (i) 4, (j), 7 Fill in each of these using > or < or =. (a) % (b) % 4 4 (c) % 7 7 (d) %, means less than. means more than (e) % 4 (f) 4 % (g) % 4 (h) % 0 (i) % (j) 4 % (k) 6 % (l) % 7 0 (m) % 4 (n) 0 % (o) 6 % 7 0 (p) % 4 (q) 4 % (r) % (s) 4 % 6 (t) 7 % 7 Put these fractions in order, placing the smallest first. Put each fraction into its simplest form first. (a),, (b), 6, 6 (c) 4, 4, (d) 0,, 7 0 (e) 4,, (f),, 4 0 (g),, 4 (h), 7 7, (i), 0, 4 (j) 4, 7, 4 4 Ali has 40 of a bowl of cherries. Zainah has 7 of the cherries and Lin has 7 0. Place them in order from smallest to largest to find who has the biggest fraction of cherries. Number 67 M0_CMC_SB_IGCSE_60_U0.indd 67 /6/0 6:4:0
9 . Finding a fraction of a quantity In mathematics, of means multiply so of Another way to do this is to work out of 0 first, then multiply by because. of So of You can see that both methods give 4 for the answer. You can choose whichever one you prefer. EXAMPLE 7 A factory employs 6 people. of them are men. (a) How many men work in the factory? (b) How many women work in the factory? (a) of men work in the factory. (b) So the number of women Number M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:4:0
10 Fractions EXERCISE d Work out: (a) 4 of 4 metres (b) of 6 (c) of 60 kg (d) 7 of 6 litres (e) 0 7 of 40 minutes (f) of 4 pages Work out the following. (a) 6 of 6 km (b) 7 of days (c) of $60 (d) of $ (e) 4 of 40 g (f) of litres Which is bigger? (a) 4 of or of 4 (b) 0 of 00 or 7 of 6 (c) of 40 or 4 of 440 (d) of 04 or of 6 4 Habib is awake for of a day. How many hours is this? Hannah spends 4 of her pocket money at the shops and saves the rest. One week she gets $0 pocket money. (a) How much does she spend? (b) How much does she save? 6 Amit s grandfather says he can either have of $7 or of $0 for his birthday. Which one should he choose? 7 A man earns $40.0 per week. He has to pay 0 of this in tax. What will he have left? Change a day into hours first.. Improper fractions and mixed numbers This is a proper fraction. The numerator is smaller than the denominator. 4 is an improper fraction. The numerator is greater than the denominator. It can be represented in picture form as numerator denominator Number 6 M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:4:04
11 You can see that this is the same as 4 squares. 4 is a mixed number because it contains a whole number part () and a fractional part ( 4 ). You need to be able to write an improper fraction as a mixed number and also write a mixed number as an improper fraction. Writing an improper fraction as a mixed number For example, to change 4 to 4: Divide the numerator by the denominator and write down the whole number part of your answer. The remainder gives you the fractional part. 4 The denominator is the same as the denominator of the improper fraction. 4 4 remainder remainder denominator of mixed number EXAMPLE Write these improper fractions as mixed numbers. (a) 7 (b) 0 (c) 6 (a) 7 4 remainder, so 7 (b) remainder, so 0 6 (c) remainder, so EXERCISE e Write these improper fractions as mixed numbers. (a) (b) (c) 4 (d) (e) 4 (f) (g) 7 4 (h) 6 (i) 0 7 Write these improper fractions as mixed numbers. (a) (b) 47 4 (c) 00 (d) 4 (e) 6 6 (f) 7 7 (g) 4 (h) 7 (i) 40 It is important not to use a calculator when you see this symbol. You may be asked to not use one in the examination. 70 Number M0_CMC_SB_IGCSE_60_U0.indd 70 /6/0 6:4:0
12 Fractions Copy and fill in the missing numbers. (a) 4 % (b) 0 % (c) 4 4 % 4 (d) 47 % (e) 04 % 4 (f) 7 6 % (g) % 7 (h) % 7 (i) 4 4 % 4 In each case (i). write the improper fraction as a mixed number. (ii) write your mixed number in its simplest form. (a) 4 4 (b) 0 (c) 6 6 (d) 0 4 (e) 7 6 (f) 0 (g) 4 6 (h) 4 (i) 00 Change the improper fractions into mixed numbers then write each pair of numbers using. or,. (a) 7 (b) 0 (c) 4 6 (d) 4 (e) 0 0 (f) Write these improper fractions in order of size, smallest first. (a) 4, 7, 6 (b),, (c) 4,, 0 7 (d) 4 6, 7, (e) 7,, (f) 0 7, 4, 7 6 Writing a mixed number as an improper fraction For example, to change 4 to 4: Multiply the whole number part by the denominator of the fractional part. For example is the same as 4. Add the fractional part of your answer to the result of step Number 7 M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:0
13 EXAMPLE Write these mixed numbers as improper fractions. (a) (b) 4 7 (c) 0 4 (a) (b) (c) EXERCISE f Write these mixed numbers as improper fractions. (a) (b) 4 (c) 4 (d) (e) 6 (f) 4 (g) 4 (h) 6 Write these mixed numbers as improper fractions. (a) 7 (b) 4 (c) (d) 7 (e) 6 (f) 7 4 (g) 7 6 (h) 0 7 Sabina ate 4 oranges. Alise ate 4 oranges. How many orange quarters did each girl eat? 4 Joshua walks km to school. Talil walks km to school. Who has further to walk? On Saturday the temperature at noon was C. On Sunday the temperature at noon was C. Which day was hotter at noon? Explain why. 6 Erich says that is bigger than 00. He is wrong. Explain why. 7 Copy and complete with the correct symbols,,,. or. (a) % 7 (b) 7 % 4 (c) % 4 4 (d) % 6 (e) 4 % (f) 00 % 0 6 Change both numbers into improper fractions or mixed numbers to compare them. 7 Number M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:06
14 Fractions.4 The four rules for fractions You can add, subtract, multiply and divide fractions. Adding fractions When fractions have the same denominator it is easy to add them. You simply add the numerators. For example, Notice that you do not add the denominators. When fractions do not have the same denominator you need to use equivalent fractions to change one or both fractions so that the denominators are the same. Equivalent fractions are in Section. For example, 0 can be written as 4 0 So the addition becomes Multiplying both numerator and denominator by. 4 0 To add fractions, you need to change them to equivalent fractions with the same denominator. EXAMPLE 0 The denominators are and. Look for a number that is in the times and times tables. is in both. Work out See Section., Improper fractions and mixed numbers. EXAMPLE Work out 4 6. Method Continued... Write mixed numbers as improper fractions first. Multiplying by and by to get a denominator of in each fraction. Always give your answer as a mixed number. Number 7 M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:0
15 Method Add whole numbers together. Multiply to get a common denominator of. Convert improper fraction to mixed number and add. EXERCISE g Work out: (a) (b) 4 7 (c) 0 Work out: (a) 4 (b) (c) (d) 4 4 (e) 6 (f) 6 Work these out, giving your answers as mixed numbers. (a) 4 (b) 6 (c) 0 (d) 4 4 (e) 0 6 (f) 4 After a party I have of one cake and 6 of another identical cake left over. How much cake do I have altogether? I have a piece of rope metres long and another piece 4 metres long. What length of rope do I have altogether? 6 Juanita travelled by bus for 4 hours and then by train for hours. How long was her journey? 7 A box has a mass of kg. Its contents have a mass of kg. What is the total mass of the box and its contents? Subtracting fractions To subtract fractions, you need to change them to equivalent fractions with the same denominator. You use the same method as for adding fractions. 74 Number M0_CMC_SB_IGCSE_60_U0.indd 74 /6/0 6:4:0
16 Fractions EXAMPLE Work out: (a) (b) 7 6 (c) 0 (d) 4 6 The denominators are the same, so you can subtract. (a) 6 (b) (c) (d) Writing 7 as 4 6 Using a denominator of 0 and writing equivalent fractions. Convert to improper fractions first. These are the same fractions as in Example. Write the answer as a mixed number. EXERCISE h Work out: (a) (b) 6 6 (c) 4 Work out: (a) 0 7 (b) 6 (c) 4 Work out: (a) 4 4 (b) 4 (c) 4 I had bars of chocolate but ate 4 of a bar after lunch. How much do I have left? I cut 4 m of copper pipe from a length m long. How much copper pipe do I have now? 6 At my party we drank litres of cola. There were 4 litres of cola to start with. How much cola remained? Number 7 M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:0
17 7 of a garden is grass and is paving. (a) How much of the garden is grass and paving together? (b) How much of the garden is left? Kai spends of his allowance at the cinema and 6 on a magazine. What fraction does he have left? Multiplying fractions To multiply two fractions you multiply the numerators, and multiply the denominators. Change mixed numbers to improper fractions before multiplying. Writing a fraction in its simplest form is in Example. You need to write the answer in its simplest form. EXAMPLE Work out: (a) 4 7 (b) (c) 4 Multiply the numerators and the denominators. (a) (b) 0 Write as. Then multiply numerators and denominators. The answer is written in its simplest form. (c) First write as an improper fraction. Write the answer as a mixed number and then in its simplest form. 76 Number M0_CMC_SB_IGCSE_60_U0.indd 76 /6/0 6:4:
18 Fractions EXAMPLE 4 Molly buys metres of material to make a skirt. She only uses 4 of the material. How many metres of material does she use? 4 of 4 4 Remember that of means multiply. Write as an improper fraction. Molly uses metres of material. Write the final answer as a mixed number. EXERCISE I Work out: (a) 7 (b) 4 (c) (d) 4 4 (e) 4 7 (f) Work out: (a) of (b) 7 of 4 (c) 4 of 4 of means multiply. Work out: (a) (b) (c) 7 Convert the mixed numbers to improper fractions. (d) 7 (e) (f) Find the area of a rectangular photograph 4 cm long and cm wide. Area of rectangle length width. One pizza has a mass of kg. What is the mass of pizzas? 6 A rectangular path is metres long and metres wide. What is its area? 7 It takes minutes to fill a bag with sand. How long will it take to fill 0 bags? Number 77 M0_CMC_SB_IGCSE_60_U0.indd 77 /6/0 6:4:4
19 Reciprocals The reciprocal of 4 is 4 and the reciprocal of 4 is 4 The reciprocal of 6 is 6, and the reciprocal of 6 is 6. Remember that 6 6 To find the reciprocal of a fraction, you invert it (turn it upside down). When you multiply a number by its reciprocal, you always get. Note: 0 has no reciprocal. EXAMPLE Find the reciprocal of (a) 7 (b). (a) 7 Change all mixed numbers into improper fractions before you find the reciprocal. Check: 7 7 (b) 6 6 Reciprocal of is Check: 4 4 EXERCISE j Invert these fractions. (a) (b) 4 (c) 7 (d) Find the reciprocals of these fractions. (a) 7 (b) (d) (d) 4 (f) (g) 7 (h) (i) 0 7 Dividing fractions To divide two fractions invert (turn upside down) the fraction you are dividing by change the division sign to a multiplication sign. So means how many halves in 4? Answer: You turn the second fraction upside down. This is the fraction you are dividing by. 7 Number M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:
20 Fractions If the division involves mixed numbers, change these to improper fractions first. EXAMPLE 6 Work out: (a) 7 4 (b) 4 4 (c) Write as, then use the rule for division. (a) (b) (c) Write 4 as an improper fraction Write the answer as a mixed number in its simplest form. EXERCISE k Work out: (a) 4 4 (b) 4 (c) 7 4 (d) 4 4 (e) 4 4 (f) 4 Work out: (a) 4 4 (b) (c) 4 4 To wrap a parcel takes m of string. How many parcels can I wrap with 7 m of string? 4 litres of orange juice is poured into litre glasses. How many full glasses are there? How many pieces of ribbon m long can I cut from a piece 4 m long? 6 Share 4 cakes between 6 people. How much will they each receive? Give your answer in its simplest form. Number 7 M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:6
21 . Fractions, decimals and percentages You will need to know: about place value how to multiply and divide by 0, 00, 000 Changing a percentage into a fraction or a decimal Percentage (%) means out of a hundred. So if you score 60 marks out of 00 in a mathematics test your mark can be written as or as 60%. There is more on percentages in Chapter 4. To change a percentage to a fraction, you write it as a fraction with a denominator of 00. So 7% 7 00, 4% 4 00, 7% 7 00 When you have written a percentage as a fraction with denominator 00, you can change it to its simplest form. Always use a denominator of 00 to change a percentage into a fraction. EXAMPLE 7 Write these percentages as fractions in their simplest form. (a) % (b) % (c) % (d) % (a), (b) and (c) use the simplifying skills from Section.. (a) % (b) % (c) % (d) % Multiply by first to make the numerator a whole number. Don t stop at 00, you must simplify as much as possible. 0 Number M0_CMC_SB_IGCSE_60_U0.indd 0 /6/0 6:4:7
22 Fractions You already know how to divide by 00 using decimals. You move the digits places to the right. You can use this to change a percentage to a decimal. 7% See Chapter, Section means To change a percentage to a decimal: Write it as a fraction with a denominator of 00. Divide the numerator by 00 (move the digits places to the right). EXAMPLE Change these percentages to decimals. (a) 4% (b) 7% (a) 4% (b) 7% EXERCISE l Write these percentages as fractions in their simplest form. (a) 0% (b) 0% (c) 40% (d) 0% (e) % (f) 6% (g) % (h) 6% (i) % (j) 6% (k) 6% (l) 6% Copy and complete each of these. (a) % = % 40 (b) % = % 0 (c) 7 % = 7 % (d) 4% = % (e) % = 7 % (f) 7 % = % (g) % = % 60 (h) 4 7% = % 7 (i) % = % Copy and complete with the correct sign,,,. or. (a) 7% % 4 (b) % % 0 (c) 6% % (d) 4% % (b) 7 % % 7 40 (e) 66 % % Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
23 4 Write these percentages as decimals. (a) 6% (b) 4% (c) % (d) % (e) 0% (f) 60% (g) 0% (h) % (i) % (j) 7.% (k) 7.% (l).% Write these percentages as: (i) fractions in their simplest form (ii) decimals. (a) 0% (b) 0% (c) % (d) % (e) % (f) % (g) % (h).% 6 Two friends were comparing test results. Yvan got 70% and Jacques got 0. By changing them both into fractions in their simplest form, find which one scored the highest in the test. 7 Which of these percentages is equivalent to % 6% 40%? Changing a decimal or a fraction into a percentage Changing a decimal into a percentage is the reverse of changing a percentage to a decimal. 00 % So 0. (0. 00)% % To multiply by 00, move the digits places to the left. See Section.. EXAMPLE Change into percentages: (a) 0.07 (b).64 (a) 0.07 ( )% 7.% (b).64 (.64 00)% 64%.64., so the answer is more than 00%. Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
24 Fractions To change a decimal to a percentage, multiply by 00. To change a fraction into a percentage you multiply by 00. To do this you divide the numerator of the fraction by the denominator. EXAMPLE 0 Write these fractions as percentages. (a) 0 (b) 40 You can go directly from a fraction to a % like this: % When using a calculator, you can either use the fraction button as in part (a) or the division button as in (b). (a) 0 ( 0 00)% 0% (b) 40 ( )% 77.% To change a fraction to a percentage multiply by 00. Now that you can change between percentages, fractions or decimals, you can make comparisons between all three. EXAMPLE Write these in order of size, smallest first. 0. 4% % Since 0., 0., 0.4, 0.4 The order is 7 0, 0.,, 4% A calculator has been used to change the quantities into decimals. Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
25 EXERCISE m Write these decimals as percentages. (a) 0. (b) 0.7 (c). (d) 0. Write these fractions as decimals and then as percentages. (a) 0 7 (b) 4 (c) 4 (d) Write these as percentages and then put them in order from smallest to largest Write the following fractions as percentages. (a) 7 4 (b) (c) (d) 0 Copy and complete the following table of equivalent fractions, decimals and percentages. Percentage Fraction Decimal 60% % Yvette was comparing her test results. She had 6 0 in History, 7% in Mathematics and 40 in English. By changing all the results into percentages, put her results in order from lowest to highest. 7 In a sale you can buy a TV with % off or one with a 4 off the original price. Which one is the best offer? Write these in order from smallest to largest: (a) 64% (b) % 0.4 Change both into percentages to compare them. 4 Number M0_CMC_SB_IGCSE_60_U0.indd 4 /6/0 6:4:0
26 Fractions Arrange these in order of size starting with the smallest. (a) 7% (b) 4% % 0.4 (c) 0% 0 (e) 6 0. % 7 0 (d) 4% 00 0 Changing a decimal into a fraction To write a decimal as a fraction you need to look at the place value of the last significant figure. th h T t H U. (a) (b) For significant figures see Section.. For place value see Section.. (c) (d). In (a) the last significant figure is in the hundredths column. So simplifies to In (b) the last significant figure is in the thousandths column. hundredths have denominator So In (c) the last significant figure is in the thousandths column. So (d) is a mixed number it has a whole number part and a fractional (decimal) part. The last significant figure of the decimal part is in the hundredths column, so. 00. Decimals such as 0.4, 0. and 0.07 are terminating decimals. They end, or terminate, rather than going on for ever. thousandths have denominator cannot be simplified further. Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:0
27 EXERCISE n Change the following decimals into fractions. Give each answer in its simplest form. (a) 0. (b).7 (c) 0.0 (d) (e) 0.7 (f) 0.0 (g) 0.4 (h) 0.6 (i).0 (j) 6. (k) 4.0 (l).4.6 Definitions of rational and irrational numbers All numbers in our number system are either rational or irrational. A rational number is one that can be written in the form a b where a and b are integers and b 0. An irrational number is one that cannot be written in this way. You will see more of these later in the chapter. All integers are rational. For example 7 7 and All fractions are rational. For example 4,, Some decimals are rational. For example , 0. 0 Some, as you will see, are not. 6 Number M0_CMC_SB_IGCSE_60_U0.indd 6 /6/0 6:4:
28 Fractions.7 Terminating and recurring decimals All rational numbers can be written as decimals, some of which are terminating decimals and some of which are not. Here are two examples of terminating decimals: Terminating means that the decimal ends there is a finite number of decimal places. To change a fraction into a decimal you can do the division on your calculator. For example, 0.6 Only those fractions that have a denominator with prime factors of only and/or can be converted to terminating decimals. Fractions that do not satisfy this condition convert to decimals that repeat and do not terminate. Look at these fractions written as decimals When you work out the decimal answer on your calculator, the decimal places fill the whole of the calculator display. If you do the division by short or long division methods you will see that the decimals never end. The decimals do not terminate they are called recurring decimals This is what your calculator display will show when you work out 4 7. Recurring decimals have either a repeating digit or a repeating pattern of digits. Because they never end, you need to have a way of writing them which makes the repeating pattern clear. When a single digit repeats, you put a dot over this digit. 0. is written as is written as 0.46 Number 7 M0_CMC_SB_IGCSE_60_U0.indd 7 /6/0 6:4:
29 When more than one digit repeats, you put a dot over the first digit of the pattern and a dot over the last digit of the pattern is written as is written as Some fractions give decimals where the pattern does not start repeating immediately which is written as EXERCISE o Use any method to change these fractions into decimals. Then say if they are terminating or recurring decimals. (a) (b) 4 (c) 4 (d) (e) 7 (f) 7 (g) 6 (h) 6 (i) 6 (j) 7 Without converting these fractions into decimals, decide which of the denominators indicate terminating decimals and which indicate recurring decimals. (a) (b) (c) 6 7 (d) (e) 64 (f) 0 (g) (h) 7 4 (i) (j) Convert the following fractions into decimals, and write down how many numbers there are in the recurring patterns. (a) 7 (b) (c) 7 (d) 4 7 (e) 4 4 Investigate when happens when you multiply: (a) a fraction that gives a terminating decimal by another fraction that gives a terminating decimal. Will your answer always be a terminating decimal? (b) a fraction that gives a recurring decimal by another fraction that gives a recurring decimal. Will your answer always be a recurring decimal? (c) a fraction that gives a terminating decimal by another fraction that gives a recurring decimal. Will the answer always be terminating, recurring or can t you tell? For a terminating decimal, the prime factors of the denominator of the fraction can only be or. Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
30 Fractions Write out the full calculator display for the following recurring decimals, and the number of digits in the recurring pattern. Use them to answer the following questions. 6 7 (a) Is the number of digits in any of the recurring patterns larger than the denominator? (b) How many recurring digits are there in the decimals of fractions with an even number denominator? (c) Is there a connection between the denominators of the fractions with recurring digit? (d) Is there any connection between the denominators of the fractions with 6 recurring digits? (e) Do your observations work with other fractions that give recurring decimals? Test them out on at least another four fractions each. What can you say about finding patterns from a small set of numbers?. Converting recurring decimals into fractions To turn a decimal into a fraction is easy if the decimal terminates. Look at the place value of the last significant digit and write the fraction with this denominator. Simplify the fraction if possible. decimal places means hundredths, decimal places means thousandths, and so on. For example, When the decimal is a recurring decimal you need a special technique to find the equivalent fraction. EXAMPLE Write each of these recurring decimals as fractions in their simplest form. (a) 0. (b) (c) Continued... Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
31 (a) Let x 0. 0x. Subtracting: x Dividing both sides by : x When digit recurs multiply both sides by 0. When digits recur multiply both sides by 00. When digits recur multiply both sides by 000. (b) Let x Subtracting: x 4 Dividing both sides by : x 4 Dividing top and bottom by : x 00x (c) Let x x Subtracting: x 60. Dividing both sides by : x 60. Multiplying top and bottom by 0: x 60 0 Dividing top and bottom by : x 67 0 Dividing top and bottom by 6: x Example (c) can be done another way: Let x then 0x 6.00 and 0 000x Subtracting: 0x 60 Dividing both sides by 0: x 60 0 x You must not leave your answer as 60. because a fraction should consist of whole numbers. Notice that there is still quite a lot of work to do to get the answer into its simplest form. You will not score full marks if you do not simplify fully. This method requires two multiplications at the start but then has the advantage that the recurring decimal pattern is the same. When you subtract you automatically get whole numbers on the top and bottom of your fraction. Both methods give you the same answer. You can choose whichever method you prefer, but if you choose the second one you must remember to multiply x by multiples of 0 which will give you the same pattern of recurring decimals. EXERCISE p Write these recurring decimals using the dot notation. (a) (b) 0. (c) Number M0_CMC_SB_IGCSE_60_U0.indd 0 /6/0 6:4:
32 Fractions (d) 0. (e) (f) 0. (g) 0. (h) 0.77 (i) (j) 0.0 (k) (l) Write down the first 0 digits of these recurring decimals. (a) 0. (b) 0. 7 (c) (d) 0.7 (e) 0. (f) 0. 7 (g) 0.0 (h) Change these fractions to decimals. Write: (i) the full calculator display. (ii) the recurring decimal using the dot notation. (a) (b) 6 (c) 7 (d) 4 (e) (f) 7 (g) 7 (h) (i) (j) 4 4 Write these fractions as recurring decimals using the dot notation: (a) (b) (c) 7 (d) 4 (e) 6 (f) 4 (g) 4 (h) 6 (i) 7 (j) (k) 6 6 (l) 7 Find the fractions that are equivalent to the following terminating decimals. Express the fractions in their simplest terms. (a) 0.4 (b) 0.7 (c) Find the fractions that are equivalent to the following recurring decimals. (a) 0. (b) 0. (c) 0. (d) (e).666 (f) (g) (h) (i) 0. (j) (k) (l) (m).6 66 (n).0 (o) 4.0 Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
33 EXAMInATIOn QUESTIOnS Show how to work out 4 4 without using a calculator. [] (CIE Paper, Jun 000) (Write down 6 and 7%. Put one of the symbols,,, or. between 6 and 7% to make a correct statement. [] (CIE Paper, Jun 000) Showing all your working, calculate 4 4. [] (CIE Paper, Nov 000) 4. Write (a) 7 0 as a decimal, [] (b) 000 as a percentage. [] (CIE Paper, Jun 00) Copy and fill in the missing numbers in the following statement. = % = %. [?] (CIE Paper, Jun 00) 6 Showing your working, calculate 4. [] (CIE Paper, Nov 00) 7 Last week, Mr and Mrs Hernandez spent $0. This was of their earnings What were their earnings? [] (CIE Paper, Nov 00) Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
34 Fractions 6.% Five values are listed above. Find (a) which two are equal, [] (b) which one is the smallest, [] (c) which one is the largest. [] (CIE Paper, Nov 00) Write as a decimal 7 (a) 0, [] (b) 7%. [] (CIE Paper, Jun 00) 0 Write 4% as a fraction in its lowest terms. [] (CIE Paper, Nov 00) Ian and Joe start to dig a garden. They both dig at the same rate. (a) When they are halfway through the job, what fraction of the garden has Ian dug? [] (b) Keith arrives to help. All three dig at the same rate until the job is finished. (i) What fraction of the garden did Ian dig after Keith arrived? [] (ii) What fraction of the garden did Ian dig altogether? [] (CIE Paper, Nov 00) (a) Write in order of size, smallest first % [] (b) Convert 0.6 into a fraction in its lowest terms. [] (CIE Paper, Jun 00) Show all your working in the following calculations. The answers are given so it is only your working that will be given marks. (a) = 6, [] (b) 4 = 0. [] (CIE Paper, Jun 00) Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:6
35 4 Work out each of the following as a decimal. (a) (i) % [] (ii) [] (iii) 7 [] (b) Write %, 7/000 and 7 in order of size, smallest first. [] (CIE Paper, Nov 00) Write as a fraction in its lowest terms (a) 7%, [] (b) [] (CIE Paper, Jun 004) 6 Without using a calculator, work out 4 4 as a single fraction. Show all your working. [] (CIE Paper, Jun 004) Show how this calculation is done without using a calculator. Write down the working. [] (CIE Paper, Nov 004) Anne took a test in Chemistry. She scored 0 marks out of 0. Work out her percentage mark. [] (CIE Paper, Jun 00) Alphonse spends $ on food. This amount is 4 of his allowance. Calculate his allowance. [] (CIE Paper, Jun 00) 4 Number M0_CMC_SB_IGCSE_60_U0.indd 4 /6/0 6:4:
36 Fractions 0 Work out 6. Give your answer as a fraction in its lowest terms. You must show all your working. [] (CIE Paper, Jun 00) Number M0_CMC_SB_IGCSE_60_U0.indd /6/0 6:4:
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