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3Fractions, Decimals and Percentages Fractions, decimals and Chapter3percentages What Australian you will learn curriculum 3A Equivalent fractions (Consolidating) NUMBER AND ALGEBRA 3B Operations with fractions (Consolidating) Real numbers 3C Operations with negative fractions Investigate terminating and recurring decimals (ACMNA184) 3D Understanding decimals (Consolidating) Solve problems involving the use of percentages, including 3E Operations with decimals percentage increases and decreases, with and without digital (Consolidating) technologies (ACMNA187) 3F Terminating, recurring and rounding Money and nancial mathematics decimals Solve problems involving pro16x16 t and loss, with and without 32x32digital 3GUNCORRECTEDConverting fractions, decimals and technologies (ACMNA189) percentages (Consolidating) 3H Finding a percentage and expressing as a percentage 3I Decreasing and increasing by a percentage SAMPLE PAGES 3J Calculating percentage change 3K Percentages and the unitary method (Extending) Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses Phi and golden rectangles NUMBER AND ALGEBRA Phi is a unique and mysterious most visually appealing proportion Real numbers decimal number approximately to the human eye. Proportions Investigate terminating and recurring decimals (ACMNA184) equal to 1.618. The value of phi has using the decimal phi are found Solve problems involving the use of percentages, including now been calculated to one trillion in an astounding variety of places percentage increases and decreases, with and without digital (1000 000 000 000) decimal places including ancient Egyptian pyramids, technologies (ACMNA187) (2010 record). Phi’s decimal places Greek architecture and sculptures, Money and nancial mathematics continue forever; no pattern has art, the nautilus shell, an ant, a Solve problems involving pro16x16 t and loss, with and without 32x32digital been found and it cannot be written dolphin, a beautiful human face, an technologies (ACMNA189) UNCORRECTEDas a fraction. ear, a tooth, the body, the graphic Golden rectangles are design of credit cards, websites and rectangles that have the proportion company logos, DNA and even the of length to width equal to phi : shape of the Milky Way galaxy. 1. This ratio isSAMPLE thought to be the PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 100 Chapter 3 Fractions, decimals and percentages 3A Equivalent fractions CONSOLIDATING Fractions are extremely useful in all practical situations whenever a proportion is required. Fractions are used by a chef measuring the ingredients for a cake, a builder measuring the ingredients for concrete and a musician using computer software to create music. A fraction is formed when a whole number or amount is divided into equal parts. The bottom number is referred to as the denominator (down) and tells you how many parts the whole is divided up into. The top number is referred Aerial view of farmland. The paddocks show the farmer’s to as the numerator (up) and tells you how land divided into parts. Each paddock is a particular fraction of the farmer’s land. many of the parts you have selected. 4 parts selected Numerator 4 Denominator 7 The whole is divided into 7 parts. Equivalent fractions are fractions that represent equal portions of a whole amount and so are equal in value. The skill of generating equivalent fractions is needed whenever you add or subtract fractions with different denominators. Let’s start: Know your terminology It is important to know and understand key terms associated with the study of fractions. Working with a partner and using your previous knowledge of fractions, write a one-sentence definition or explanation for each of the following key terms: · Numerator · Denominator · Equivalent fraction · Proper fraction · Improper fraction · Mixed numeral UNCORRECTED· Multiples · Factors · Reciprocal · Highest common factor · Lowest common multiple · Descending · Ascending · Lowest common denominator · CompositeSAMPLE number PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 Number and Algebra 101 Equivalent fractions are equal in value. They mark the same place on a number line. Key 3 6 For example: and are equivalent fractions. ideas 5 10 0 1 2 3 4 5 5 5 5 5 5 5 011 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 Equivalent fractions are formed by multiplying or dividing a fraction by a number equal to 1, a which can be written in the form . a 3 3 2 6 3 6 For example: × 1 = × = ) and are equivalent fractions. 4 4 2 8 4 8 Equivalent fractions are therefore produced by multiplying the numerator and the denominator by the same whole number. ×5 2 10 2 10 For example: = ) and are equivalent fractions. 7 35 7 35 ×5 Equivalent fractions are also produced by dividing the numerator and the denominator by the same common factor. ÷3 6 2 6 2 For example: = ) and are equivalent fractions. 21 7 21 7 ÷3 The simplest form of a fraction is an equivalent fraction with the lowest possible whole numbers in the numerator and denominator. This is achieved by dividing the numerator and the denominator by their highest common factor (HCF). In the simplest form of a fraction, the HCF of the numerator and the denominator is 1. 12 For example: The HCF of 12 and 18 is 6. 18 ÷6 12 2 12 2 = ) written in simplest form is . 18 3 18 3 ÷6 This technique is also known as ‘cancelling’. The HCF is cancelled (divided) 2 × 6¡1 6 UNCORRECTED12 2 from the numerator and the ‘cancels’ to 1 = = 18 3 × 6¡1 3 6 denominator. because 6 ÷ 6 = 1 Two fractionsSAMPLE are equivalent if they have the same simplest form. PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 102 Chapter 3 Fractions, decimals and percentages Example 1 Generating equivalent fractions Rewrite the following fractions with a denominator of 40. a 3 b 1 c 7 d 36 5 2 4 120 SOLUTION EXPLANATION 3 24 a = Denominator has been multiplied by 8. 5 40 Numerator must be multiplied by 8. 1 20 b = Multiply numerator and denominator by 20. 2 40 7 70 c = Multiply numerator and denominator by 10. 4 40 36 12 d = Divide numerator and denominator by 3. 120 40 Example 2 Converting to simplest form Write the following fractions in simplest form. a 8 b 25 20 15 SOLUTION EXPLANATION 8 2 × 4¡1 2 a = = The HCF of 8 and 20 is 4. 20 5 × 4¡ 5 1 Both the numerator and the denominator are divided by the HCF of 4. 25 5 × 5¡1 5 b = = The HCF of 25 and 15 is 5. 15 3 × 5¡ 3 1 The 5 is ‘cancelled’ from the numerator and UNCORRECTEDthe denominator. SAMPLE PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 Number and Algebra 103 Exercise 3A 1–3 3 — 1 Fill in the missing numbers to complete the following strings of equivalent fractions. 3 12 120 4 8 80 a = = = b = = = 5 10 7 70 3 2 14 18 6 15 UNDERSTANDING c = = = d = = = 9 3 24 4 2 2 Which of the following fractions are equivalent to ? 3 2 4 4 20 1 4 10 20 30 9 6 30 2 3 15 36 3 Are the following statements true or false? 1 1 3 1 a and are equivalent fractions. b and are equivalent fractions. 2 4 6 2 8 14 2 c The fraction is written in its simplest form. d can be simplified to . 9 21 3 11 1 2 4 2 e and and are all equivalent fractions. f can be simplified to . 99 9 18 5 5 4–10(½) 4–10(½) 4–10(½) Example 1 4 Rewrite the following fractions with a denominator of 24. a 1 b 2 c 1 d 5 FLUENCY 3 8 2 12 e 5 f 5 g 3 h 7 6 1 4 8 5 Rewrite the following fractions with a denominator of 30. a 1 b 2 c 5 d 3 5 6 10 1 e 2 f 22 g 5 h 150 3 60 2 300 6 Find the missing value to make the equation true. 2 7 14 7 1 21 a = b = c = d = 5 15 9 14 30 10 4 8 80 3 7 28 e = f = g = h = UNCORRECTED3 21 5 12 60 11 7 State the missing numerators for the following sets of equivalent fractions. 1 a = = = = = = 2 4 6 10 20 32 50 3 b = = = = = = 2 SAMPLE4 8 20 30 100 200 PAGES 2 c = = = = = = 5 10 15 20 35 50 75 Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 104 Chapter 3 Fractions, decimals and percentages 3A 8 State the missing denominators for the following sets of equivalent fractions. 2 4 8 10 25 100 1 FLUENCY a = = = = = = 3 1 2 3 4 5 10 12 b = = = = = = 5 5 10 15 35 55 100 500 c = = = = = = 4 Example 2 9 Write the following fractions in simplest form. a 3 b 4 c 10 d 15 9 8 12 18 e 11 f 12 g 16 h 25 44 20 18 35 i 15 j 22 k 120 l 64 9 20 100 48 10 Using your calculator, express the following fractions in simplest form. a 23 b 34 c 375 d 315 92 85 875 567 e 143 f 707 g 1197 h 2673 121 404 969 1650 11, 12 11–13 11–13 11 Three of the following eight fractions are not written in simplest form.
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