345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 7890123456789012 34567890123456 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 567890123456789012345678901 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 23456789012 45678901234 6789012345678901 34567890123456789012345678901234567 012345678901234567890123456756789012 56789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 01234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901 567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456 90123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 45678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 9012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 89012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234 78901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 23456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 67890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 1234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456789012 56789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 01234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901 56789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 90123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 45678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 89012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 234567890123456789012345678901234567890123456 7890123456789012 345678901 9012345 678901234567890123456789012345678901234567890123456789012345678901 34 6789012345678901234567890123456789012345678901234567890123 456789012345678901234567890123456789012345678 56789 7890123456789012 3456789012345678 901234567890123456789012345678901234567890123456789012345 567890123456789012345678901234567890123456789012345 56789012345678901234567890 123456789012345678 5678 234567890123 45678901234 0123456789 4567890123456789012345 123456789012345678 234567890123 4567890123 567890123 4567890123 8901234 0123456789 7890123 8901234 23456789 2345 3 1 01234 67890
1
5 4567890123 8901234 48901234 NUMBERMathematics began with people counting, and many civilisations came up with symbols to represent numbers. As people around the world started to cross paths, a common number system was needed. Eventually the Hindu–Arabic system was adopted all over the world. It is important to understand how our number system works and the rules it follows.
89012345678 345678901234567890123 0123 756789012345678901234 901234567890123456789 456789012345678901234 890123456789012345678 345678901234567890123 890123456789012345678 234567890123456789012 789012345678901234567 234567890123456789012 678901234567890123456 123456789012345678901 678901234567890123456 012345678901234567890 567890123456789012345 012345678901234567890 456789012345678901234 901234567890123456789 456789012345678901234 890123456789012345678 345678901234567890123 890123456789012345678 234567890123456789012 789012345678901234567 234567890123456789012 45678901 456789012 56789012 8 789012345678 0123456789 78901234 012345 56789012345678 0123456789 5678 678901234 0123456789 3 8901234 345678 9012345 67890123456789012345678 90123456789012345678901234567890123456 7890123456789012 901234567890123 01234567890 123456789012345678901234 5678 012345678901234 5678901 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 45678901234567890123456 789012345678901234567 8901234 678901234567890 2345678901 234567 45678901234 6789012345678901 34567890123456789012345678901234567 0123456789012345 5678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456 901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 4567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 9012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890 3456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 8901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234 7890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 2345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 6789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 1234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012 5678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 56789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234567890123456789012345678901234567890123456 901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 4567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 901234567890123456789012345678901234567890 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 3456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 8901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 456789 3456789012345678901234567890123456 7890123456789012 34567890 9012345 6789012345678901234567890123456789012345678901234567890123456789 6789012345678901234567890123456789012345678901234567890123 456789012345678901234567890123456789012345678 56789012 7890123456789012 3456789012345678 901234567890123456789012345678901234567890123456789012345678 567890123456789012345678901234567890123456789012345 56789012345678901234567890 1234567890123456789 5678 234567890123 45678901234 0123456789 4567890123456789012345 56789012345678 1234567890123456789 5678 234567890123 45678901234 0123456789 567890123 4567890123 8901234 0123456789 567890123 4567890123 8901234 0123456789 0123456789 48901234 0123456789 0123456789
In this chapter you will: Wordbank
• compare the Hindu–Arabic number system with number systems from different societies, past and present
• recognise, read and convert Roman numerals • state the place value of any digit in large numbers • order numbers of any size, in ascending and
descending order
• record large numbers using expanded notation • revise the four basic operations on whole numbers • apply order of operations to simplify expressions • divide digit and three-digit numbers by a
two-digit number
• use the symbols of mathematics, including and . 3
• cube root The value which, if cubed, will give the required number, for example = 4 because 43= 64.
• evaluate To find the value of a numerical expression.
• expanded notation A way of writing a number that shows the place valueof every digit. • Hindu–Arabic number system The number
system we use, with the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
• numeral A symbol that stands for a number, such as 8 or X.
• order of operations The rules for calculating an expression containing mixedoperations, such as 14 − 2 × 4 + 1.
• place value The way that the position of a digit in a number tells us its value.
64 3 01 NCM7 2nd ed SB TXT.fm Page 3 Saturday, June 7, 2008 2:30 PM
Start up
1 Write the answers to the following.
a 10 × 10 b 4 × 7 c 900 + 30 d 7 + 9 e 10 × 10 × 10 f 35 ÷ 5 g 9 × 9 h 26 − 8 i 1000 + 200 + 50 j 6 × 5 k 99 ÷ 11 l 75 − 16 m18 × 3 n 7 × 12 o 128 − 24 p 128 ÷ 4 q 137 + 45 r 35× 12 s 452 − 140 t 280× 10 u 3601− 59
2 Write each of these numbers in words.
a 45 b 120 c 138
d 3680 e 5001 f 47 613
3 Write each of the following numbers using numerals.
a sixty-eight b seven hundred
c two thousand and four d eight hundred and ninety-nine
e ten thousand, four hundred and ninety-two
1-01 The ancient Egyptian number system
The ancient Egyptians used one ofthe earliest number systems about 5000 years ago. Pictures called hieroglyphs represented words or sounds. They were written on papyrus (a type of paper made from reeds) or painted on walls.
The hieroglyphic symbols used by the Egyptians were:
Worksheet 1-01 Brainstarters 1 Worksheet 1-02 Multiplication facts
Reading and writing large numbers Skillsheet 1-01 10 1 2 3 4 5 6 . . . 9 20 . . . 100
(coiled rope) (lotus flower)
200 . . . 1000
10 000 100 000
(bent reed) (fish)
1 000 000 (million)
(man with hands raised in surprise) 01 NCM7 2nd ed SB TXT.fm Page 4 Saturday, June 7, 2008 2:30 PM
1 If you were an ancient Egyptian student, how would you write these numerals?
a 7 b 37 c 165
d 268 301 e 3 251163 f 1253
2 Use our numerals to write the numbers represented by these Egyptian numerals.
3 Write the answer to these in Egyptian numerals.
Exercise 1-01
Example 1
Show how an ancient Egyptian would have written each of these numbers.
a 25 b 126 c 3468
Solution
a b c
Example 2
If ancient Egyptian numerals could be written in any order, how could 125 be written? Solution or or Ex 1 a b c d e plus plus minus a b c d minus minus minus 01 NCM7 2nd ed SB TXT.fm Page 5 Saturday, June 7, 2008 2:30 PM
4 State one advantage and one disadvantage of working with ancient Egyptian numerals. 5 Why do you think a picture of a surprised man was used by the ancient Egyptians to
represent a million?
1-02 Australian Aboriginal number systems
The Australian Aboriginal way of life had no need for a complicated number system. Their society relied on story-telling, using the spoken language rather than writing, andAboriginal people did not have symbols for numbers. Different regions had their own names for numbers.
The Belyando River people of central Queensland used only two words to name their numbers:
1 = wogin 2 = booleroo
3 = booleroo wogin 4 = booleroo booleroo
The Kamilaroi people lived in northern New South Wales, including the regions surrounding Moree and Tamworth. They used three words to name their numbers.
1 = mal 2 = bularr
3 = guliba 4 = bularr bularr
5 = bularr guliba 6 = guliba guliba
1 How did the Belyando River people form words for the numbers 3 and 4? 2 How did the Kamilaroi people form words for 4, 5 and 6?
3 Answer the following, using the correct Aboriginal words: a wogin + booleroo wogin b guliba × bularr c bularr + bularr + mal d booleroo × booleroo e guliba guliba − guliba f bularr bularr − mal
4 State one advantage and one disadvantage of working with Aboriginal numbers.
1-03 The Babylonian number system
The ancient kingdom of Babylon existed from about 3000 to 200 BC where Iraq is today.
Babylonian writing used wedge shapes called cuneiform.The wedges were stamped into clay tablets which were then baked. Babylonian numerals also used cuneiform.
While our number system is based on 10 and 100, the Babylonian number system was based on 10 and 60. This wedge stood for 1: A sideways wedge stood for 10:
A larger wedge stood for 60:
Exercise 1-02
10 20 30 . . . 60 70 80 . . . 120 130
1 2 3 4 5 . . . 9
1 How would you write each of these numerals using our numerals?
Notice that there was no need for a zero.
2 Use Babylonian numerals to write each of these amounts.
a 26 b 58 c 107
d 300 e 144 f 401
3 State one advantage and one disadvantage of working with the Babylonian number system.
1-04 The Roman number system
The Roman empire was one of the greatestempires. Roman numerals were invented about 2000 years ago. They were used until the end of the 16th century. Today they are used mainly in clocks and for some page numbers in books.
The Romans used the following numerals:
1 2 3 4 5 I II III IV V 6 7 8 9 10 VI VII VIII IX X 50 100 500 1000 L C D M
Exercise 1-03
Example 3
Show how a Babylonian would have written each of these numbers.
a 15 b 252
Solution a
b For numbers greater than 60, we need to find how many 60s divide into them. 252 ÷ 60 = 4 and remainder 12 because 4 × 60 = 240
So 252 = (4 × 60) + 10 + 2. In Babylonian numerals, 252 is:
a b c d Ex 3 Worksheet 1-03 Roman numerals Skillsheet 1-02 Roman numerals
The Romans had an unusual method of writing certain numbers:
• Instead of writing 4 asIIII, they wrote IVmeaning V − I (that is 5 − 1 = 4). • Instead of writing 9 asVIIIIthey wroteIX meaning X − I(that is 10 − 1 = 9). • For 90, they wrote XC (that is 100 − 10 = 90).
1 Titus, a student in ancient Rome, wrote these numerals. Change them into our numbers.
a XXVI b XL c CCLXIV d LIV
e MMCLIX f MCMXC g XCVIII h MDVII
2 What would Titus have written for these numbers?
a 365 b 36 c 79 d 97
e 2600 f 344 g 999 h 3473
3 Why do you think Roman numerals are no longer widely used?
4 The Roman word for hundred was ‘centum’ which is why C stands for 100. List some words beginning with ‘cent’ that mean one hundred of something.
1-05 The modern Chinese number system
Chinese people today use the numerals below.• The Chinese write from top to bottom. • The symbols in a number are grouped in
pairs and the numbers in each pair are multiplied together.
• The products are added to give the number.
Exercise 1-04
Example 4
Write each of the following in Roman numerals.
a 23 b 46 c 101 d 249
Solution
a 23 is XXIII b 46 is XLVI c 101 is CI d 249 is CCXLIX
Ex 4
Ancient Chinese rod numerals Worksheet 1-04 1 2 3 4 5 6 7 8 9 10 100 1000 10000 Worksheet 1-05 Mayan numerals
1 Use our numerals to rewrite these Chinese numerals Zhang Li wrote.
2 If you were writing to Zhang Li, how would you write each of these numbers using Chinese numerals?
a 13 b 46 c 175 d 999
3 What are the difficulties in working with modern Chinese numerals?
Exercise 1-05
Example 5
Write each of these Chinese numbers using our number system.
Solution a b 3 × 100 = 300 7 × 10 = 70 + 5 = 5 375 6 × 10 = 60 + 4 = 4 64 a b Ex 5 a b c d e
Working mathematically
Calendar month
Make a calendar for the month of your birthday using a different type of number system. Are some number systems easier to use than others? Why?
1-06 The Hindu–Arabic number system
Our number system goes back to the Hindus (who lived in India) and came to Europe through the Middle East/Arabia. Our system needs only ten symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is easier to use because it has a zero and the position of each numeral determines its value. This is called place value. The numerals first appeared in Europe in the 10th century, but were different to the ten numerals we use today.
The following table shows how our numerals have changed over time.
The Hindus called the zero ‘sunya’ meaning a void. Other names used were ‘cipher’, ‘nought’ and the Arabic ‘sifr’.
Even today, different cultures use different symbols:
Place value
We can write any number using only ten symbols or digits. When we write numbers, each column has a special value called the place value.
Ancient number systems Worksheet 1-06 Hindu Hindu Hindu Arabic Spanish Italian Caxton (Printer) 200 BC AD 2 AD 800 AD 900 AD 976 AD 1400 AD 1480 1 2 3 4 5 6 7 8 9 0 10 Origin Date Numerals or or or Skillsheet 1-03 Place value Worksheet 1-07 Big numbers
Example 6
Write the value of each of the digits in 4625. Solution
In 4625: 5 has a value of 5 or 5 × 1 2 has a value of 20 or 2 × 10 6 has a value of 600 or 6 × 100 4 has a value of 4000 or 4 × 1000
1 Write the value of each digit in the following numbers, then write each number in words.
a 609 b 1039 c 70 104
d 504 860 e 9 134 671 f 5 837 000
g 4001 h 205 689 i 34 000 036
2 Write each of the following using numerals. a eight thousand, seven hundred and ninety-six b three million and eighty-eight
c two thousand, three hundred and eighty-five d six thousand, nine hundred and seven
e four hundred and twenty thousand, eight hundred and thirty f three hundred and nine thousand, two hundred and eleven
g one million, two hundred and eighty thousand, four hundred and sixty h twelve million, nine hundred and one
Exercise 1-06
Example 7
What is the value of each of the digits in 501? Solution
In 501: 1 has a value of 1
0 means there are no tens (zero used to mark a place)
5 has a value of 500
Another way to show the meaning of each digit in a number is with a place-value table.
Ten thousands Thousands Hundreds Tens Ones
1 3 8 138
4 6 2 5 4625
5 0 1 501
8 2 3 5 0 82 350
Example 8
What value does the digit 5 have in:
a 57? b 235?
Solution
a In 57, the 5 has a value of 50 (or 5 tens). b In 235, the 5 has a value of 5 (or 5 units).
3 What are the advantages of using a Hindu–Arabic number system? 4 What is the value of 7 in 237 601? Select A, B, C or D.
A 7 hundred B 7 thousand
C70 thousand D7 hundred thousand
5 In 2 982 645, which digit is in the ten thousands place? Select A, B, C or D.
A 2 B 9 C8 D6
6 Place these numbers in a place-value table, as shown on the previous page.
a 48 b 382 c 2751
d 3020 e 15 364 f 44 040
7 What is the value of the digit 5 in each of these numbers?
a 45 b 1057 c 1526
d 12 345 e 65 013 f 51 480 260
8 What is the value of the digit 3 in each of these numbers?
a 123 b 2356 c 32 185
d 85 532 e 1 385 264 f 3 485 260
9 What is the value of the digit 4 in each of these numbers?
a 4281 b 124 386 c 6004
d 4 316 725 e 362 154 f 1 426 813
10 Arrange the numbers in each of these sets in order, from smallest to largest.
a 321, 17, 8000 b 17, 707, 27, 63
c 246, 3596, 5369, 432, 16, 6125 d 123, 321, 132, 231, 213
e 1045, 450, 145, 82 f 721, 243, 43, 4372, 722
g 380 211, 308 022, 300 806, 392 084 h 4 856 231, 4 766 372, 1 429 950, 3 006 853 11 How many times is the first 3 bigger than the second 3 in each of these numbers?
a 1433 b 1343 c 3143 d 2 352 312
1-07 Expanded notation
One way to show the place value of each digit in a number is to use expanded notation.
Ex 7
Ex 8
Base 8 number system
Worksheet
1-08
Example 9
Write each of these numbers using expanded notation.
a 345 b 3287 Solution a 345= (3 × 100) + (4 × 10) + (5 × 1) = 3 × 102 + 4 × 10 + 5 × 1 b 3287= (3 × 1000) + (2 × 100) + (8 × 10) + (7 × 1) = 3 × 103 + 2 × 102 + 8 × 10 + 7 × 1
1 Write each of these numbers using expanded notation.
a 56 b 3562 c 416 d 502 e 1001
f 10 253 g 38 002 h 59 644 i 3809 j 120 435
2 Write each of these as a single number. a (5 × 100) + (2 × 10) + (4 × 1) b (6 × 1000) + (5 × 100) + (3 × 10) + (7 × 1) c (4 × 102) + (2 × 10) + (9 × 1) d (6 × 103) + (4 × 102) + (7 × 10) + (3 × 1) e 8 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 3 × 1 f 3 × 103 + 0 × 102 + 5 × 10 + 7 × 1 g 7 × 104 + 6 × 103 + 0 × 102 + 0 × 10 + 1 × 1 h 1 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1 i 3 × 105 + 4 × 104 + 4 × 103 + 2 × 102 + 2 × 10 + 0 × 1 j 9 × 105 + 0 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1 3 What is 9047 in expanded notation? Select A, B, C or D.
A 9 × 1000 + 4 × 100 + 7 × 10 B 9 × 1000 + 4 × 10 + 7 × 1 C9 × 1000 + 4 × 100 + 7 × 1 D9 × 100 + 4 × 10 + 7 × 1
4 Find out what ‘to expand’ means. Is the dictionary meaning the same as the one in mathematics?
Exercise 1-07
102 10 squared means 10 × 10 = 100 103 10 cubed means 10 × 10 × 10 = 1000
104 10 to the power of 4 means 10 × 10 × 10 × 10 = 10 000 The power of 10 shows how many zeros follow the 1 in the number.
!
Ex 9
Just for the record
Googol-plexing
The number 10100, the googol, is 1 followed by one hundred zeros. The name ‘googol’ was created by the 9-year-old nephew of American mathematician Dr Edward Kasner. The number 10googol, that is 1 followed by a googol zeros, is called the googolplex. The googol is a very big number but it is rarely used for practical purposes. Even the number of particles in the observable universe, estimated at being between 1072 and 1087, is less than a googol!
The Internet search engine Google was named after the googol, to reflect the huge size of the world wide web. It was invented in 1996 by two Stanford University students, Larry Page and Sergey Brin. Google is a powerful search engine because it can find information from at least 25 billion web pages in less than 1 second. How many googols are there in a googolplex?
1-08 The four operations
There are four basic operations in our number system:
+ addition × multiplication − subtraction ÷ division
The old symbols for writing these operations are:
We will now review these operations.
Mental skills 1A
Multiplying by a multiple of 10
Place value allows us to simply add zeros to the end of a number whenever we multiply by a power of 10. The zeros at the end shift all the other digits one or more places to the left which results in them having higher place values.
1 Examine these examples.
a 37 × 10 = 370 b 45 × 100 = 4500 c 16 × 1000 = 16 000 d 100 × 1000 = 100 000 e 7 × 90 = 7 × 9 × 10 = 63 × 10 = 630 f 5 × 400 = 5 × 4 × 100 = 20 × 100 = 2000 g 12 × 300 = 12 × 3 × 100 = 36 × 100 = 3600 h 40 × 800 = 4 × 10 × 8 × 100 = 4 × 8 × 10 × 100 = 32 × 100 = 32 000 2 Now simplify these.
a 18 × 100 b 26 × 1000 c 77 × 10 000 d 10 × 100
e 315 × 1000 f 1000 × 1000 g 294 × 10 h 475 × 100
i 3 × 80 j 8 × 200 k 6 × 50 l 7 × 30
m2 × 6000 n 11 × 900 o 4 × 400 p 5 × 700
q 5 × 80 r 25 × 20 s 300 × 60 t 900 × 4000
Maths without calculators
Worksheet
1-09
Four operations
Example 10
Copy and complete this number grid.
Solution + 5 14 8 12 + 5 14 8 12 + 5 14 8 13 22 12 17 26 + 5 14 8 13 22 12 17 26 5 + 8 14 + 8 14 + 12 5 + 12
Use the link to Worksheet 1–10 to print the number grids in this exercise. 1 Copy and complete these number grids.
2 Copy and complete these number grids.
a top row − side column b top row − side column c top row − side column
3 Copy and complete these number grids.
Exercise 1-08
Example 11
Copy and complete this number grid.
Solution + 12 30 20 65 + 12 30 20 65 + 12 45 18 30 63 20 32 65 + 12 45 18 30 63 20 32 65 30 − 12 18 + 45 65 − 20 20 + 12 Worksheet 1-10 Number grids Ex 10 Worksheet 1-11 Arithmagons a + 3 4 b c 7 2 + 15 41 28 19 + 11 9 8 5
The take-away bar: go figure
L 102
TLF
The multiplier: go figure L 90
TLF
− 19 25 7 12 − 54 78 37 26 − 243 412 128 239The divider: with or without remainders
L 2006
TLF
The multiplier: make your own hard multiplications
L 82
TLF
a × 11 9 b c 8 5 × 2 5 15 23 × 12 20 10 174 Copy and complete these number grids.
a top row ÷ side column b top row ÷ side column c top row ÷ side column
5 Copy and complete these number grids.
6 Find the missing numbers (top row − side column).
7 Find the missing numbers.
8 Find the missing numbers (top row ÷ side column).
÷ 36 48 4 3 ÷ 32 64 8 4 ÷ 60 100 4 5 Ex 11 a + 10 b c 50 80 100 + 16 26 28 13 + 22 33 6 14 a − 20 15 b c 8 9 − 17 9 15 11 − 7 9 12 11 a × 5 b c 3 12 28 × 5 56 40 7 × 10 6 90 4 a ÷ 24 b c 6 3 4 ÷ 8 4 2 24 ÷ 72 24 10 5
Using technology
What is a spreadsheet?
A spreadsheet is like a calculator. We can enter data and solve many problems more easily, using an Excel spreadsheet.
Spreadsheets are made up of many cells. As we go across the page, we change the column (A, B, C, D, etc.). As we go down the page, the row changes (1, 2, 3, 4, etc.). Using formulas
To write a formula in a cell, always start with an equal sign ‘=’. A spreadsheet uses special symbols to do calculations. Consider these basic operations:
a =A1+A2+A3 or =sum(A1:A3) means add the values in cells A1, A2 and A3 b =A5-A4 means subtract the value in cell A4 from the value in cell A5 c =A1*A3 means multiply the value in cell A1 by the value in cell A3 d =A5/4 means divide the value in cell A5 by 4 (/ is used instead of ÷) e =A2^2 means square the value in cell A2 (instead of (A2)2)
f =average(A1:A5) means find the average of all values in cells A1 to A5
1 Enter the following numbers into cells as shown below, where m represents the value in cell B1, n is the value in cell B2, p is the value in cell B3, and so on.
2 Enter the following formulas into the given cells.
a C1, q − 7 (means enter the formula into cell C1 as shown above)
b C2, n − m c C3, 2 × r − 7
d C4, 3 × ( p + q) e C5, p × q × r
f C6, p2 g C7,
h C8, i C9, m + n + p + q + r
j C10, average of m, n, p, q and r k C11, −
3 Choose different values and enter them into cells B1 to B5. Consider the new answers obtained in column C, for the formulas entered from question 2.
3×m 2 ---r–p 3 ---q m ---- r p ----Skillsheet 1-04 Spreadsheets
Working mathematically
Double-digit dice game
This is a game for two or more players using one die. Instructions
Step 1: Copy the scoresheet shown on the right. Step 2: Each player rolls the die seven times and,
for each roll, can choose to write the number in either the tens column or the units column of his or her scoresheet.
Step 3: Each player finds the total of his or her
seven numbers. The winner is the person with a total closest to 99.
Step 4: Play the game again and work out a
strategy to improve your score.
Scoresheet
Roll Tens Units
1st 2nd 3rd 4th 5th 6th 7th Total
Applying strategies and reasoning
Mental skills 1B
Dividing by a multiple of 10
Place value allows us to remove zeros from the end of a number when we divide by a power of 10. The deleted zeros shift all the other digits one or more places to the right which results in them having lower place values.
1 Examine these examples.
a 2000 ÷ 10 = 2000 ÷ 10 = 200 b 1800 ÷ 100 = 1800 ÷ 100 = 18 c 37 000 ÷ 100 = 37 000 ÷ 100 = 370 d 6 000 000 ÷ 1000 = 6 000 000 ÷ 1000 = 6000 e 6000 ÷ 200 = 6000 ÷ 100 ÷ 2 = 60 ÷ 2 = 30 f 350 ÷ 70 = 350 ÷ 10 ÷ 7 = 35 ÷ 7 = 5 g 2800 ÷ 40 = 2800 ÷ 10 ÷ 4 = 280 ÷ 4 = 70 h 40 000 ÷ 5000 = 40 000 ÷ 1000 ÷ 5 = 40 ÷ 5 = 8 2 Now simplify these.
a 200 ÷ 10 b 6000 ÷ 100 c 45 000 ÷ 100 d 30 000 ÷ 1000
e 1900 ÷ 10 f 2600 ÷ 100 g 530 ÷ 10 h 720 000 ÷ 1000
i 180 ÷ 30 j 300 ÷ 50 k 1600 ÷ 400 l 45 000 ÷ 5000
m4200 ÷ 60 n 21 000 ÷ 700 o 44 000 ÷ 2000 p 1600 ÷ 200
q 24 000 ÷ 600 r 24 000 ÷ 3000 s 64 000 ÷ 80 t 5400 ÷ 900
1-09 Dividing by a two-digit number
In primary school, you studied division by a single-digit number. We will now divide numbers by a two-digit number using two different methods.
1 Find the answers for the following.
a 180 ÷ 15 b 462 ÷ 22 c 731 ÷ 17 d 666 ÷ 18 e 992 ÷ 31 f 78 ÷ 13 g 900 ÷ 25 h 667 ÷ 23 i 85 ÷ 17
Exercise 1-09
Worksheet 1-09 Four operationsExample 12
Divide $312 among 12 people. Solution
Method 1: Long division Method 2: Preferred multiples
Each person receives $26. 26 12 312 12 into 31 is 2 −24↓ 72 12 into 72 is 6 −72 0 26 12 312 −120 10 times 192 −120 10 times 72 −72 6 times 0 26 times
Example 13
Simplify 296 ÷ 21. Then complete: 296 = × + . Solution
Method 1: Long division Method 2: Preferred multiples
Answer = 14 , so 296 = 14 × 21 + 2 14 remainder 2 21 296 21 into 29 is 1 −21↓ 86 21 into 86 is 4 −84 2 14 remainder 2 21 296 −210 10 times 86 −84 4 times 2 14 times 2 21 ---Ex 12
2 Carry out these divisions and write your answers in the form: = × + .
a 304 ÷ 12 b 505 ÷ 14 c 99 ÷ 26
d 917 ÷ 19 e 958 ÷ 34 f 869 ÷ 28
g 594 ÷ 27 h 79 ÷ 13 i 815 ÷ 40
3 At a party 275 lollies are shared equally among 25 children. How many lollies does each child get?
4 A piece of wood 390 cm in length is to be cut into 15 equal pieces. How long is each piece?
Ex 13
Working mathematically
Magic squares
Magic squares have every row, column and diagonal adding to the same magic sum. The Lo-Shu magic square dates back to about 2200 BC. It appeared on an ancient Chinese tablet and was first drawn on a tortoise shell given to the Emperor Yu.
Reasoning
1 a Draw a 3 × 3 magic square frame. Write the Lo-Shu magic square into your frame
using the numbers 1 to 9. (Hint: Count the dots. Top left-hand corner is a 4.)
b What is the magic sum for the Lo-Shu square?
2 Which of these squares are not magic?
3 Make these squares magic by finding the missing numbers.
42 14 34 22 30 38 26 46 18 21 0 15 12 6 18 3 30 5 a b c 38 8 28 16 24 32 20 30 12 29 19 33 21 35 44 39 49 34 a b c 21 6 12 45 48 27 33 3 42 Worksheet 1-12 Magic squares
4 Another famous magic square appears in a woodcut by the German artist Albrecht Dürer, who lived from 1471 to 1528. It is called the magic square of Jupiter.
a Find the 4-digit numeral contained within the square that identifies a year that occurred during Dürer’s lifetime.
b What is the magic sum for this 4 × 4
square?
c Find five 2 × 2 squares within the
magic square for which the numbers have the same total as the magic sum.
d Apart from the two diagonals, find four numbers each from a different row and column that add to the magic sum. There are more than two solutions.
Using technology
Sorting data
Sort the set of numbers {60, 107, 85, 6, 28, 45, 265} using a spreadsheet, by following the instructions shown below.
1 a Enter the numbers, in the given order, into column A.
1-10 Order of operations
c Choose Sort by Column A and Ascending as shown below.
d The data should now be sorted from smallest number (cell A1) to the largest number (cell A7).
e A set of numbers can also be sorted in descending order. Highlight the cells and choose Sort by Column A and Descending.
2 Now sort these sets of numbers in the columns given, by repeating this method. a Enter {55, 89, 36, 21, 19, 4, 95} in column B
b Enter {263, 141, 940, 508, 836, 392, 1063} in column C c Enter {4987, 4200, 8740, 9005, 2601, 2514, 4810} in column D
d Enter {16 101, 12 167, 10 010, 11 412, 10 107, 10 761, 11 214} in column E 3 Sort the data from question 2 in descending order, for each of columns B to E.
The order of operations rules
First: Work out the value within any grouping symbols, starting with the innermost grouping symbols:
parentheses or round brackets ( ) square brackets [ ] braces { }.
Second: Work out multiplication or division as you come to it, going from left to right. Third: Work out addition or subtraction as you come to it, going from left to right.
!
Example 14
Find the value of (5 + 13) ÷ 2. Solution
(5 + 13) ÷ 2 work out grouping symbols
= 18 ÷ 2 division
= 9 answer
{
1 Evaluate (find the value of) each of the following.
a 12 × (3 + 5) b (16 − 3) × 2 c (60 + 12) ÷ 6
d (3 − 2) × 5 e (2 + 5) × 6 f (12 − 4) ÷ 4
g 7 × (25 − 12) h 36 ÷ (14 − 10) i (5 × 7) − 16
j 120 ÷ (34 − 24) k 5 + 6 × (50 − 10) l (77 ÷ 11) − 7
2 Evaluate the following.
a 3 + 5 × 2 b 20 − 2 × 5 c 5 + 3 × 2 − 7 d 19 − 4 × 4 − 1 e 24 − 5 ÷ 5 + 7 f 17 + 8 − 3 × 2 g 2 × 10 − 9 + 28 h 42 ÷ 7 − 5 i 9 + 28 − 12 j 4 × 8 − 3 × 3 k 109 + 36 ÷ 4 l 60 − 8 × 4 + 20 3 12 ÷ 4 + 8 × 5 = ? Select A, B, C or D. A 5 B 16 C43 D55
4 Find the answer to each of the following.
a (24 − 4) ÷ 5 + 7 b 2 × (10 − 9) + 28 c (8 + 2) × (17 − 7) d 7 + 7 + (11 − 8) e (16 − 5 + 8) × 9 f (8 + 8 − 5) × (7 + 4) g 9 + 3 × (15 − 4) − 5 × 6 h 16 × 3 − 4 × (15 − 6 × 2) + 7 i (5 + 8) × 2 − (25 ÷ 5) j 4 × [(5 + 11) ÷ 2] − (15 × 2)
Exercise 1-10
Example 15
1 Find the value of 15 ÷ 5 × 8.
Solution 15 ÷ 5 × 8 division = 3 × 8 multiplication = 24 answer
{
{
2 Find the value of 5 + 6 × 2 − 7.
Solution 5 + 6 × 2 − 7 multiplication = 5 + 12 − 7 addition = 17 − 7 subtraction = 10 answer
{
{
{
Example 16
Find the value of 25 − [7 × (5 − 3) + 4]. Solution
25 − [7 × (5 − 3) + 4] innermost grouping symbols
= 25 − [7 × 2 + 4] grouping symbols: inside multiplication first
= 25 − [14 + 4] grouping symbols = 25 − 18 subtraction = 7 answer
{
{
{
{
Ex 14 Ex 15 Ex 16k 100 − [12 + (3 × 5) ÷ 3] l 120 ÷ {16 + [(2 × 5) + 4]} m{15 − [3 × (12 − 9) + 1]} − [(44 × 2) + 12] ÷ 50 n [(16 − 4) × 10] ÷ [(45 ÷ 3) + 25] o 86 + [(15 ÷ 3) + (65 ÷ 5)] × 2 p [20 ÷ (5 − 4) × 2] − {[(4 + 5) × 3] ÷ [15 − (30 ÷ 5)]}
5 Put grouping symbols where necessary to make each of the following statements true. The first one has been done for you.
a 5 − 2 × 4 = 12 becomes (5 − 2) × 4 = 12 b 3 + 8 − 7 = 4
c 15 − 3 × 5 = 60 d 15 − 3 × 5 = 0 e 8 + 4 − 3 × 2 = 10
f 8 + 4 − 3 × 2 = 6 g 8 + 4 − 3 × 2 = 18 h 6 + 4 × 0 = 6
i 6 + 4 × 0 = 0 j 100 ÷ 10 + 10 = 5 k 100 ÷ 10 + 10 = 20
6 Put grouping symbols where necessary to make each of the answers correct. a 84 ÷ 3 + 9 × 15 − 11 = 152 b 84 ÷ 3 + 9 × 15 − 11 = 64 c 84 ÷ 3 + 9 × 15 − 11 = 94
7 Use the four numbers in each set only once (in any order), with the operations +, −, ×, ÷ or grouping symbols, to make an equation that equals the number in the red box.
a 2, 7, 8, 9 b 1, 2, 3, 5 c 3, 4, 6, 8
d 2, 6, 8, 1 e 2, 4, 6, 8 f 2, 5, 8, 10
g 3, 5, 7, 9 h 4, 5, 7, 9 i 2, 5, 7, 10
1-11 The symbols of mathematics
Mathematics does not only involve numbers. It has a language of its own and uses symbols recognisable throughout the world. This table shows some of the most common symbols.
The square root of a given number is the positive value which if squared will give that number.
The cube root of a number is the value which if cubed will give the number.
Symbol Meaning Symbol Meaning
+ plus, add, sum square root ( = 5)
− minus, subtract, difference cube root ( = 2)
× multiply, times, product ∴ therefore
÷ divided by, quotient or approximately equal to
= equal to 32 squared (3 × 3)
≠ not equal to 53 cubed (5 × 5 × 5)
less than ( ) parentheses or brackets
less than or equal to [ ] square brackets
greater than { } braces
greater than or equal to
12 18 41
21 10 44
2 8 60
Cross number puzzle
Worksheet
1-13
25
1 Here is a list of words that relates to the four basic operations +, −, × and ÷.
plus minus times multiply and divide subtract share decrease product difference less increase total lots of quotient take away more than
Draw a table with column headings as shown below in your notebook, and write each of the given words in the appropriate column.
2 Rewrite these questions using mathematical symbols.
a 15 minus 6 b 48 plus 12 c 12 is greater than 5
d 5 is not equal to 3 plus 6 e the product of 7 and 8 f the square root of 16
g 36 divided by 4 h 5 squared i 8 more than 12
j 6 less than 13 k increase 3 by 13 l the quotient of 39 and 3
mthe difference between 25 and 8 n the cube root of 125
o 13 is not equal to 3 p 999 is approximately equal to 1000
3 Write the answer to each of the following.
a the number 6 less than 18 b the sum of 26 and 14 c the total of 6, 8 and 22
d 9 times 8 e 7 squared f the quotient of 36 and 4
g the number 14 more than 8 h decrease 33 by 11 i increase 83 by 27
j 7 lots of 13 k the cube root of 64
l the difference between 135 and 29
Exercise 1-11
+ − × ÷
Example 17
Find the answer for each of the following.
a 62 b c
Solution
a 62 = 6 squared = 6 × 6 b = the square root of 9
= 36 = 3 since 32 = 3 × 3 = 9
c = the cube root of 125
= 5 since 53 = 5 × 5 × 5 = 125 9 3 125 9 125 3
Example 18
Write the meaning of each of the following.
a 3 7 b 5 5
Solution
4 Which of these statements is true? Select from A, B, C or D.
A = 18 B 18 ÷ 2 ≠ 9 C 6 × 4 15 D 72 12
5 Write whether each of the following is true (T) or false (F).
a 16 2 b 42 = 8 c 300 5 × 100 d 3602 = 3600 e = 5 f 8 × 201 8 × 200 g 2 h product of 2 and 15 = 17 i 63 ÷ 3 60 ÷ 5 j 33 = 27 k 52 − 3 = 7 l 72 73 m16 × 0 7 × 0 n (30 − 6) × 5 12 × 10 o = 6 p = 1 q 53 = 15 r 4
6 Complete the blank with or to make each statement true.
a 7130 860 b 2001 2010
c 352 140 4 082 716 d 2651 2561
e 3602 3206 f 13 253 1353
g 8079 8097 h 1432 1483
7 For each of the following statements, select all the numbers from this list of seven numbers that make the statement true: 2, 3, 7, 8, 11, 36, 41.
a 13 b 5 c 8 d 42 e 3= 8 f 11 g = 2 h 5 + 8 36 Ex 17 25 27 3 36 1 3 24 Ex 18 3
Working mathematically
The four 4s puzzle
Form 10 groups (Group A, Group B, Group C, etc.). Use only four 4s and any of the mathematical symbols =, −, ×, ÷, brackets, a decimal point (.), factorial (!) or square root ( ) to make expressions for all the numbers from 1 to 100. Group A does the numbers 1 to 10, Group B does 11 to 20, … Group J does 91 to 100.
Here are some suggestions:
• 4 + 4 × 4 + 4 = 4 + 16 + 4 = 24 • 4 × 4 − 4 ÷ 4 = 16 − 1 = 15 • 4! + 4 × 4 ÷ 4 = 24 + 4 = 28 • 4 × 4 + 4 × 4 = 16 + 16 = 32
(Hint: 4! = 4 × 3 × 2 × 1)
Brain bender
Various forms of ‘brain benders’ are common in daily newspapers and magazines. Here is one for you. Copy the grids and fill in the six gaps to complete each of the lines, using the remaining digits from 1 to 9 only once. Be sure to use the ‘order of operations’ rules. The aim is to make the sum of the answers for the three lines total 45.
5 + × =
× 3 − =
− + 4 =
45 Applying strategies and reflecting
Using technology
Fruit picking
An orchardist employed people to pick fruit in his orchard over the summer. The table below shows the types of fruit grown and numbers of bins of fruit picked each day in a particular week.
1 Copy the table, as shown, into a spreadsheet.
2 To find the total number of bins of fruit picked on Monday, type the formula =sum(B2:B5) in cell B6.
3 To copy this formula into cells C6 to F6, click on cell B6 and Fill Right by grabbing the bottom right-hand corner of the cell and dragging across to cell F6. Let go of the mouse and you will see the totals for each day.
4 Use the sum formula in cell G2 to find the number of bins of apples picked in this particular week. Use Fill Down to copy the formula into cells G3 to G6. Centre the totals calculated in the ‘G’ column.
5 Answer the following questions in the given cell. In cell:
a A8, type the number of bins of fruit pieces picked on Wednesday
b A9, write a formula to find how many more bins of oranges than apples were picked in this week.
c A10, write a formula to find how much more fruit was picked on Wednesday compared to Monday in this week.
d A11, write a formula to find how many bins of lemons and mandarins in total were picked in this week.
e A12, write the day of the week on which the most fruit was picked. f A13, write the day of the week on which the least fruit was picked. g A14, type the total number of bins of fruit picked in this particular week.
Skillsheet
1-04
Power plus
Cryptic arithmetic
Simple codes can be made by replacing letters with other letters, symbols or numbers. Number codes are studied in a branch of mathematics called cryptic arithmetic. Your challenge is to figure out which letter replaces which number.
The addition: 99 could become: KK
+ 22 + DD
121 RDR
where K = 9, D = 2 and R = 1.
Note that K + D gives an answer bigger than 10 so carrying will be involved. To solve cryptic arithmetic problems, you need to know about carrying digits when adding. Choose any of the following problems from 1 to 7.
1 ON + ON + ON + ON = GO Hint: Set it out as a column sum.
2 N I NE Hint: Try R = 0 and N = 5
−F OUR F I V E
There are 71 other possible solutions. In many of these (but not all) R = 0 and N = 5. Can you find two other solutions? How many different solutions can the class find?
3 FORT Y Hint: T = 8 and Y = 6
T E N + T EN S I X T Y
The key to this problem is to decide what value is N + N and what value is E + E.
4 THRE E
+ FOUR S E V E N
For this puzzle there are 38 possible solutions.
Hint: Try E = 6 and V = 0 for one solution. Try E = 5 and V = 1 for another solution. Try H = 9 and R = 4 for another.
How many different solutions can the class find?
5 On a holiday, Carlos ran short of money. He sent an email to his parents: S E ND
+MOR E MON E Y
The value of ‘MONEY’ is the amount Carlos asked for. If Carlos asked for more than $10 000 and less than $20 000, find out how much money he asked for.
6 a RE AD b READ
+ TH I S − TH I S
P AG E P AG E
These are two different problems, so R and the other letters have a different value in each problem.
Chapter 1 review
Language of maths
braces cube root difference digit
evaluate expanded notation grouping symbols Hindu–Arabic long division million number system numeral
order of operations parentheses place value preferred multiples product quotient square brackets square root sum
1 What is ‘expanded notation’? Explain in your own words. 2 What is a thousand thousands?
3 What is the Roman numeral for 500?
4 Write and name the three types of grouping symbols.
5 With which arithmetic operation would you associate the word:
a quotient? b difference?
6 What is the meaning of each of these symbols?
a b
Topic overview
• In your own words, write what you have learnt about the history of numbers. • Is there anything you did not understand? Ask a friend or your teacher for help. • Copy this overview into your workbook and complete it using what you have learnt in
this chapter. Ask your teacher to check your overview.
Number find-a-word Worksheet 1-14 3 Order of operations • • • Four operations • • • • Hindu–Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Place value • • • Symbols •+, −, ×, ÷ • , • • 3
Early number systems • Egyptian • Aboriginal • • •
H
I
S
T
O
R
Y
OF
NUMBERS
Chapter revision
1 Write these using Egyptian numerals.
a 13 b 2402
2 Write these using the words of the Kamilaroi Aboriginal people.
a 3 b 5
3 Write these using Babylonian numerals.
a 32 b 110
4 Write each of the following in Roman numerals.
a 12 b 40
c 179 d 2004
5 Write these using modern Chinese numerals.
a 17 b 82
6 Write each of the following using numerals. a six hundred and twelve
b nine hundred and forty-three
c five thousand, four hundred and ninety-nine d six thousand and two
e nine million, seven hundred and fifty thousand and seventy-six
7 Arrange the numbers in each of these sets in order, from largest to smallest. a 16, 21, 38, 19, 14
b 89, 36, 101, 98, 88
c 2356, 2534, 2635, 2300, 2533
d 12 391, 12 913, 11 990, 11 391, 12 300 8 What is the place value of the digit 4 in:
a 47? b 3024?
c 8412? d 146 235?
9 Write each of these using expanded notation.
a 19 b 283
c 665 d 42 891
10 Find the answers to these.
a 36 + 58 b 127 + 81 c 39 − 17 d 78 − 39 e 2501 + 58 f 26 × 9 g 123 × 5 h 36 × 11 i 36 ÷ 4 j 252 ÷ 7 k 750 ÷ 6 l 3500 ÷ 10 Exercise 1-01 Exercise 1-02 Exercise 1-03 Exercise 1-04 Exercise 1-05 Exercise 1-06 Exercise 1-06 Exercise 1-06 Exercise 1-07 Exercise 1-08 Topic test 1
11 Find the answers to these. Write your answer in the form:
= × + .
a 384 ÷ 16 b 912 ÷ 19
c 784 ÷ 17 d 877 ÷ 23
12 Find the value of each of these.
a 16 − (5 × 3) b 6 + 5 × 3 c 30 − 10 ÷ 2 d (16 ÷ 2) + (18 − 11) e (320 − 120) × 12 f 35 × (19 − 17) × 20 g (36 − 14) × 2 ÷ 4 h 36 − (28 − 13) + (20 − 3 × 5) i (256 − 120) ÷ 17 j [394 + (30 ÷ 5)] ÷ (440 ÷ 11) k 36 − (4 × 3) ÷ (35 − 23) l 2 000 000 − [(300 × 100) + 1] 13 Use ‘order of operations’ to calculate:
a 12 + 7 − 2 × 3 b 15 − 2 × 4 + 6 ÷ (8 − 5)
c 24 + 16 ÷ 4 × 16 − 4 + 9 d 15 + (64 + 2) ÷ 3 − 16
e 18 + 6 ÷ 3 − 3 + 2 × 5 f 166 + 12 × 3 − 48 ÷ 4
14 Use grouping symbols and operations signs (+, −, ×, ÷) to make each of these true.
a 7 ? 3 ? 1 = 9 b 10 ? 5 ? 5 = 10
c 8 ? 3 ? 6 ? 2 = 8 d 28 ? 4 ? 7 = 49
e 6 ? 4 ? 3 ? 5 = 40 f 19 ? 1 ? 5 ? 3 ? 1 = 0
15 Write whether each of these is true (T) or false (F).
a 5 8 b 7 2 + 4 c 5210 d 6 × 7 43 e 23 5 + 1 f = 6 Exercise 1-09 Exercise 1-10 Exercise 1-10 Exercise 1-10 Exercise 1-11 36
