WHOLE NUMBERS AND DECIMALS

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1 WHOLE NUMBERS AND DECIMALS 2 IN THIS CHAPTER YOU WILL: WHAT S IN CHAPTER 2? 2 01 Mental addition 2 02 Mental subtraction 2 03 Rounding decimals and money 2 04 Adding and subtracting decimals 2 05 Mental multiplication 2 06 Multiplying decimals 2 07 Mental division 2 08 Dividing decimals 2 09 Terminating and recurring decimals add, subtract, multiply and divide mentally with whole numbers round decimals and money amounts add, subtract, multiply and divide decimals understand and use terminating and recurring decimals convert fractions to recurring decimals Shutterstock.com/Tatiana53 ISBN Chapter 2 Whole numbers and decimals 21

2 2 01 Mental addition WORDBANK sum The answer to an addition (+) of two or more numbers. mental Using the mind, not a calculator. evaluate To find the value or amount. EXAMPLE 1 Find each sum using mental calculation. a b a = = adding the tens and units separately = = 48 b = = = = 93 or using the answer 48 from part a or When adding numbers mentally: remember that numbers can be added in any order if one of the numbers is close to 10, 20, 30,, split it up look for unit digits that add up to 10, such as 6 and 4 EXAMPLE 2 Evaluate each sum mentally by splitting up the second number. a b c a = = = 36 b = = = 155 c = = = 445 EXAMPLE 3 Evaluate each sum mentally by pairing up numbers that have units digits adding to 10. a b a = ( ) + (46 + 4) = = 90 b = ( ) = = 1086 Developmental Mathematics Book 3 ISBN

3 exercise Which expression gives the same answer as ? Select the correct answer A, B, C or D. A B C D Evaluate each sum using mental calculation. a b c d e f g h i j k l Write down which sums in question 2 are the same. 4 Evaluate each sum mentally. a b c d a How could you use your answer to 4a to find 4b? b How could you use your answer to 4c to find 4d? 6 Is each statement true or false? a = b = c = d = Copy and complete each expression. a 502 = b 903 = c 498 = 500 d 299 = 1 8 Which expression gives the same answer as ? Select A, B, C or D. A B C D Evaluate each sum mentally. a b c d e f g h Evaluate each sum mentally by pairing up numbers that have units digits adding to 10. a b c d e f g h A magic square has all rows, columns and diagonals adding to the same number. Copy and complete each magic square. a 3 6 b ISBN Chapter 2 Whole numbers and decimals 23

4 2 02 WORDBANK Mental subtraction difference The result of subtracting two numbers. EXAMPLE 4 Find each difference using mental calculation. a b a = subtracting the tens and units separately = = = 61 b = or = = or / gives a negative answer, so change 40 to 140 by taking 100 from 200 = = 185 EXAMPLE 5 Evaluate each difference mentally by splitting up the second number. a 36 9 b c a 36 9 = b = c = To subtract 9, subtract 10 and add 1 = = 27 To subtract 21, subtract 20 and subtract 1 = 24 1 = 23 To subtract 28, subtract 30 and add 2 = = 818 EXAMPLE 6 Use a number line to evaluate Draw a number line and jump along it from 284 to 365 using bridges. Write the size of each bridge and add the sizes So = = Developmental Mathematics Book 3 ISBN

5 exercise Which expression is the same as 57 12? Select the correct answer A, B, C or D. A B C D Evaluate each difference using mental calculation. a 15 4 b c d e f g h i j k l Copy and complete each line of working. a 84 9 b = = = 74 + = = 318 = 4 Which expression is the same as ? Select A, B, C or D. A B C D Evaluate each difference mentally by splitting up the second number. a 72 9 b c d 97 8 e f 96 9 g h i j k l Use the number line below to jump from 164 to 203 and complete the statement below = =. 7 Use a number line to evaluate each difference. a b c d e f g h Is each statement true or false? a = 35 b = 44 c = 47 d = Evaluate mentally and write down your answer: ISBN Chapter 2 Whole numbers and decimals 25

6 2 03 WORDBANK Rounding decimals and money rounding decimals To write a number with approximately the same value using fewer digits. The place value for decimals is: Thousands (1000s) Hundreds (100s) Tens (10s) Units (1s) Decimal point Tenths 1 10 s Hundredths s Thousandths s 1 decimal place is 0. = 10 1 number after the point, rounding to the nearest tenth. 2 decimal places is 0. = numbers after the point, rounding to the nearest hundredth. 3 decimal places is 0. = numbers after the point, rounding to the nearest thousandth. To round decimals to a certain number of decimal places, look at the next digit. If it is 5 or more, round the decimal up. If it is less than 5, round the decimal down. EXAMPLE 7 Round each decimal to the number of decimal places shown in square brackets. a [1] b [2] c [3] a For 1 decimal place, look at the next digit 1. It is less than 5, so leave the digit 4 as it is b For 2 decimal places, look at the 6. It is more than 5, so round the digit 8 up to c For 3 decimal places, look at the 5. It is 5 or more, so round the digit 2 up to EXAMPLE 8 Round each money amount as required. a $30.75 (nearest dollar) b $2.483 (nearest cent) c $0.568 (nearest cent) a $30.75 $31 b $2.483 $ c is more than 50c, half a dollar, so round up. 3 is less than 5, so leave the 8 alone. c $ c 8 is more than 5, so take the digit 6 up to Developmental Mathematics Book 3 ISBN

7 exercise Round to 2 decimal places. Select the correct answer A, B, C or D. A 3.42 B 3.4 C 3.43 D Write the decimal 0.63 as a fraction. Select A, B, C or D. A B C Is each statement true or false? a 4.6 < 4 b 2.8 > 2 c 7.4 > 7 d 3 < 3.1 e 9.4 < 9 D Round each decimal to the nearest whole number. a 3.2 b 6.8 c 12.7 d 21.3 e 25.4 f 7.28 g h i j Round each decimal to 1 decimal place. a 0.42 b 1.68 c d 1.73 e 3.22 f g 2.17 h i j Round each decimal to 2 decimal places. a b c d e f g h i j Round each decimal to 3 decimal places. a b c d e Round each amount of money to the nearest dollar. a $4.72 b $52.48 c $26.92 d $45.23 e $28.99 f $10.42 g $ h $ i $ j $ Round each amount of money to the nearest cent. a $5.712 b $ c $ d $4.231 e $ f $ g $1.684 h $ i $ j $ Round $ to the nearest: a ten dollars b dollar c ten cents d cent ISBN Chapter 2 Whole numbers and decimals 27

8 2 04 Adding and subtracting decimals TO ADD OR SUBTRACT DECIMALS: write the decimals underneath each other in columns make sure the decimal points line up underneath each other: points under points (PUP) fill in the gaps with 0s add or subtract the digits in columns place the decimal point directly underneath check your answer by estimating EXAMPLE 9 Evaluate each expression. a b a Gaps filled in with 0s b Use trading to subtract Check by estimating: points under points or PUP = = 247 ( is close to 18) ( is close to 247) EXAMPLE 10 Find the change given from a $50 note if I buy food costing $ To calculate $50 $42.75, use a number line to jump from $42.75 to $50. $0.25 $2 $5 $42.75 $43 $45 $50 So the change from $50 is $ $2 + $5 = $ Developmental Mathematics Book 3 ISBN Shutterstock.com/Dmitry Kalinovsky

9 exercise Find the sum of 5.2 and Select A, B, C or D. A B C D Find the difference between 18.6 and Select A, B, C or D. A 9.37 B 9.47 C 8.37 D What must be added to: a 0.01 to make 0.04? b 0.62 to make 0.88? c 1.25 to make 1.78? d 4.64 to make 8? 4 What must be subtracted from: a 9.5 to make 9.3? b 12.6 to make 10.4? c 18.6 to make 15.7? d 24.8 to make 22? 5 Evaluate each expression. a b c d e f g h i j k l Find each missing number. a = 5 b = 7.9 c = 6 d = 3.6 e = 1.29 f = 7.86 g = 3.1 h = i 1.2 = 2.6 j 5.5 = 6.1 k = l = Use a number line to evaluate each difference. a $150 $ b $80 $62.70 c $15 $7.25 d $100 $42.60 e $180 $ f $58 $26.99 g $186 $ h $200 $ Emma was shopping in the supermarket and bought the following items. 1 box of chocolates for $ litres of milk for $ kg apples for $5.26 a Estimate the cost of the groceries. b If Emma s mother gave her a $20 note, would she have enough money to pay for the items? c Find the total cost of the groceries, correct to the nearest 5 cents. d Calculate Emma s change from $20. 9 A 5 kg tin of biscuits costing $22.95 was reduced by $8.56. What was the new price? 10 Find the cost of a European holiday correct to the nearest dollar if the flights are $ , the Eurail pass is $ and the accommodation and meals are $ ISBN Chapter 2 Whole numbers and decimals 29

10 2 05 Mental multiplication WORDBANK product The answer to a multiplication ( ) of two or more numbers. Multiplication table Mental multiplication strategies Multiplying by Strategy 2 Double 4 Double twice 5 Multiply by 10, then halve 8 Multiply by (10 2) 9 Multiply by (10 1) 10 Add a 0 to the end 11 Multiply by (10 + 1) 12 Multiply by (10 + 2) EXAMPLE 11 Evaluate each product mentally. a 32 5 b 23 9 c d e 58 4 f a 32 5 = because 5 is 1 of 10 2 = = 160 c = numbers can be multiplied in any order = = 1800 e 58 4 = double twice = = 232 b 23 9 = 23 (10 1) = = = 207 d = = = 1280 f = 43 (10 + 1) = = = Developmental Mathematics Book 3 ISBN

11 exercise Which expression can be used to evaluate 47 8 mentally? Select the correct answer A, B, C or D. A 47 (10 1) B 47 (10 2) C 47 (10 + 1) D 47 (10 + 2) 2 Which expression can be used to evaluate mentally? Select A, B, C or D. A 19 (10 1) B 19 (10 2) C 19 (10 + 1) D 19 (10 + 2) 3 Write the answer to each product. a 5 6 b 8 3 c 7 4 d 9 2 e 5 8 f 4 6 g 9 7 h 4 9 i 6 7 j 10 3 k 3 7 l 8 9 m 8 6 n 9 3 o 7 5 p 8 2 q 5 9 r Is each statement true or false? a 27 4 = b 54 9 = 54 (10 + 1) c = 42 (10 + 1) d = 36 (10 + 2) e 73 5 = f 93 8 = 93 (10 1) 5 Evaluate each product using a mental strategy. a 28 5 b c 63 9 d 82 4 e f 52 8 g 75 2 h i j k l m n o p Copy and complete each line of working. a = 33 2 = 33 = b = 42 4 = 42 = 7 Evaluate each product by changing the order. a b c d e f g h ISBN Chapter 2 Whole numbers and decimals 31

12 2 06 Multiplying decimals To multiply a decimal by 10, 100 or 1000, move the decimal point to the right, since the number is getting larger. To multiply by 10, move the decimal point 1 place to the right To multiply by 100, move the decimal point 2 places to the right To multiply by 1000, move the decimal point 3 places to the right EXAMPLE 12 Evaluate each product. a b c a = 64.2 point moves 1 right b = point moves 2 right c = = Adding zeros to the decimal to allow the point to move point moves 3 right TO MULTIPLY DECIMALS: multiply the numbers without the decimal points count the total number of decimal places in the numbers write the answer using this number of decimal places check your answer by estimating EXAMPLE 13 Evaluate each product. a b c a 6 4 = 24 b c has 1 decimal place 2.41 has 2 decimal places 3.86 has 2 decimal places 4 has no decimal places 5 has no decimal places 0.7 has 1 decimal place Total decimal places = 1 Total decimal places = 2 Total decimal places = 3 So = 2.4 So = So = Check by estimating: = 2 (2.4 is close to 2) = 10 (12.05 is close to 10) = 4 (2.702 is close to 4) 32 Developmental Mathematics Book 3 ISBN

13 exercise When multiplying a decimal by 100, where does the decimal point move? Select the correct answer A, B, C or D. A 1 place right B 2 places right C 1 place left D 2 places left 2 Evaluate the product Select A, B, C or D. A 235 B C 23.5 D Evaluate each product. a b c d e f g h Is each statement true or false? a 3.27 has 3 decimal places c 9.04 has 1 decimal place b 0.47 has 2 decimal places 5 Count the total number of decimal places in each expression. a b c d Evaluate each product in question 5. 7 Evaluate each product. a b c d e f g h i j k l If = 1664, evaluate each product, then check your answer by estimating. a b c d If Ruby worked in a flower shop and earned $21.80 per hour, how much would she earn in 6 hours? Estimate your answer first, then calculate the exact amount. 10 Jonah planted a vegetable garden that was 8.6 m long and 0.9 m wide. Estimate the area of his garden before calculating the exact area. Shutterstock.com/Gvictoria ISBN Chapter 2 Whole numbers and decimals 33

14 2 07 WORDBANK quotient The result of dividing ( ) a number by another number. For example, for 8 4 = 2, the quotient is 2. Mental division strategies Dividing by Strategy 2 Halve 4 Halve twice 5 Divide by 10, then double 8 Halve 3 times Mental division 10 Move the decimal point 1 place left, or for a whole number ending in 0, drop a 0 from the end of the number 20 Divide by 10, then halve 100 Move the decimal point 2 places left, or for a whole number ending in 0s, drop two 0s from the end of the number EXAMPLE 14 Evaluate each quotient mentally. a b c a = b = c = halving 624 twice = = 156 divide by 10 and double = 84 2 = 168 divide by 10, then halve = = 64 EXAMPLE 15 Evaluate by short division. The steps of short division are shown below = 2 remainder 2 ) ) ) ) So = = 4 remainder = 4 remainder = 5 no remainder Shutterstock.com/ninikas 34 Developmental Mathematics Book 3 ISBN

15 exercise What is the answer to a division called? Select A, B, C or D. A difference B quotient C product D sum 2 What is a quick way to divide a number by 20? Select A, B, C or D. A halve it B double it C divide by 10 twice D divide by 10 then halve 3 Write the answer to each quotient. a 15 3 b 24 6 c 32 8 d 25 5 e 42 7 f 56 8 g 45 9 h 20 4 i 22 2 j k 63 9 l 64 8 m 30 6 n 27 3 o 35 7 p 72 8 q 28 4 r Is each statement true or false? a = 64 8 b 96 4 = 48 2 c 96 8 = 24 2 d = 15 e = 2.8 f = 0.54 g = Evaluate each quotient using a mental strategy. a 86 2 b c d e f g 80 5 h i j k l Evaluate each quotient using a mental strategy. a b c d e f g h i j k l Use short division to find each quotient, then check your answer using a calculator. a b c d At a Christmas sale, all decorations were reduced by half. On Boxing Day, these same decorations were reduced by half again. a What number should you divide by to find the final sale price? b Find the final sale price of each item below. Tinsel $4.20 Baubles $2.60 Gift wrap $6.80 Cards $3.60 Shutterstock.com/Rtimages ISBN Chapter 2 Whole numbers and decimals 35

16 2 08 Dividing decimals To divide a decimal by 10, 100 or 1000, move the decimal point to the left since the number is getting smaller. To divide by 10, move the decimal point 1 place to the left To divide by 100, move the decimal point 2 places to the left To divide by 1000, move the decimal point 3 places to the left EXAMPLE 16 Evaluate each quotient. a b c a = 3.25 b = point moves 1 left point moves 2 left c = = Adding zeros to the left of the decimal to allow the point to move point moves 3 left To divide a decimal by a whole number, use short division, then write the answer with the decimal point in the same column as the original decimal To divide a decimal by another decimal, move the points in both decimals the same number of places to the right so that you are dividing by a whole number Check your answer by estimating EXAMPLE 17 Evaluate each quotient. a b a b = ) move both points 1 right so 0.5 is a whole number, 5 decimal points in the same column = 10.7 Check by estimating: ) = = 10 (10.7 is close to 10) = 260 (257.3 is close to 260) 36 Developmental Mathematics Book 3 ISBN

17 exercise When dividing a decimal by 100, where does the decimal point move? Select the correct answer A, B, C or D. A 1 place right B 2 places right C 1 place left D 2 places left 2 Evaluate the quotient Select A, B, C or D. A 3.64 B C D Evaluate each quotient. a b c d e f g h Copy and complete this division. 5) Evaluate each quotient. a b c d e f g h Copy and complete this division = 3= 3 ) 7 Evaluate each quotient by making the second decimal a whole number. a b c d e f g h i j k l Sam made a mistake when calculating this quotient and put the decimal point in the wrong place: = What is the correct answer? 9 A length of timber 4.76 m long is cut into 4 equal pieces. What is the length of each piece? 10 Corrine spent $31.40 at the school canteen last week. What was her average spending per day if she spent at the canteen every school day? ISBN Chapter 2 Whole numbers and decimals 37

18 2 09 WORDBANK Terminating and recurring decimals recurring decimal A decimal with digits that repeat or recur, such as 0. 3 = and = terminating decimal A decimal that ends or terminates, such as or Recurring decimals are written with dots placed over the repeating digits. If a series of digits are repeated, then a dot is placed over the first and last digits in the series. EXAMPLE 18 Write each recurring decimal using dot notation. a b c Where a group of digits recur, just put a dot above the first and last digits. a = 0.42 b = 2.36 c = When using short division, if there are still remainders after dividing into the last digit, you need to add 0s to the end of the decimal to continue dividing, until there are no remainders. EXAMPLE 19 Evaluate each quotient and state the type of decimal in each answer. a b a b ) ) It was necessary to add 2 zeros. It was necessary to add 3 or more zeros to see a repeating pattern = 7.075, a terminating decimal = , a recurring decimal. 38 EXAMPLE 20 Convert each fraction to a decimal. 2 a 5 a 2 5 means ) 2 5 = 0.4 Developmental Mathematics Book 3 ISBN b 5 6 b 5 6 means ) =

19 exercise Write as a recurring decimal. Select the correct answer A, B, C or D. A 3.54 B C D What type of decimal is the answer to ? Select A, B, C or D. A non-terminating B recurring C terminating D none of these 3 Is each statement true or false? a = b = c = Write each recurring decimal using dot notation. a b c d e f g h Evaluate each quotient using short division, adding zeros to the decimal if necessary. a b c d e f g h State the type of decimal for each quotient found in question 5. 7 a How do you change 3 into a decimal without the use of a calculator? 8 b Convert 3 to a decimal and check your answer using a calculator. 8 c What type of decimal is it? 8 Convert each fraction below to a decimal, then check your answer using a calculator a b c d e f g h Jamiela had a $900 gift card and her sister had spent one-third of it. a What fraction of the money is left on the gift card? b Write this fraction as a decimal. c How much money does Jamiela have left to spend on her gift card? istockphoto/yinyang ISBN Chapter 2 Whole numbers and decimals 39

20 LANGUAGE ACTIVITY WORD PUZZLE List the letters in order, for the clues below to spell out a phrase relating to this topic. The first 0.2 of DESCENDING The first 0.1 of CELLOPHANE The first 0.6 of IMAGE The first 0.2 of LUNCH The first 0.4 of DIGIT The last 0.25 of DEAL The last 0.5 of STEM The first 0.2 of MANAGEMENT 123RF/konstantin32 40 Developmental Mathematics Book 3 ISBN

21 PRACTICE TEST 2 Part A General topics Calculators are not allowed. 1 Evaluate Complete: 12 km = m. 8 Find the value of d if the perimeter of this rectangle is 46 m. 3 List all the factors of 6. 4 Simplify x+ y y+ 2x 5 Find the range of the scores: 12, 5, 7, 9, 8, 7, Given that 15 6 = 90, evaluate Find 5% of $ m 9 Simplify 4 r 9 v d m 10 Copy and complete 4 = Part B Whole numbers and decimals Calculators are not allowed Mental addition 11 Use a mental strategy to evaluate each sum. a b c Which equation is correct? Select the correct answer A, B, C or D. A = B = C = D = Mental subtraction 13 Use a mental strategy to evaluate each difference. a 58 9 b Rounding decimals and money 14 Round each decimal to the required number of decimal places in the square brackets. a [2] b [3] 2 04 Adding and subtracting decimals 15 Evaluate each expression. a b ISBN Chapter 2 Whole numbers and decimals 41

22 PRACTICE TEST Mental multiplication 16 Evaluate each product using a mental strategy. a b Multiplying decimals 17 Evaluate each product. a b c Is this statement true or false? = Mental division 19 Evaluate each quotient using a mental strategy. a b c Dividing decimals 20 Evaluate each quotient. a b c Is this statement true or false? = Terminating and recurring decimals 22 Write as recurring decimals using dot notation. a b c Convert 2 to a decimal Developmental Mathematics Book 3 ISBN