Chapter 7
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07 NCM7 2nd ed SB TXT.fm Page 214 Saturday, June 7, 2008 4:59 PM
NUMBER
You are already familiar with decimals because you use them whenever you pay money to buy something. Decimals are used to measure all sorts of things—how fast a race is run, the lengths of a long jump, how much timber is needed. Building, banking and many other businesses need decimals.
7
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07 NCM7 2nd ed SB TXT.fm Page 215 Saturday, June 7, 2008 8:22 PM
0123456789
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0123456789 567890123 3 8901234 456789 012345 0123456789 678901234
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In this chapter you will: 56789012345678 5678
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• decimal A number that 5678 • revise place value and order for decimals includes a decimal point 1234567890123456789 234567890123 45678901234and place value to indicate parts of a whole. • 0123456789 add and subtract decimals 0123456789 456789012345678901234567890123456789012345678 56789012 56789012 3456789012345678 8 • estimate An educated guess concerning the • multiply901234567890123456789012345678901234567890123456789012345678 and divide 7890123456789012 a decimal by a power of 10, a 789012345678 answer to a calculation. 567890123456789012345678901234567890123456789012345 whole number or another decimal 56789012345678901234567890 1234567890123456789 0123456789 • • convert terminating decimals to fractions 5678 234567890123 45678901234 number of decimal places The number of digits 78901234 0123456789 after the decimal point. • convert fractions to terminating or recurring 45678901 890123456789012345678901234567890123456 7890123456789012 345678901 456789012 9012345 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07 NCM7 2nd ed SB TXT.fm Page 216 Saturday, June 7, 2008 4:59 PM
Start up Worksheet 7-01 Brainstarters 7
Skillsheet 7-01 Rounding whole numbers
1 Arrange the numbers in each of these sets in ascending order. a 21, 32, 34, 17, 8 b 23, 26, 54, 66, 33 c 101, 100, 110, 99, 102, 111 d 251, 247, 256, 254, 244 2 Arrange the numbers in each of these sets in descending order. a 44, 39, 42, 45, 38, 40 b 55, 56, 53, 50, 49, 52 c 1001, 1000, 1100, 1010, 1111 d 301, 310, 299, 318, 295 3 a Write 27 to the nearest ten. c Write 9079 to the nearest thousand. e Write 29 537 to the nearest hundred.
b Write 752 to the nearest hundred. d Write 16 837 to the nearest thousand. f Write 40 219 to the nearest ten.
4 Evaluate the following. a 123 + 32 + 456 d 47 852 + 365 + 1458 + 81 g 894 − 35 j 126 × 8 m 2920 ÷ 8
b e h k n
5 Evaluate the following. a 32 × 10 d 400 × 10 g 1 000 000 ÷ 1000
b 578 × 100 e 400 ÷ 10 h 81 000 ÷ 100
432 + 45 + 2341 + 7 234 − 45 114 782 − 36 989 589 × 13 2163 ÷ 7
c f i l o
6 + 78 + 32 + 9 + 199 1138 − 445 58 × 3 789 × 26 10 836 ÷ 21
c 325 × 1000 f 1200 ÷ 100
6 What is the value of 7 in each of these numbers? a 725 b 1207 c 670 d 7189
e 972 150
f 878
7-01 Place value Skillsheet 7-02 Decimal fractions
You are familiar with numbers that have decimal points, for example: • money $3.52 • measurements 6.2 m. The digits after the decimal point indicate a part of a whole. The position of a digit in a number shows its size. This is known as place value.
Example 1 What does the number 532.81 mean? Solution Hundreds
Tens
Unit
tenths
hundredths
5
3
2
8
1
decimal point So 532.81 is 5 × 100 + 3 × 10 + 2 × 1 + 8 ×
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NEW CENTURY MATHS 7
1 -----10
1 -. + 1 × -------100
07 NCM7 2nd ed SB TXT.fm Page 217 Saturday, June 7, 2008 4:59 PM
Example 2 Write each of these decimals in expanded notation. a 604.75 b 29.04 Solution 1 1 - + 5 × --------- . a 604.75 = 6 × 100 + 0 × 10 + 4 × 1 + 7 × ----b 29.04 = 2 × 10 + 9 × 1 + 0 ×
1 -----10
+4×
10 1 --------- . 100
100
Example 3 What is the value of the digit 8 in the number 612.87? Solution 8 - ). The 8 in 612.87 is in the tenths column, so it has a value of 8 tenths (or ----10
Example 4 Write the following in decimal form. 81a 6 + ----+ --------
1b 2 × 10 + 4 × --------
Solution a 6.18
b 20.04
10
100
100
Exercise 7-01 1 Copy this place-value table. Put the 12 given numbers in the table, with the digits in their correct columns. Hundreds
Tens
Units
tenths
hundredths
thousandths
Ex 1
TLF
L 1997
Scale matters: range of numbers
a d g j
14.82 70.8 503.92 200.047
b e h k
6.014 297.86 8.3 4.025
c f i l
931.02 11.14 0.375 0.81
2 Write each of the following (a to p below and on the next page) as a decimal. a five and four-tenths b six and fifteen-hundredths c eight and three tenths d eleven and thirty-eight hundredths e fourteen and six-hundredths f four hundred and two and three-thousandths CHAPTER 7 DECIMALS
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g 6 + 2 tenths + 3 hundredths i nineteen and nine-tenths k 2 tens + 4 + 0 tenths + 9 hundredths m 8 + 7 tenths + 5 hundredths o nine-thousandths Ex 2
h j l n p
3 + 7 tenths 14 + 3 tenths + 9 hundredths 2 + 6 hundredths seventy-three hundredths 8 tenths + 5 hundredths + 9 thousandths
3 Write the following in expanded notation. a 1.234 b 102.34 c 30.12 f 12.71 g 8.003 h 4.509
d 0.751 i 0.04
e 2.09 j 0.386
4 Which digit is in the hundredths place in the decimal 251.945? Select A, B, C or D. A2 B9 C4 D5 Ex 3
5 What is the value of the digit 4 in each of these numbers? a 431.70 b 31.047 c d 114.37 e 3.734 f g 0.064 75 h 42 376 i j 26.74 k 100.0407 l
761.04 907.431 72.314 94.071
6 What is the value of the digit 7 in each number listed in Question 5? Ex 4
7 Write the following in decimal notation. 1a 4 + ----c 4+ e 5+
10 3----10 2----10
+ +
5-------100 7-------100
8b 3 + -----
d 9 + -----------
f
1000
9 g 19 + -----------
h
i 11 +
j
1000 2 6----+ ---------------10 000 10
11k 2 × 100 + 7 × 10 + 6 × ----+ 1 × --------
l
m3 ×
n
o 2× q 9+
10 1 1 1 ------ + 4 × --------- + 1 × -----------1000 100 10 1110 + 7 + 2 × ----+ 3 × -------100 10 1 14 × -------+ 1 × ----------1000 100
100
p r
10 3912 + ----+ -------100 10 6 41 0 + -------+ ----------+ ---------------1000 100 10 000 3 72 + ----+ ----------1000 10 113 × 1 + 2 × ----+ 7 × -------100 10 17 × 10 + 6 × 1 + 5 × -------100 1 19 × ----+ 7 × ----------1000 10 14 × 10 + 9 × -------100 1 15 × 100 + 4 × ----+ 3 × ----------1000 10
7-02 Understanding the point Example 5 Where would you place the decimal point in this weather report? The day was fine and warm, with a maximum temperature of 245˚C. Solution Since a warm day has a temperature in the mid-twenties, we would place the decimal point after the 4 so that it reads 24.5˚C.
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NEW CENTURY MATHS 7
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Exercise 7-02 Read the following story carefully. List the numbers appearing in as a to r and place a decimal point in each so that the story makes sense. Maria filled her car, which was nearly empty, with a 434 litres of petrol at b 1399 cents per litre and handed the attendant a c $5000 note and a d $2000 note. After she received change of e $930 she picked up her friend Fred, a tall man of f 188 metres in height. Into the car jumped Charlie the dog, weighing g 144 kilograms, happily wagging his h 150 centimetres of tail. Maria and Fred had each packed a i 200 kg backpack. They stopped for a snack at McDougall’s and bought two beefburgers each, so that they could have their daily intake of j 250 kg of meat. After they had driven for several hours, averaging k 850 kilometres per hour, they were within sight of Mt Kosciuszko, the highest mountain in Australia, about l 23000 m above sea level. They enjoyed seeing the small eucalypt trees about m 105 m high. They also saw a kangaroo that jumped about n 80 metres, which was o 110 times the Olympic record for the long jump (set in Mexico City). After a hike of p 134 km they arrived at their campsite. After hiking back to the car next day, they drove the q 2500 kilometres home in about r 300 hours.
Ex 5
7-03 Ordering decimals The number of digits after the decimal point tells us the number of decimal places in the decimal number.
Worksheet 7-02 Dewey decimals
Example 6 How many decimal places are there in: a 3.6567?
b 15.801?
Solution a 3.6567 has four decimal places.
b 15.801 has three decimal places.
1234
123
Example 7 Arrange these numbers in ascending order (smallest to largest). 67.41 67.14 6.714 67.04 Solution To compare decimals more easily, insert zeros at the end to give all the numbers the same amount of decimal places. 67.14 6.714 67.04 becomes 67.410 67.41 67.140 6.714 67.040 In ascending order: 6.714, 67.04, 67.14, 67.41
CHAPTER 7 DECIMALS
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Example 8 Arrange these numbers in descending order (largest to smallest): 0.5 0.08 1.7 0.85 Solution 0.5
0.08
1.7
0.85
becomes
0.50 0.08 1.70 0.85
In descending order: 1.7, 0.85, 0.5, 0.08
Exercise 7-03 Ex 6
1 How many decimal places does each of these numbers have? a 1.65 b 3.881 c 15.3062 d 0.005 e 7.045 73 f 814.3 g 9.100 001 h 203.222 22 i 0.040 400 4 2 Which of these is the smallest number? Select A, B, C or D. A 1.07 B 1.7 C 1.077
Ex 7
3 Arrange these numbers in ascending order (smallest first). a 43.89, 56.324, 9.998, 80.879, 400, 23.89, 56.314 b 0.568, 0.684, 0.099, 1.002, 0.586, 5.608, 0.0586 c 1.23, 0.891, 1.814, 0.222, 7.007, 0.89 d 0.5, 0.05, 0.005 e 3.441, 3.404, 3.4, 3.44, 3.004, 3.044 f 0.2, 0.202, 0.22, 0.022 g 1.01, 1.002, 1.012, 1.21 h 0.07, 0.67, 0.71, 0.007, 7
Ex 8
4 Arrange these numbers in descending order (largest first). a 570.25, 125.63, 0.9899, 4000.99, 1256.3, 400.099 b 5.37, 6.539, 5.639, 5.367, 3.659, 3.66, 5.369 c 1.6, 1.61, 1.599, 1.601, 1.509 d 6, 0.06, 0.6, 6.6 e 0.7, 0.07, 0.707, 0.77, 0.007, 7.07 f 0.4004, 0.044, 0.404, 0.44 g 0.1, 0.08, 0.65, 0.029 h 0.92, 0.921, 0.09, 0.099
220
NEW CENTURY MATHS 7
D 1.77
07 NCM7 2nd ed SB TXT.fm Page 221 Saturday, June 7, 2008 4:59 PM
5 Insert or to make each of these true. a 0.2 0.25 b 0.731 c 0.035 d 1.59
0.305 1.059
g 44.44
4.444
0.73
d 0.007
0.070
f 0.099
0.99
h 0.7932
0.7239
6 Copy each of those number lines carefully and fill in the values of the points marked with dots. a 1.9
1.6
2.3
b 4.09
4.07
4.11
c 0.67
0.65
0.71
d 8.0
8.1
8.16
e
0.23
0.07
f
2.6
2.0
g
4.4
4.3
8.2
4.5
4.7
4.6
7 Look at the following diagram. a Find a path from the START circle to the TOP circle. You can make your first move in either direction, but then you can only move to a circle with a larger number. TOP 0.029 0.12 0.09
0.2 0.121
0.071 0.081
0.17 0.11 0.3
0.05 0.005
0.14
0.015 START
b Now write out the number sequence for: i the longest path ii a path requiring seven moves iii the route requiring nine moves c There are several paths that use the smallest number of moves. i Find and write the number sequence for all the shortest paths you can find. ii How many moves are in the shortest path?
CHAPTER 7 DECIMALS
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7-04 Decimals and fractions Worksheet 7-03 Decimals squaresaw 1
!
This shape has been divided into 10 equal parts. Nine out of the 10 parts are coloured blue. 9 - or nine-tenths of the Writing this as a fraction, ----10 shape is coloured. Writing it as a decimal, 0.9 or ‘zero-point-nine’ of the shape is coloured. A decimal number has a decimal point to mark where the whole part is separated from the fraction part of the number.
Note: 1 • tenths have one decimal place ( ------ has 1 zero) 10 1 • hundredths have two decimal places ( ---------- has 2 zeros) 100 1 • thousandths have three decimal places ( ------------- has 3 zeros) 1000 1 • ten thousandths have four decimal places (----------------- has 4 zeros) 10 000
Example 9 Convert each fraction to a decimal. 7b 3 -----
22a --------
10
100
c eighty-five thousandths
Solution 22a -------= 0.22
b
100 73 ----10
2 zeros ➞ 2 decimal places
= 3.7
1 zero ➞ 1 decimal place
c eighty-five thousandths =
85 ----------1000
= 0.085
3 zeros ➞ 3 decimal places
Example 10 Convert each decimal to a fraction. a 0.09 b 0.273
c 8.1
Solution 9a 0.09 = --------
2 decimal places means hundredths ⎛⎝ ---------⎞⎠ 100
b
3 decimal places means thousandths ⎛⎝ ------------⎞⎠
c
100 2730.273 = ----------1000 18.1 = 8 + ----= 10
1000
18 ----10
1 decimal place means tenths
⎛ ------⎞ ⎝ 10⎠
Note: The number of zeros in the denominator (bottom number) of the fraction is the same as the number of decimal places in the decimal.
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NEW CENTURY MATHS 7
07 NCM7 2nd ed SB TXT.fm Page 223 Saturday, June 7, 2008 4:59 PM
Exercise 7-04 1 What part of each of these shapes has been shaded? (Give your answers as decimals.) a b
c
d 23 - equals: 2 Select A, B, C or D to complete this statement. As a decimal, ----------1000
A 0.23
B 0.023
C 0.0023
D 0.00023
79-------100 6----10
60-------100 411----------1000 7-------100 247----------1000 493----------1000
3 Convert each fraction to a decimal. a
9 -----10
b
15 --------100 23-------100
Ex 9
c
e four-tenths
f
704i -----------
j
l
8m -----
235n -----------
o
q
r
s
p
1000 14-------100 17-------100
g
d h
eighty-seven hundredths 10 368 -----------1000
k
1000 9345 ---------------10 000
4 Write each of these as a decimal. 67a --------
100 11 d ----------1000 1g -------100 17 j ---------------10 000 33 m ---------------10 000
3b -----
4c --------
e
f
10 8-------100
100 6 ----------1000
h thirty-two hundredths
i five-thousandths
2 k --------
57 l -----------
n
o seven-millionths
1000
100 46 ------------------100 000
5 Write each of these as a decimal. 67a 1 -------100
47b 4 -------100
9c 23 -----
e forty-five and twenty-three thousandths f 143h 89 ----------1000
0i 42 ----10
6 Convert each decimal to a fraction. a 0.7 b 0.4 e 0.003 f 0.05 i 0.9 j 0.999 m 0.0471 n 0.3333 q 0.087 r 1.9 u 10.349 v 7.41
j
10 3 2 -----10 -------5 0100
8d 6 -----
g k
10 11 6 ----------1000 6 4 ----------1000 Ex 10
c g k o s w
0.39 0.11 0.013 0.5001 27.33 101.3
d h l p t x
0.572 0.309 0.0004 0.91 2.007 6.0102
CHAPTER 7 DECIMALS
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Working mathematically
Applying strategies and reflecting
Decimal distances Equipment: A tape measure and some coloured chalk. Step 1: Form a group of three or four students and find a flat area such as a path, basketball court or the classroom floor. Step 2: Mark a starting point on your flat area. Step 3: Choose a distance between 1 and 4 metres, for example 3.25 m. Step 4: Using chalk, take turns to mark your estimate of 3.25 m from the starting point. Measure the exact length and mark it with an X. Step 5: Give points to each person. Score 5 points for best estimate, 3 points for next best, 2 points for next and 1 point for next. Take turns to choose distances, and play the game until someone reaches 20 points and wins. Sample score sheet for one person. Trial
Distance
Guess
1
3.25
2.57
2
4.50
3.16
3
3.50
3.41
4
1.75
1.82
5
2.20
2.38
Points
Total
Mental skills 7A
Maths without calculators
Multiplying an even number by 5 To multiply a number by 5, it’s sometimes easier to halve it, then multiply by 10 (by inserting a 0 at the end). This is because 1--- × 10 = 5. 2
1 Examine these examples. a 14 × 5 = 14 × 1--- × 10 = 7 × 10 = 70 b 36 × 5 = 36 ×
2 1 --2
× 10 = 18 × 10 = 180
c 22 × 5 = 11 × 10 = 110 d 18 × 5 = 9 × 10 = 90 2 Now simplify these. a 32 × 5 e 42 × 5 i 60 × 5
224
b 26 × 5 f 54 × 5 j 34 × 5
NEW CENTURY MATHS 7
c 12 × 5 g 38 × 5 k 16 × 5
d 28 × 5 h 44 × 5 l 58 × 5
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7-05 Adding and subtracting decimals When adding and subtracting whole numbers, we must write them under one another in their correct place-value columns: 67 Not: 67 486 486 9 9 302 302 + 59 + 59 923 ??? The same is true when adding and subtracting decimals.
Example 11 Add 16.27, 10.92, 4.03, 0.89, 32, 0.6 Solution Estimate: 16 + 11 + 4 + 1 + 32 + 1 = 65 The answer should be about 65. 16.27 10.92 4.03 0.89 32.00 + 0.60
Fill any spaces with zeros. Remember that 32 is the same as 32.00.
64.71
!
When adding or subtracting decimals, keep decimal points below one another.
Example 12 1 Subtract 8.914 from 46.029. Solution Estimate: 46 − 9 = 37 The answer should be about 37.
46.029 − 8.914 37.115
CHAPTER 7 DECIMALS
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2 Find the answer to 4.31 − 2.183. Solution Estimate: 4 − 2 = 2.
4.310 − 2.183
Fill any spaces with zeros.
2.127
Exercise 7-05 Ex 11
TLF
L 874
Swamp survival: thousandths patterns
Ex 12
TLF
L 869
Wishball challenge: thousandths
TLF
L 875
Wishball challenge: ultimate
1 Copy and complete these addition calculations: a 5.3 b 4.723 c 43.5 6.2 0.01 116.29 + 0.5 + 12.2 7.3 + 0.227
d
0.0076 1.23 + 0.9
2 Copy and complete these subtraction calculations. a 57.703 b 6.1 c 23.57 d 22.6 − 16.21 − 0.2 − 16.88 − 13.54
e 37 3.45 1.98 + 548.7
e 48.6 − 9.951
3 Which of the following is the answer to 0.61 + 12.345? Select A, B, C or D. A 73.345 B 18.445 C 12.955 D 12.406 4 Find the answers to the following. a 103.67 + 9.81 + 0.24 + 3.7 c 98.64 − 41.09 e 38.624 + 1.109 + 23.7 + 0.65
b 4.6 + 2.9 + 15 + 0.16 + 32.32 d 7.41 − 3.95 f 58.94 − 2.31 − 46.13
5 An electrician needed these lengths of cable to complete a wiring job: 12.3 m, 4.8 m, 18.7 m, 7.98 m, 13.65 m and 23.6 m. a How many metres of cable did the electrician use? b If the full spool of cable was 100 m long, how many metres of cable were left in the spool after the electrician completed the job? 6 Add each set of prices. Calculate the exact change from the amount shown in brackets. a $7.01 $0.34 $2.19 ($10) b $0.85
$4.34
$1.17
$8.79
($20)
c $11.34
$9.15
$3.95
$7.92
$2.36
($50)
7 To keep fit, Angela runs each day. Last week she ran 3.8 km, 4.1 km, 2.3 km, 2.6 km, 3.0 km, 0.9 km and 1.8 km. How far did she run last week? 8 A truck carrying sand had a total mass of 13 248 kg. If the truck alone had a mass of 5210.8 kg, what is the mass of the sand?
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9 Five runners in the school’s 100 m race recorded the following times: 13.5 s, 13.81 s, 12.7 s, 14.62 s, 12.45 s a Place these times in order, from fastest to slowest. b What is the time difference between the first runner to finish and the last? c If the runner in second place had run 0.3 seconds faster, would she have won the race? Explain your answer. 10 Krysten’s expenses for one week are shown in the table below. a How much did she spend? b How much did she have left out of her salary of $620.80? Item
Cost ($)
Food
128.80
Clothing
88.45
Car
58.35
Rent
185.00
Entertainment
78.95
Savings
66.00
11 A block of wood 11.27 cm thick has 0.34 cm shaved off one side and 0.55 cm shaved off the other side. How thick is the block of wood then? 12 Wendy was making teddy bears. She needed these amounts of material for five bears: 2.6 m, 0.8 m, 1.2 m, 0.75 m and 0.88 m. How much material did she need altogether?
13 The odometer in a car measures the total distance the car has travelled. The odometer below reads 21 456.9 km. The purple digit shows tenths of a kilometre. 2
1
4
5
6
9
Find the distance travelled during a holiday if the odometers below give the readings at the start of the holiday and at the end of the holiday. 2
1
5
7
6
4
2
2
3
1
5
3 CHAPTER 7 DECIMALS
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Working mathematically
Applying strategies
Comparing heights 1 Use the clues below to find each girl’s name and height. • Mandy is taller than Sarah. • Sarah is shorter than Yin. • Kelly is taller than Sarah but shorter than Mandy. • Mandy is not the tallest. • The heights of the girls are 168.5 cm, 166.3 cm, 164.2 cm and 160.7 cm. A
B
C
D
2 Use the clues below to find each boy’s name and height. • Steve is 164.7 cm tall. • Mike is 14.7 cm taller than Milof. • Steve is 3.9 cm shorter than Milof. • Ganesh is 1.6 cm taller than Mike. A
B
C
D
3 Use the clues below to find each student’s name and height. • Jade is 13.7 cm shorter than Yoko. • Yoko is 15.1 cm taller than Peter. • Karl is 20.6 cm taller than Jade. • Yoko is 6.9 cm shorter than Karl. • Peter is 163 cm tall. A
B
C
D
4 Devise your own problem using four to six class members. Estimate their heights and height differences. Write a set of clues. (Don’t forget to change their names!)
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Working mathematically
Reasoning
Decimals and powers of 10
th t ou en sa nd th s
dt hs an us
3
th o
2
ed th s
7
hu nd r
nt hs
6
te
3
ts Un i
s
Hu nd re ds
nd sa ou
Te ns
36.7
Th
th T ou en sa nd s
Ca l
cu
la
tio
n
You will need a copy of the table below and a calculator. 1 Complete each calculation in the left-hand column and write the answer in the correct place in the table.
36.7 × 10 36.7 × 100 36.7 × 1000 2.35
5
2.35 × 10 2.35 × 100 2.35 × 1000 36.7 ÷ 10 36.7 ÷ 100 36.7 ÷ 1000 2.35 ÷ 10 2.35 ÷ 100
2 Copy and complete the following. a When multiplying by powers of 10, the decimal point moves to the b When dividing by powers of 10, the decimal point moves to the
. .
3 Write your own rule to work out how many places the decimal point is moved.
7-06 Multiplying and dividing by powers of 10 Exercise 7-06 1 Copy and complete each of these statements. a To multiply by 10, move the decimal point place to the b To multiply by 100, move the decimal point places to the c To multiply by 1000, move the decimal point places to the d To divide by 10, move the decimal point places to the e To divide by 100, move the decimal point places to the f To divide by 1000, move the decimal point places to the
. . . . . .
CHAPTER 7 DECIMALS
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2 Copy and complete the following. a 46.3 ÷ 10 = b 507 ÷ 100 = d 36.4 ÷ 100 = e 705 ÷ 1000 = g 6.43 ÷ 1000 = h 64.3 ÷ 1000 = j 66 ÷ 100 = k 0.31 ÷ 1000 = m 24.9 × 100 = n 0.81 × 10 = p 0.416 × 100 = q 0.81 × 100 = s 60 451 × 100 = t 6.02 × 100 =
c f i l o r u
1203 ÷ 1000 = 3102 ÷ 10 = 4.28 ÷ 1000 0.02 ÷ 10 37.42 × 1000 = 2.192 × 1000 = 0.031 × 10 =
3 Evaluate the following, using the rules that you have found. a 45.213 × 100 b 10.64 × 1000 c 6304 ÷ 100 d 5.98 ÷ 1000 e 847.612 × 100 f 0.0592 × 10 g 36.28 ÷ 100 h 519.4 × 1000 i 40.075 ÷ 10 j 81.348 ÷ 1000 k 502 × 100 l 0.61 ÷ 100 m 0.4 ÷ 1000 n 17.01 × 100 o 12.3 × 10 000 p 66 ÷ 10 000 q 5.2 × 10 ÷ 100 r 14.71 ÷ 100 × 10 4 Evaluate the following. a 469 187 ÷ 100 000 c 27.43 ÷ 1 million e 235 000 137 ÷ 10 000
Worksheet 7-04 Where’s the point? Worksheet 7-05
b 437.421 ÷ 1000 d 1 200 000 ÷ 1 000 000 f 137.429 × 1000 ÷ 10 000
7-07 Using estimation to multiply decimals If you know the result of a whole number multiplication, you can use estimation to work out the answer to similar decimal multiplications and be able to correctly position the decimal point.
Multiplication estimation game
Example 13 1 Given that 17 × 12 = 204, find: a 1.7 × 12
b 1.7 × 120
Solution Multiplication
Estimate
17 × 12
Answer 204
(given)
a
1.7 × 12
2 × 10 = 20
20.4
use the digits 204 to make a number near 20
b
1.7 × 120
2 × 100 = 200
204
use the digits 204 to make a number near 200
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2 Given that 23 × 47 = 1081, find: a 2.3 × 4.7 b 230 × 4.7
c 23 × 0.47
Solution Multiplication
Estimate
Answer
23 × 47
1081
a
2.3 × 4.7
2 × 5 = 10
b
230 × 4.7
200 × 5 = 1000
c
23 × 0.47
20 × 0.5 = 10
10.81 1081 10.81
(given) use the digits 1081 to make a number near 10 use the digits 1081 to make a number near 1000 use the digits 1081 to make a number near 10
Exercise 7-07 1 Copy and complete each of the following tables. a
Multiplication
Estimate
69 × 18
Ex 13
Answer 1242
690 × 180 6.9 × 180 6.9 × 1.8 690 × 1.8
b
Multiplication
Estimate
104 × 42
Answer 4368
10.4 × 42 1.04 × 4.2 104 × 4.2 0.104 × 4.2
c
Multiplication 38 × 92
Estimate
Answer 3496
3.8 × 92 38 × 920 38 × 0.92 380 × 0.92
2 Given that 63 × 34 = 2142, use estimates to find: a 630 × 340 b 0.63 × 3.4 d 3.4 × 630 e 6.3 × 34 000
c 0.63 × 3400 f 63 × 340 CHAPTER 7 DECIMALS
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3 Given that 1.7 × 1.2 = 2.04, use estimates to find: a 1.7 × 12 b 17 × 12 d 120 × 170 e 0.12 × 17
c 0.17 × 1.2 f 17 000 × 0.12
4 Given that 7.2 × 3.4 = 24.48, use estimates to find: a 7.2 × 34 b 72 × 3.4 d 72 × 34 e 7.2 × 0.34
c 0.72 × 3.4 f 720 × 340
5 Given that 1.26 × 6 = 7.56, use estimates to find: a 12.6 × 6 b 126 × 6 d 0.126 × 6 e 0.126 × 0.6
c 1.26 × 0.6 f 126 000 × 600
6 Use the fact that 0.3 × 0.24 = 0.072 to find: a 3 × 0.24 b 0.3 × 2.4 d 30 × 0.24 e 300 × 2.4
c 0.3 × 24 f 0.03 × 0.24
Working mathematically
Reflecting and reasoning
Decimal places in multiplication answers 1 What happens when you multiply by a number less than one? As a group activity, consider the product of 12 × 0.8. a Is the answer more or less than 12? Why? b Estimate the answer to 12 × 0.8. c How many decimal places have 12 and 0.8? d Use a calculator to evaluate 12 × 0.8. How many decimal places has the answer? 2 a b c d
Is the answer to 0.7 × 0.3 more or less than 0.7? Why? Estimate the answer to 0.7 × 0.3. How many decimal places have 0.7 and 0.3? Use a calculator to evaluate 0.7 × 0.3. How many decimal places has the answer?
Is the answer to 2.5 × 4.1 more or less than 2.5? Why? Estimate the answer to 2.5 × 4.1. How many decimal places have 2.5 and 4.1? Use a calculator to evaluate 2.5 × 4.1. How many decimal places has the answer? e What is the relationship between the number of decimal places in the question and the number of decimal places in the answer?
3 a b c d
4 a b c d
If 82 × 6 = 492, what do you think is the answer to 82 × 0.6? Why? If 4 × 17 = 68, what do you think is the answer to 0.4 × 1.7? Why? If 367 × 51 = 18 717, what do you think is the answer to 3.67 × 5.1? Why? What is the answer to 0.5 × 0.9?
5 Make up a question about multiplying decimals. Swap questions with all members of the group. If you disagree about the correct answers, check with your teacher.
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7-08 Multiplying decimals When multiplying decimals, the number of decimal places in the answer equals the total number of decimal places in the question.
!
Example 14 1 Evaluate 3.06 × 4.8.
Worksheet 7-06
Solution Step 1: Do the multiplication without decimal points. 306 × 48
Which decimals?
2448 12 240 14 688 Step 2: Decide where to place the decimal point: 3.06 has 2 decimal places 4.8 has 1 decimal place So the answer has 3 decimal places: 14.688 OR Estimate: 3 × 5 = 15 Place the decimal point to make a number near 15: 14.688 2 Evaluate 0.6 × 4.1. Solution 41 ×6 246 0.6 has 1 decimal place 4.1 has 1 decimal place So the answer has 2 decimal places: 2.46 OR Estimate: 1 × 4 = 4 So the answer must be 2.46 (near 4)
Exercise 7-08 1 How many decimal places will the answers to the following have? a 0.25 × 11 b 10.2 × 4 c 0.5 × 10 d 7 × 2.193 e 0.9 × 0.75 f 8.06 × 4.1 g 0.11 × 1.01 h 6.3 × 0.04 i 2.95 × 5.13 j 0.237 × 1.2 k 0.023 × 0.042 l 321.2 × 8.1 CHAPTER 7 DECIMALS
Ex 14
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Ex 14
2 Calculate the following. a 3.05 × 4 b 1.02 × 7 e 10 × 2.25 f 3 × 4.20 i 0.18 × 5 j 0.4 × 12 m 10.941 × 3 n 492 × 0.12
c g k o
2.001 × 9 6.95 × 5 6 × 0.002 0.11 × 365
d 17.1 × 2 h 1.0004 × 8 l 3 × 4.2
3 Calculate the following. a 0.4 × 0.8 b 3.9 × 0.5 e 0.08 × 0.04 f 3.1 × 0.4 i 0.39 × 9 j 2.93 × 0.3
c 0.8 × 0.6 g 12.6 × 0.06 k 6.80 × 4
d 0.3 × 0.24 h 0.28 × 3 l 0.54 × 20
4 Evaluate the following. a 47.9 × 23 d 94.6 × 5.1
b 6.43 × 7.2 e 9.2 × 7.9
c 83.4 × 6.3 f 0.7521 × 3.6
7-09 Calculating change Example 15 Worksheet 7-06 Which decimals?
Worksheet 7-07 Shopping and change
Find the change from $50 if the following items are bought. • 3 kg of butter at $2.50 per kilogram • 500 g of cheese at $12.88 per kilogram • 2 kg of meat at $6.90 per kilogram • 2 dozen eggs at $4.26 per dozen. Round the total to the nearest 5 cents. Solution 3 0.5 2 2
× 2.50 × 12.88 × 6.90 × 4.26 Total
= 7.50 = 6.44 = 13.80 = + 8.52 36.26
Rounded to the nearest 5 cents, this total is $36.25. Change from $50: 50.00 − 36.25
SALESPERSON 31 MILK 1LT BUTTER BREAD
1.53 1.98 1.64
RND TOTAL CASH CHANGE
5.15 6.00 0.85
10:52
12/12/2008
13.75 Change will be $13.75.
Exercise 7-09 Ex 15
Calculate the change from $50 for each of the following purchases. Round totals to the nearest 5 cents. 1 • meat $25.36 • 2 dozen eggs at $4.65 per dozen • 2.5 kg of apples at $4.28 per kilogram
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2 • 3 kg of tomatoes at $2.45 per kilogram • 10 kg of potatoes at $1.98 per kilogram • 4 L of milk at $1.48 per litre • 2 kg of butter at $3.08 per kilogram 3 • 1.5 kg of chops at $7.98 per kilogram • 0.5 kg of T-bone steak at $15.90 per kilogram • 5 L of soft drink at $1.68 per litre 4 • • • • •
toothpaste $2.56 jam $1.13 cheese $3.08 4 kg of oranges at $3.99 per kilogram 3 kg of sausages at $5.40 per kilogram
5 • 2 kg of nails at $2.51 per kilogram • 2 m of wood at $13.48 per metre • 2.5 L of glue at $6.74 per litre 6 • 7 pens at $2.15 each • 12 glue sticks at $1.99 each
• 4 erasers at $0.35 each
7 • 2 tubes of toothpaste at $3.15 each • 4 boxes of tissues at $3.29 each
• 2 bottles of drink at $1.37 each • 1.5 kg of tomatoes at $5.98 per kilogram
8 • 31.5 litres of petrol at 139.9 cents per litre • 0.5 litres of premium oil at $6.60 per litre
Working mathematically
Applying strategies and reflecting
Multiplication maze 1 Can you work out how to get through the maze below? Start with 1 in your calculator display and, as you travel along each path, multiply the number in the display by each number you pass. You may travel along each path only once, but it can be in any direction. You must finish with a 5 in the display. 2 How many steps are needed to complete the maze? 3 Is there only one solution to this maze? × 0.1
× 2.5
× 0.5
× 0.2
× 10
1 START × 1.5
×5 ×2
×4
×2
FINISH 5
× 0.4
× 0.25
CHAPTER 7 DECIMALS
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Using technology
Fruit and vegetables In this activity, a spreadsheet is used to show a shopping list of items and to complete calculations involving their cost. 1 Zoe did her fruit and vegetables shopping for the week. Enter the items into a spreadsheet, as shown below. (Centre values, bold headings, include cell borders and $ signs for column B values.)
2 Write a formula in cell D2 to calculate the cost of the oranges. 3 Use Fill Down to calculate the cost of each item purchased. 4 In cell C13, enter the label ‘Total cost’. In cell D13, write a sum formula to find the cost of Zoe’s shopping. 5 a In cell C14, enter the label ‘Cash’. In cell C15, enter the label ‘Change’. b If Zoe paid $50 for her shopping, enter this value into cell D14 and in cell D15 write a formula to calculate the amount of change Zoe will receive. c Zoe paid cash, but because the change is an irregular amount, she could not be given the amount in cell D15 in coins. In cell C16, enter the label ‘Rounded change’. In cell D16, enter the amount of change Zoe will actually receive. 6 Return to Exercise 7-09 and, using a spreadsheet with similar formatting to the above, check your answers to Questions 1 to 8.
7-10 Dividing decimals by whole numbers
!
When dividing a decimal by a whole number: • rewrite the question in ‘short division’ form • make the decimal point in the answer line up with the decimal point in the question • add zeros to the end of the decimal being divided, if needed.
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Example 16 Evaluate each of the following. a 10 ÷ 4 b 13.62 ÷ 3
c 0.018 ÷ 6
d 2.63 ÷ 4
Solution 2.5 a 4 10.0 4.54 b 3 13.62
Write a zero after the decimal point so that you can to complete the division.
0.003 c 6 0.018 0.6575 d 4 2.6300
Write two zeros so that you can to complete the division.
Exercise 7-10 1 Evaluate each of the following. a 4.8 ÷ 2 b 18.6 ÷ 3 e 29.3 ÷ 2 f 8.79 ÷ 4 i 195.6 ÷ 8 j 7.35 ÷ 2
Ex 16
c 20.8 ÷ 5 g 0.056 ÷ 7 k 4.15 ÷ 8
2 Evaluate each of the following. a 12 ÷ 5 b 13.56 ÷ 12 d 88.88 ÷ 11 e 107.1 ÷ 9 g 177 ÷ 12 h 2075.6 ÷ 8 3 Evaluate each of the following. a 0.651 ÷ 3 b 37 ÷ 8 e 89.341 ÷ 7 f 4.275 ÷ 5 i 117.09 ÷ 9 j 0.471 ÷ 3
d 32.8 ÷ 8 h 10.71 ÷ 4 l 0.318 ÷ 3 c 23 ÷ 4 f 82.5 ÷ 6 i 0.732 ÷ 6
c 0.078 ÷ 6 g 2.75 ÷ 4 k 256.84 ÷ 4
d 675 ÷ 12 h 0.0913 ÷ 11 l 0.696 ÷ 12
7-11 Dividing decimals by decimals Look at this pattern:
18 ÷ 3 = 6 180 ÷ 30 = 6 1800 ÷ 300 = 6 When dividing, if we multiply both numbers by the same number first, the answer stays the same. We can use this property to help us divide decimals. For example: 9.8 ÷ 0.08 = 980 ÷ 8 (multiplying both numbers by 100) = 122.5 The answer to 980 ÷ 8 is easier to find than 9.8 ÷ 0.08 because 8 is a whole number. CHAPTER 7 DECIMALS
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To divide a decimal by a decimal: Step 1: Make the second decimal a whole number by moving the decimal point the appropriate number of places to the right. Step 2: Move the decimal point in the first decimal the same number of places to the right. Step 3: Divide the new first number by the new second number.
!
This works because in Steps 1 and 2 we multiply both decimals by the same power of 10.
Example 17 Evaluate each of the following: a 0.4 ÷ 0.2 b 1.75 ÷ 0.5
c 122.4 ÷ 0.04
Solution a 0.4 ÷ 0.2 =4÷2 =2
Move the decimal points one place to the right (multiplying both decimals by 10).
b 1.75 ÷ 0.5 = 17.5 ÷ 5 = 3.5
Move the decimal points one place to the right (multiplying both decimals by 10).
c 122.4 ÷ 0.04 = 12 240 ÷ 4 = 3060
Move the decimal points two places to the right (multiplying both decimals by 100).
Always check by estimation that your answers sound reasonable.
Exercise 7-11
Ex 17
1 a 18 ÷ 0.5 means ‘how many times does 0.5 go into 18’. Is the answer more or less than 18? Why? b Estimate the answer to 18 ÷ 0.5. c Use the method from Example 17 to evaluate 18 ÷ 0.5. 2 a 20.4 ÷ 0.3 means ‘how many times does 0.3 go into 20.4’. Is the answer more or less than 20.4? b Estimate the answer to 20.4 ÷ 0.3. c Find the exact answer to 20.4 ÷ 0.3. 3 What happens when you divide by a number less than 1? Is the answer more or less than the number? (Check your answers to Questions 1 and 2). 4 Which of the following is the answer to 13.59 ÷ 0.03? Select A, B, C or D. A 45.3 B 453 C 4.53 D 4530
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5 Rewrite each of the following divisions so that the second decimal is a whole number. a 508.8 ÷ 1.2 b 17.82 ÷ 0.11 c 333 ÷ 4.5 d 1.725 ÷ 2.5 e 129.2 ÷ 0.38 f 49.5 ÷ 1.5 g 168 ÷ 0.75 h 14.823 ÷ 0.61 i 0.66 ÷ 0.022 6 Evaluate each of these, and check that your answers seem reasonable by estimating. a 3.48 ÷ 0.4 b 7.32 ÷ 0.2 c 2.94 ÷ 0.6 d 16.28 ÷ 2.2 e 27 ÷ 0.25 f 10.08 ÷ 2.4 g 10.4 ÷ 0.05 h 5.6 ÷ 0.07 i 1.71 ÷ 0.3 j 40.82 ÷ 5.2 k 532 ÷ 3.5 l 78.12 ÷ 6.2 m 0.272 ÷ 0.08 n 98.4 ÷ 0.08 o 465 ÷ 7.5
Mental skills 7B
Maths without calculators
Multiplying by 9, 11 or 12 To multiply a number by 9, multiply by 10 and then subtract the number. 1 Examine these examples. a 14 × 9 = 14 × 10 − 14 = 140 − 14 = 126 b 25 × 9 = 25 × 10 − 25 = 250 − 25 = 225 c 18 × 9 = 18 × 10 − 18 = 180 − 18 = 162 2 Now simplify these. a 12 × 9 b 27 × 9 e 34 × 9 f 63 × 9
c 46 × 9 g 21 × 9
d 19 × 9 h 15 × 9
To multiply a number by 11, multiply by 10 and then add the number. This is because 10 times a number plus the same number equals 11 times the number. 3 Examine these examples. a 26 × 11 = 26 × 10 + 26 = 260 + 26 = 286 b 13 × 11 = 13 × 10 + 13 = 130 + 13 = 143 c 35 × 11 = 35 × 10 + 35 = 350 + 35 = 385 4 Now simplify these. a 17 × 11 b 22 × 11 e 25 × 11 f 19 × 11
c 38 × 11 g 54 × 11
d 40 × 11 h 31 × 11
To multiply a number by 12, multiply by 10, then add double the number. This is because 10 times a number plus double the same number equals 12 times the number. 5 Examine these examples. a 22 × 12 = 22 × 10 + 22 × 2 = 220 + 44 = 264 b 16 × 12 = 16 × 10 + 16 × 2 = 160 + 32 = 192 c 70 × 12 = 70 × 10 + 70 × 2 = 700 + 140 = 840 6 Now simplify these. a 44 × 12 b 15 × 12 e 52 × 12 f 18 × 12
c 29 × 12 g 26 × 12
d 31 × 12 h 37 × 12
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Using technology
Calculations involving decimals There are three types of entries we can make into the cells of spreadsheets. Values: these are numerical values we enter into cells Labels: these are words Formulas: these begin with ‘=’. Some examples include: =sum(A1:A3) =A3-A2 =B1*C1 =D2/E1 1 Open a new spreadsheet and enter the four headings in the cells as shown below.
Addition of decimals 2 The decimals in our calculation are: 12.3 + 14 + 7.29 + 8.05. Enter them in cells A3 to A6.
3 In cell A7, enter =sum(A3:A6). Highlight your answer in bold. 4 Complete this addition: 56.2 + 188.65, by entering the decimals into cells A9 and A10. Enter the formula into cell A11. Highlight your answer in bold. Subtraction of decimals 5 a Enter 175.4 into cell C3, ‘-’ into cell D3 and 80.9 in cell C4. b Enter =C3-C4 into cell C5 and bold your answer. 6 a Enter 1011.111 into cell C8, ‘-’ into cell D8 and 99.34 into cell C9. b Enter a formula into cell C10 to subtract C9 from C8. c Bold your answer. Multiplication of decimals 7 a Enter 72.6 into cell E3, ‘×’ into cell F3 and 10.3 in cell E4. b Enter =E3*E4 into cell E5 and bold your answer. 8 a Enter 825.5 into cell E8, ‘×’ into cell F8 and 2.2 into cell E9. b Enter a formula into cell E10. Remember to bold your answer. Division of decimals 9 a Enter 189.3 into cell G3, ‘/ ’ into cell H3 and 3 in cell G4. b Enter =G3/G4 into cell G5 and bold your answer.
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Rounding answers 10 a Enter 748.6 into cell G8, ‘×’ into cell H8 and 3.6 into cell G9. b Enter a formula into cell G10. Right click on cell G10, choose Format Cells, Number, 1 decimal place and bold your answer. 11 Complete these calculations using your spreadsheet and creating appropriate formulas, as you have practised in the examples above. a 11.2 + 54.87 + 2.3 + 9.65 b 675.8 − 224.1 c 196.52 + 223.08 − 56.33 d 12.73 × 4.4 e 210.2 × 81.6 f 909.8 × 1.712 g 5.4 ÷ 1.8 h 284.796 ÷ 32.4 i 1217.9 ÷ 45.6 (round to 3 decimal places) j 92.7 × 1.15 ÷ 1.5 k 1604.12 ÷ 0.02 × 1.234 l (8756.32 − 9025.198 + 1023.5697) ÷ 1.444 (round to 2 decimal places) m (12.3 + 6.59) ÷ (56.4 × 0.04) (round to 5 decimal places)
Working mathematically
Reflecting and applying strategies
Back-to-front problems The cards for this set of questions have been printed without any decimal points. Insert the decimal points so that the numbers on the cards fit the clues. 1 The difference between these two numbers is 53.7. The sum of the numbers is 58.5.
561
24
2 The difference between the numbers is 178.8. When you divide the greater number by the smaller number, the quotient is between 43 and 44.
183
42
3 The sum of the three numbers is 5.36. The product of the numbers is 1.2096.
8
42
36
4 The sum of the three numbers is 4.61. The product of the numbers is 0.9135.
15
29
21
31
14
14
5 The sum of the four numbers is 2.55. The product of two of the numbers is 0.196. The product of the other two numbers is 0.217.
7
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7-12 Fractions and decimals It is useful to recognise fractions in their decimal form.
Example 18 Write each of these as a decimal. a 3---
b 5---
5
8
Solution a 3--- means 3 ÷ 5 5
b 5--- means 5 ÷ 8 8
0.6 5 3.0
0.625 2 4 8 5.000
3 --5
5 --8
= 0.6
= 0.625
Remember to add zeros, if necessary, to complete the division.
Worksheet 7-08
Decimals such as 0.625 are called terminating decimals. ‘Terminate’ means ‘to stop’. Sometimes when a common fraction is converted to a decimal, we get a repeating or recurring decimal. One or more of the digits in the decimal repeat forever.
Fraction families
Example 19 Worksheet 7-09 Decimals squaresaw 2
1 Change 1--- to a decimal. 3
Solution 1 --3 1 --3
means 1 ÷ 3
0.333… 3 1.000…
= 0.333… = 0. 3 or 0. 3
2 Change 5--- to a decimal. 6
Solution 5 --6 5 --6
means 5 ÷ 6
0.8333… 6 5.0000…
= 0.8333… = 0.8 3 or 0.8 3
2 - to a decimal. 3 Change ----11
Solution 2 -----11 2 -----11
means 2 ÷ 11
0.181 81… 11 2.000 00…
= 0.181 818… = 0.18 or 0.18
Note the repeating or recurring pattern in the numbers following the decimal point. To show that the pattern goes on forever, we use dots or a line to identify the repeating section: for example, 0.259 259 259… = 0. 259 or 0. 259 .
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Exercise 7-12 1 Copy and complete this table. Use a calculator if you need to. Common fraction
Meaning as division
Decimal
a
3 --5
3÷5
0.6
b
1 --2
1÷2
c
1 --4
d
4 --5
e
2 --5
Ex 18
0.25
f
3÷4
g
1÷5
h
1÷8
0.75
2 Use your completed table from Question 1 to help you find decimals equal to the following fractions. a 2---
b 3---
c 3---
d 2---
e
f
g
h
i
5 2 --4 5 --8
8 6 --8 7 --8
j
k
4 3 --5 4 --8
l
2 2 --8 5 --5
3 Explain why some of the fractions in Question 2 have the same decimal value. 4 Write the following as decimals.
e
10 57 2--5
5
8
4 1 19 --8
f
d 6 3---
c 12 5---
b 23 3---
8a 4 -----
g
h 80 1---
110 7--8
4
5 Change these fractions to repeating decimals.
Ex 19
a 1---
b 1---
c 1---
d 1---
e 2---
f
g
h
i
j
k
9 2 --7 5 --7
l
3 2 --9 5 --9
m
6 3 --7 6 --7
n
7 4 --7 7 --9
o
3 4 --9 8 --9
6 Copy and complete the following pattern: Fraction:
1 --9
2 --9
Decimal:
0. 1
0. 2
3 --9
4 --9
5 --9
6 --9
7 --9
8 --9
7 Perform these calculations and write the answers as recurring decimals. a 11 ÷ 3 b 11.3 ÷ 7 c 58.43 ÷ 9 d 1.9 ÷ 6 e 76 ÷ 9 f 0.67 ÷ 6 g 2÷7 h 13.4 ÷ 6 i 149 ÷ 7 CHAPTER 7 DECIMALS
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7-13 Rounding decimals Skillsheet 7-01 Rounding whole numbers
!
Sometimes, to approximate an answer with many decimal places, we round to fewer decimal places. We need to be able to round when working with money, measuring quantities or writing answers to division calculations.
To round a decimal: • cut the number at the required decimal place • look at the digit immediately to the right of the specified place • if this digit is 0, 1, 2, 3 or 4, leave the number in the specified place unchanged (round down) • if the digit is 5, 6, 7, 8 or 9, add 1 to the number in the specified place (round up).
Example 20 1 Round 86.246 correct to one decimal place. Solution 86.2 46 cut the next digit is 4, so the number 2 does not change So 86.245 is 86.2 (correct to one decimal place). 2 Round 86.246 correct to two decimal places. Solution 86.24 6 cut the next digit is 6, so add 1 to 4 to give 5 So 86.246 is 86.25 (correct to two decimal places).
Example 21 The number 0.087 1245 has seven decimal places. Round it to: a one decimal place b two decimal places c the nearest thousandth Solution a 0.1
!
b 0.09
Rounding to: • the nearest tenth = one decimal place • the nearest hundredth = two decimal places • the nearest thousandth = three decimal places
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Exercise 7-13 1 Write each of the following correct to one decimal place. a 0.35 b 0.47 c 0.81 e 2.55 f 0.32 g 0.90
d 0.69 h 2.88
2 Round each of the following to two decimal places. a 0.481 b 0.736 c 0.069 e 0.309 f 0.655 g 2.096
d 0.293 h 3.995
Ex 20
3 Write each of the following correct to two decimal places. a 25.3456 b 341.7675 c 321.3333 d 734.6541 e 27.757 575 f 1314.123 45 4 Copy this table into your book. Use a calculator to help you complete it. Question 12.19 ÷ 3
Calculator display 4.0633333333
Rounded to 1 decimal place 4.1
Ex 21
Rounded to 2 decimal places 4.06
12.32 ÷ 6 19.82 ÷ 9 56.85 ÷ 11 17.13 ÷ 4 12.65 ÷ 12 4.875 ÷ 21 27.45 ÷ 8 17 ÷ 12 254.678 ÷ 32
5 Use a calculator to find the value of each of the following divisions. Give your answers to the nearest hundredths. a 2.89 ÷ 3 b 8.57 ÷ 6 c 0.812 ÷ 9 d 7 ÷ 11 e 5.12 ÷ 6 f 11.71 ÷ 7 g 8÷6 h 12 ÷ 7 i 18.87 ÷ 11 j 12.62 ÷ 13 k 7 ÷ 12 l 9.38 ÷ 15 6 Round each of these numbers to four decimal places. a 10.33374 b 431.543 27 d 3217.654 061 e 4.670 89
c 1.444 95 f 0.888 88
7 The answers to the following are whole numbers but, for particular reasons, some need to be rounded up and some need to be rounded down. Find the answers. a A box of chocolates with 33 chocolates is shared among a family of five people. How many chocolates does each person receive? b A new bathroom requires 32 square metres of tiles. The tiles come in boxes containing 1.5 square metres. How many boxes are needed to tile the bathroom? CHAPTER 7 DECIMALS
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c A team of four golfers wins 27 new golf balls in a competition. How many does each person receive? d Some timber comes in 1.8 m lengths. How many lengths are needed to build a chicken house needing 23 m of timber? e Each blouse requires 1.3 m of material. How many blouses can be made from a 5 m length of material? 8 Give a reason for: a rounding up b rounding down.
Working mathematically
Questioning and reflecting
Rounding prices Since the use of 1c coins and 2c coins stopped in 1990, prices have been rounded to the nearest 5 cents. Find out: a when amounts are rounded b how amounts are rounded (check the major stores in your area) c what happens with bills such as electricity, phone, etc. d who decides how the rounding will work.
Just for the record
The salami technique When banks started using computers to keep track of customers’ accounts, they left themselves open to a new type of crime: computer theft. One such crime employs the salami technique, where computer hackers steal a cent or a fraction of a cent from many bank accounts. They round down the decimal amount of an account balance (for example $234.6523 would become $234.65) and the stolen fraction of a cent ($0.0023) is deposited into the hacker’s account, with no one noticing it missing. When this is done to thousands of bank customers over a number of years, a considerable amount of money can be accumulated. If the salami technique is applied to $1723.35631, $456.3277, $6701.2315 and $488.29891, how much will the computer criminal have in his or her account? Why do you think this type of crime is difficult to detect? Why do you think it is called the ‘salami technique’?
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7-14 Applying decimals
Worksheet 7-10
Decimals are used in many everyday situations.
Decimals review
Exercise 7-14 1 Find the cost of 352 units of electricity at 12.3 cents per unit. 2 A farmer wants to fence a rectangular paddock. The paddock is 35.6 metres long and 20.85 metres wide. How many metres of fencing will be needed? 3 Mark buys golf balls for $4.85 each and sells them for $5.15 each. How much money does he make if he sells 30 golf balls? 4 A drink bottle holds 0.75 litres. How many drink bottles can be filled from a tub that holds 4.5 litres? 5 A car travels 220.6 kilometres on 14 litres of petrol. How many kilometres would the car travel on one litre of petrol? (Give the answer to one decimal place.) 6 Pamela runs 3.8 kilometres each day of the week. How far does she run in one week? 7 A long distance train is made up of a diesel engine, two dining cars and 15 passenger carriages. The engine has a mass of 20.2 tonnes, each dining car has a mass of 14.35 tonnes and each passenger carriage has a mass of 13.96 tonnes. How heavy is the entire train?
8 George is cutting shelves from a board which is 4.6 metres long. Each shelf needs to be 1.25 metres long. How many shelves can be cut? 9 Samantha ran 100 metres in 15.21 seconds. How long would Samantha take to run 400 metres at this pace? 10 Henry has a faulty calculator; it does not show the decimal point. For each of the calculations in the table shown on the right, write the correct answer.
Calculation
Answer
3.42 × 12
4104
4.145 × 0.2
829
37.3 × 8.8
32824
0.03 × 157.64
47292
8.3902 × 0.3
251706
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11 Mick walks to work and back each day. He works six days a week and, in one week, walks 16.8 kilometres. How far is Mick’s apartment from work? 12 The following are calculator displays for amounts in dollars and cents. Rewrite each amount in dollars and cents, to the nearest cent. a b c 487.759318 d 1234.047126 15.236 6.8412 13 A sheet of paper is 0.01 cm thick. How many sheets would be in a stack 4 cm high? 14 FM radio In the radio and television guide you will find a list of FM stations and their allocated frequencies measured in megahertz (MHz). a Locate the stations on a number line. 90
100
110
b What is the frequency difference in megahertz between 2DAY and WS-FM? c Find the smallest frequency difference between adjacent stations. (‘Adjacent’ means side-by-side.) d What is the largest difference in frequency between adjacent stations?
Station
Frequency (MHz)
ABC Classic
92.9
Vega
95.3
The Edge
96.1
NOVA
96.9
SBS Radio 2
97.7
WS-FM
101.7
2MBS
102.5
2DAY
104.1
MMM
104.9
JJJ
105.7
MIX
106.5
2SER
107.3
15 Decimal currency a Find out about the history of the decimal money system in Australia. i When was it introduced? ii What system did it replace? iii Why was the change made? iv What coins and notes were introduced? v What changes have occurred since? b Design a set of notes ($5, $10, $20, $50, $100) that blind and sighted people could both use. 16 Decimal time a Investigate the idea of decimal time. How would you measure time in this system? b Design a decimal time calendar. c How would decimal time affect your birthday and age? d Do you think decimal time is possible? Explain.
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9
10 1
8
2
7
3 6
5
4
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Using technology
Rainfall figures The daily rainfall for three NSW towns in the last week of June 2007 is given below. 1 Enter the rainfall data shown below into a spreadsheet.
Source: www.bom.gov.au
2 a In cell A11, enter the label ‘Total’. In cells A12 and A13 enter the labels ‘Maximum’ and ‘Minimum’ respectively. b In cell B11, write a formula to find Condobolin’s total rainfall for the week. Use Fill Right to copy the formula into cells C11 and D11. Centre the three totals. c In cell B12, enter a formula to find Condobolin’s maximum rainfall for the week. Use Fill Right to copy the formula into cells C12 and D12. Centre the three totals. d In cell B13, enter a formula to find Condobolin’s minimum rainfall for the week. Use Fill Right to copy the formula into cells C12 and D12. Centre the three totals. 3 In cell E4, enter a formula to find the total rainfall for Condobolin, Goulburn and Katoomba on 24 June 2007. Use Fill Down to sum the rainfall for each of the following 6 days. 4 a Which of the three towns had the most rain for the week? Bold the cell with the highest total rainfall. b Which of the three towns had the highest rainfall, on a single day, in this week? Bold the cell with the highest rainfall. c i In this particular week, on which day was the highest total rainfall recorded? Bold this cell. ii In this particular week, on which day/s was the lowest total rainfall recorded? Bold this cell. iii In cell F4, enter a formula to find the difference between the answers to i and ii above. 5 Using the website www.bom.gov.au, search for New South Wales temperature and rainfall data. Use the data to write a paragraph comparing the rainfall patterns of June 2007 with the 12 previous months.
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Power plus 1 Code breaker A long time ago, the warrior Wolf was trying to break a code that would open some dungeon doors. He had to free the prisoners before midnight so that they would not be turned into frogs by an evil spell. Each castle door was operated by a combination lock. a Use the following clues to match the various combinations with the doors to the different rooms in the castle. Combinations Rooms 8.262 Queen’s chamber 9.24 armoury 9.96 throne room 8.07 banquet hall 8.16 kitchen 8.79 dungeon Clues • Combination 9.24 opens a door to a room that deals with food. • The combination to the armoury has a 6 in the hundredths place. • The combinations of the throne room and the banquet hall add to 18.03. • The Queen’s chamber has a combination that is bigger than 3.78 × 2.1 but smaller than 25.11 ÷ 3.1. (Hint: The doors in the dungeon and the throne room remained locked when Wolf tried 9.96 and 8.262.) • The kitchen combination is one of the three largest combination numbers. b Invent another story that sets a code-breaking task involving decimals and ask your classmates to solve it. 2 Calculate the following. a 0.02 × 0.02 d
0.04
b (0.02)3
c (1.1)3
e
f 3 0.027
0.36
3 Decide where the decimal point should be so that the numbers in the ovals fit the clues. a The product of the two numbers is 91.02. 246 37 The sum of the numbers is 28.3. b The sum of the three numbers is 14. The product of the three numbers is 78.66. c The sum of the four numbers is 28.32. The product of two of the numbers is 3.672. The product of the other two is 81.
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23
57
51
72 45
6
18
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Chapter 7 review Language of maths approximate decimal point hundredth round
ascending descending place value terminating decimal
Worksheet 7-11
decimal estimate power of 10 tenth
decimal place fraction recurring decimal thousandth
Decimals crossword
1 Use the words ‘ascending’ and ‘descending’ in a non-mathematical way. 2 ‘The bushfire decimated the possum population of the forest.’ Look up the meaning of the word ‘decimate’. 3 What is a recurring dream or a recurring back pain? What does ‘recurring’ mean? 4 What happens when a train terminates at a station? 5 What is a ‘decimetre’? 6 Find examples of words beginning with ‘deci’ that mean a tenth of something.
Topic overview • What parts of this topic were new to you? What parts did you already know? • Write any rules you have learnt about working with decimals. • What parts of this topic did you not understand? Be specific. Talk to a friend or your teacher about them. • Give three examples of where decimals are used. • Copy this overview into your workbook and complete it. Check it with a friend or your teacher, to be sure nothing is left out. Decimals and fractions Place value
Operations
DECIMALS Rounding
Problems Ordering decimals
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Topic test 7
Exercise 7-01
Chapter revision 1 Write each of the following in decimal form. 1 1b 5 × 100 + 2 × 10 + 7 × ----+ 3 × -----------
1 1 - + 6 × --------a 4 × 10 + 3 × -----
d 8+6×
c twelve thousandths Exercise 7-02
1000
10
100
10
1 --------100
+9×
1 -----------1000
2 Read the story below and write the number in each part with a decimal point so that the story makes sense. a Angelo and Loren decided to go to the motorbike races. They caught the train at 12:45pm after walking 123 km to the station. b They gave the station master a $2000 note c and received $460 change. d The train trip took 15 hours. e Angelo said that each motorbike weighed 1400 kg. f There were 4 500 000 people at the race meeting g who paid a total of $50 850 000 in entrance costs. h The average price of a ticket was $113. i A mechanic told Loren that the price of race fuel was 1349 cents per litre.
Exercise 7-03
3 Draw a decimal number line for each of these sets of decimals and mark them in their correct locations. a 1.6 b 0.63
Exercise 7-03
Exercise 7-04
2.0 0.69
2.2 0.71
2.3
2.4
a 34.98
56.86
3.998
50.141
340
34.89
3.099
b 0.136
0.86
0.652
0.662
0.23
1.006
0.086
c 1.015
1.293
1.1015
1.239
8.6
5 Convert each fraction to a decimal. 10
13b -------100
7c -------100
111d -----------
45 e -----------
d 0.444
e 4.051
1000
1000
6 Convert each decimal to a fraction. a 0.5
Exercise 7-05
1.9 0.67
4 Arrange each group of numbers in descending order.
4a ----Exercise 7-04
1.7 0.64
b 0.89
c 0.09
7 Find the value of each of the following. a b c d e f g h
252
12.35 + 4.53 + 0.56 + 3.125 + 24.7 + 20.09 214.33 − 109.84 0.568 + 23 + 4.027 − 16.28 + 0.65 1600.8 − 562.9 1453.6 + 1287.31 − 2344.4 204.9 + 23.24 + 0.015 9.23 − 6.851 16.51 + 9.003 + 125 + 0.9
NEW CENTURY MATHS 7
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8 Evaluate the following. a 7.54 × 10 d 13.9 ÷ 10
Exercise 7-06
b 7.54 × 100 e 13.9 ÷ 100
c 7.54 × 1000 f 13.9 ÷ 1000
9 Given that 42 × 76 = 3192, find: a 4.2 × 76 b 4.2 × 7.6 d 420 × 76 e 42 × 7.6
Exercise 7-07
c 0.42 × 760 f 420 × 760
10 Find the answer to each of the following. a 2.75 × 6 b 0.5 × 1.2 d 6.1 × 1.2 e 0.92 × 10 g 3.25 × 0.41 h 0.05 × 0.02
Exercise 7-08
c 12.23 × 4 f 100 × 6.7 i 4.67 × 1.1
11 Calculate the change from $50 if all of the following are purchased. • 3 litres of milk at $1.40 per litre • 2 loaves of bread at $2.80 per loaf
Exercise 7-09
• 1--- kg of tomatoes at $4.50 per kg 2
• 2 kg of sausages at $6 per kg 12 Find: a 762.4 ÷ 2
Exercise 7-10
b 97.6 ÷ 8
c 2.75 ÷ 4
d 195.6 ÷ 12
13 Find: a 12.5 ÷ 0.5
Exercise 7-11
b 12.72 ÷ 0.4
c 6.9 ÷ 0.03
d 27.94 ÷ 1.1
14 Change the following to decimals. a
4 --5
b
3 --8
Exercise 7-12
c
5 --9
d
6 2--3
15 Round each of these numbers to the place value shown. a c e g
456.8 to the nearest ten 2345.876 to two decimal places 78 654.056 to the nearest thousand 102.007 to the nearest hundred
b d f h
Exercise 7-13
125.84 to the nearest whole number 3.8967 to the nearest tenth 678.439 to one decimal place 102.007 to the nearest hundredth
16 The Liverpool Women’s Cricket Club is having a pizza night. They order 16 Super Supreme pizzas at $13.70 each and 10 Hawaiian pizzas at $12.10 each. How much will the club need to spend?
Exercise 7-14
17 Maria saved $90 to go to a rock concert. Her return fare cost $5.60, her concert ticket cost $48.95, the program cost $11.00 and food cost $8.70. She did not have enough to buy the band’s latest compact disc (priced $24.00) after the concert. How much did she need to borrow from her friend Sam to buy the disc?
Exercise 7-14
18 Rae bought 1200 bricks for $465.70. How much did one brick cost?
Exercise 7-14
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