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Rational Numbers
When you think of your favourite game, what comes to mind? It may be a computer game or video game. You may also enjoy playing games that have been around a lot longer. These may include the use of a game board and may involve cards, dice, or specially designed playing pieces. Examples of these games include chess, checkers, dominoes, euchre, bridge, Monopoly™, and Scrabble®.
In this chapter, you will learn more about games and about how you can use rational numbers to describe or play them. You will also design your own game.
What You Will Learn
- to compare and order rational numbers
- to solve problems involving operations on rational numbers
- to determine the square root of a perfect square rational number
- to determine the approximate square root of a non-perfect square rational number
C H A P T E R
Web Link For more information about board games invented by Canadians, go to www.mathlinks9.ca and follow the links.
Did You Know? Canadians have invented many popular board games, such as crokinole, Yahtzee®, Trivial Pursuit®, Balderdash™, and Scruples™.
42 Chapter 2 NEL
NEL
Literacy Link
A Frayer model is a tool that can help you understand new terms. Create a Frayer model into your math journal or notebook.
1. Write a term in the middle. 2. Define the term in the first box. The glossary on pages 494 to 500 may help you. 3. Write some facts you know about the term in the second box. 4. Give some examples in the third box. 5. Give some non-examples in the fourth box.
Definition Facts
Examples Non-examples
Rational Number
Key Words
rational number perfect square
non-perfect square
NEL Chapter 2^^43
Math Link
Problem Solving With Games
Millions of Canadians enjoy the challenge and fun of playing chess. Early versions of this game existed in India over 1400 years ago. The modern version of chess emerged from southern Europe over 500 years ago.
- If each of the small squares on a chessboard has a side length of 3 cm, what is the total area of the dark squares? Solve this problem in two ways.
- If the total area of a chessboard is 1024 cm^2 , what is the side length of each of the smallest squares?
- For the chessboard in #2, what is the length of a diagonal of the board? Express your answer to the nearest tenth of a centimetre.
- Compare your solutions with your classmates’ solutions.
In this chapter, you will describe or play other games by solving problems that involve decimals, fractions, squares, and square roots. You will then use your skills to design a game of your own.
NEL Math Link^^45
2.
Focus on…
After this lesson, you will be able to…
- compare and order rational numbers
- identify a rational number between two given rational numbers
Comparing and Ordering
Rational Numbers
The percent of Canadians who live in rural areas has been decreasing since 1867. At that time, about 80% of Canadians lived in rural areas. Today, about 80% of Canadians live in urban areas, mostly in cities. The table shows changes in the percent of Canadians living in urban and rural areas over four decades.
Decade
Change in the Percent of Canadians in Urban Areas (%)
Change in the Percent of Canadians in Rural Areas (%) 1966 - 1976 +1.9 - 1. 1976 - 1986 +1.0 - 1. 1986 - 1996 +1.4 - 1. 1996 - 2006 +2.3 - 2.
How can you tell that some changes in the table are increases and others are decreases?
Did You Know? An urban area has a population of 1000 or more. In urban areas, 400 or more people live in each square kilometre. Areas that are not urban are called rural. What type of area do you live in?
Cape Dorset, Nunavut
46 Chapter 2
Link the Ideas
Example 1: Compare and Order Rational Numbers
Compare and order the following rational numbers.
__^7
__ 5 - __^7 8
Solution
You can estimate the order.
- 1.2 is a little less than - 1. (^4) __ 5 is a little less than 1. (^7) __ 8 is a little less than 1.
__ 5 is a little less than - 0.5.
- __^78 is a little more than - 1. An estimate of the order from least to greatest is - 1.2, - __^78 , - 0.
__ 5 , __^45 ,^7 __ 8.
Express all the numbers in the same form. You can write the numbers in decimal form.
= 0.8 __^7
__ 5 = - 0.555… -^7 __ 8
Place the numbers on a number line.
7 8
_ 4 5
_ 7 8
The numbers in ascending order are - 1.2, -^7 __ 8
__ 5 ,^4 __ 5
, and^7 __ 8
The numbers in descending order are __^7 8
, __^4
__ 5, - __^7 8
, and - 1.2.
Compare the following rational numbers. Write them in ascending order and descending order.
__ 3 - 0.6 - __^3 4
1 __^1
Show You Know
Draw a Diagram
Strategies
What number is the opposite of - __^78? How does the position of that number on the number line compare with the position of -^7 __ 8?
48 Chapter 2
Example 2: Compare Rational Numbers
Which fraction is greater, -^3 __ 4
or - __^2 3
Solution
Method 1: Use Equivalent Fractions You can express the fractions as equivalent fractions with a common denominator.
A common denominator of the two fractions is 12.
× 3 × 4
- __^3
= - ___^9
-^2 __
= - ___^8
× 3 × 4
When the denominators are the same, compare the numerators.
= - ___^9
- ___^8
= - ___^8
- ___ 8
> - ___^9
, because - 8 > - 9.
-^2 __ 3
is the greater fraction.
Method 2: Use Decimals You can also compare by writing the fractions as decimal numbers.
-^3 __ 4
-^2 __
__ 6
__ 6 > - 0.
-^2 __ 3
is the greater fraction.
Literacy Link The quotient of two integers with unlike signs is negative. This means that
- ___ 129 = ___- 129 = ____ -^912
and
- ___ 128 = ___- 128 = ____ -^812.
Web Link For practice comparing and ordering rational numbers, go to www. mathlinks9.ca and follow the links.
Which fraction is smaller, - ___^7 10
or - __^3 5
Show You Know
How do you know 12 is a common denominator?
How does the number line show the comparison? − 12 __^9 = − 34 _ − 12 __^8 = − _^23
− 1 0
2.1 Comparing and Ordering Rational Numbers 49
Check Your Understanding
Communicate the Ideas
- Laura placed - 21 __ 2
incorrectly on a number line, as shown.
− 3 − 2 − 1 0 How could you use the idea of opposites to show Laura how to plot - 21 __ 2
correctly?
- Is Dominic correct? Show how you know.
- Tomas and Roxanne were comparing - 0.9 and - __^7 8 . Tomas wrote
- 0.9 as a fraction, and then he compared the two fractions. Roxanne wrote - __^7 8
as a decimal, and then she compared the two decimals.
a) Which method do you prefer? Explain. b) Which is greater, - 0.9 or -^7 __ 8
? Explain how you know.
Practise
For help with #4 to #9, refer to Example 1 on page 48.
- Match each rational number to a point on the number line.
− 3 − 2 − 1 0 + 1 + 2 + 3
A B C D E
a) 3 __ 2
b) - 0.7 c) - 21 __ 5 d) 14 ___ 5
e) - 11 __ 3
- Which point on the number line matches each rational number?
− 2 − 1 0 + 1 + 2
V W X Y Z
a) - 12 __ 5
b) 3 __ 4
c) 1 ___^1 20 d) - 13 __ 5
e) - 0.
__ 4
- Place each number and its opposite on a number line.
a) 8 __ 9
b) - 1.2 c) 2 ___^1 10
d) -^11 ___ 3
- What is the opposite of each rational number?
a) - 4.
__ 1 b) 4 __ 5
c) - 53 __ 4
d) 9 __ 8
- Compare 1__^5 6
, - 1 __^2
, - 0.1, 1.9, and -^1 __ 5
Write the numbers in ascending order.
- Compare -^3 __ 8
__ 8 , __^9 5
, - __^1
, and - 1. Write the numbers in descending order.
2.1 Comparing and Ordering Rational Numbers 51
For help with #10 to #13, refer to Example 2 on page 49.
- Express each fraction as an equivalent fraction. a) -^2 __ 5
b) 10 ___ 6 c) - ___^9 12
d) - ___^4 3
- Write each rational number as an equivalent fraction. a) - ___^1 3
b) - ___^4
d) ___^7
- Which value in each pair is greater?
a) 1 __ 3
, -^2 __
b) - ___^9 10
, ___^7
c) -^1 __ 2
, -^3 __
d) - 21 __ 8
, - 2 __^1
- Which value in each pair is smaller?
a) 4 __ 7
, __^2
b) -^4 __ 3
, -^5 __
c) - ___^7 10
, -^3 __
d) - 13 __ 4
, - 1 __^4
For help with #14 to #17, refer to Example 3 on page 50.
- Identify a decimal number between each of the following pairs of rational numbers. a) 3 __ 5
, __^4
b) -^1 __ 2
, -^5 __
c) -^5 __ 6
, 1 d) -^17 ___ 20
, -^4 __
- What is a decimal number between each of the following pairs of rational numbers? a) 1 __^1 2
, 1 ___^7
b) - 22 __ 3
, - 2 __^1
c) 1 __^3 5
, - 1 ___^7
d) - 3 ____^1 100
, - 3 ___^1
- Identify a fraction between each of the following pairs of rational numbers. a) 0.2, 0.3 b) 0, - 0. c) - 0.74, - 0.76 d) - 0.52, - 0.
- Identify a mixed number between each of the following pairs of rational numbers. a) 1.7, 1.9 b) - 0.5, 1. c) - 3.3, - 3.4 d) - 2.01, - 2.
Apply
- Use a rational number to represent each quantity. Explain your reasoning. a) a temperature increase of 8.2 °C
b) growth of 2.9 cm
c) 3.5 m below sea level
d) earnings of $32.
e) 14.2 °C below freezing
52 Chapter 2
Play the following game with a partner or in a small group. You will need one deck of playing cards.
- Remove the jokers, aces, and face cards from the deck.
- Red cards represent positive integers. Black cards represent negative integers.
- In each round, the dealer shuffles the cards and deals two cards to each player.
- Use your two cards to make a fraction that is as close as possible to zero.
- In each round, the player with the fraction closest to zero wins two points. If there is a tie, each tied player wins a point.
- The winner is the first player with ten points. If two or more players reach ten points in the same round, keep playing until one player is in the lead by at least two points.
5
5 represents - 5
4
4
represents 4
With a five of clubs and a four of hearts, you can make ___^4
___^4
- 5 because it is closer to zero.
- Which integers are between^11 ___ 5
and ___^15
- Which number in each pair is greater? Explain each answer. a) 0.4 and 0. b) 0.
__ 3 and 0. c) - 0.7 and - 0. d) - 0.66 and - 0.
__ 6
- Identify the fractions that are between 0 and
- 2 and that have 3 as the denominator.
Extend
- How many rational numbers are between (^2) __ 3
and 0.
__ 6? Explain.
- Replace each ■ with an integer to make each statement true. In each case, is more than one answer possible? Explain. a) ■.5 < - 1.9 b) ___■
= - 21 __
c) - ___^3 ■
= - -____^15
d) - 1.5■ 2 > - 1.
e) -^3 __ 4
< - 0.7■ f) - 51 __ 2
> 11 ___
g) - 23 __ 5
= ___■
h) __^8 ■
< - __^2
- Determine the value of x. a) ___^4
= ____ x
b) x __ 3
= ___^6
c) 5 __ x = -^20 ___ 12
d) - ___^6
= 30 ___ x
54 Chapter 2
2.
Focus on…
After this lesson, you will be able to…
- perform operations on rational numbers in decimal form
- solve problems involving rational numbers in decimal form
Problem Solving With Rational
Numbers in Decimal Form
In Regina, Saskatchewan, the average mid-afternoon temperature in January is - 12.6 °C. The average mid-afternoon temperature in July is 26.1 °C. Estimate how much colder Regina is in January than in July.
Explore Multiplying and Dividing Rational Numbers in Decimal Form
- a) Estimate the products and quotients. Explain your method. 3.2 × 4.5 3.2 × (-4.5) - 20.9 ÷ 9.5 - 20.9 ÷ (-9.5) b) Calculate the products and quotients in part a). Explain your method.
- a) Suppose the temperature one January afternoon in Regina decreased by 2.6 °C every hour for 3.5 h. What was the overall temperature change during that time? b) Suppose the temperature in Regina one July afternoon increased by 9.9 °C in 5.5 h. What was the average temperature change per hour?
Reflect and Check
- How can you multiply and divide rational numbers in decimal form?
- Create a problem that can be solved using the multiplication or division of rational numbers. Challenge a classmate to solve it.
Did You Know? As Canada’s sunniest provincial capital, Regina averages almost 6.5 h of sunshine per day. That is over 2 h per day more sunshine than St. John’s, Newfoundland and Labrador. St. John’s is the least sunny provincial capital.
2.2 Problem Solving With Rational Numbers in Decimal Form 55
Estimate and calculate. a) - 4.38 + 1. b) - 1.25 - 3.
Show You Know
Example 2: Multiply and Divide Rational Numbers in Decimal Form
Estimate and calculate.
a) 0.45 × (-1.2) b) - 2.3 ÷ (-0.25)
Solution
a) Estimate. 0.5 × (-1) = - 0.
Calculate. Method 1: Use Paper and Pencil You can calculate by multiplying the decimal numbers. 0.45 × 1.2 = 0. Determine the sign of the product. 0.45 × (-1.2) = - 0.
Method 2: Use a Calculator C 0.45 × 1.2 +^ -^ =^ –0.
b) Estimate.
- 2.3 ÷ (-0.25) ≈ - 2 ÷ (-0.2) ≈ 10
Calculate. C 2.3 +^ -^ ÷^ 0.25 +^ -^ =^ 9.
Estimate and calculate. a) - 1.4(-2.6) b) - 2.76 ÷ 4.
Show You Know
Literacy Link Parentheses is another name for brackets. They can be used in place of a multiplication sign. For example,
How do you know what the sign of the product is?
2.2 Problem Solving With Rational Numbers in Decimal Form 57
Example 3: Apply Operations With Rational Numbers in Decimal Form On Saturday, the temperature at the Blood Reserve near Stand Off, Alberta decreased by 1.2 °C/h for 3.5 h. It then decreased by 0.9 °C/h for 1.5 h. a) What was the total decrease in temperature? b) What was the average rate of decrease in temperature?
Solution
a) The time periods can be represented by 3.5 and 1.5. The rates of temperature decrease can be represented by - 1.2 and - 0.9.
Method 1: Calculate in Stages You can represent the temperature decrease in the first 3.5 h by 3.5 × (-1.2) = - 4.2.
You can represent the temperature decrease in the last 1.5 h by 1.5 × (-0.9) = - 1.35.
Add to determine the total temperature decrease.
The total decrease in temperature was 5.55 °C.
Method 2: Evaluate One Expression The total temperature decrease can be represented by 3.5 × (-1.2) + 1.5 × (-0.9).
Evaluate this expression, using the order of operations. 3.5 × (-1.2) + 1.5 × (-0.9) = - 4.2 + (-1.35) = - 5.
You can also use a calculator. C (^) 3.5 × (^) 1.2 + - + (^) 1.5 × (^) 0.9 + - = (^) -5. The total decrease in temperature was 5.55 °C.
b) The average rate of decrease in temperature is the total decrease divided by the total number of hours. The total number of hours is 3.5 + 1.5 = 5.
The average rate of decrease in temperature was 1.11 °C/h.
4 × (-1) = - 4 1.5 × (-1) = - 1.
Literacy Link Order of Operations
- Perform operations inside parentheses first.
- Multiply and divide in order from left to right.
- Add and subtract in order from left to right.
4 × (-1) + 1.5 × (-1) = - 5.
Why are the time periods represented by positive rational numbers? Why are the rates of temperature decrease represented by negative rational numbers?
58 Chapter 2
For help with #4 and #5, refer to Example 1 on page 56.
- Estimate and calculate. a) 0.98 + (-2.91) b) 5.46 - 3. c) - 4.23 + (-5.75) d) - 1.49 - (-6.83)
- Calculate. a) 9.37 - 11.62 b) - 0.512 + 2. c) 0.675 - (-0.061) d) - 10.95 + (-1.99)
For help with #6 and #7, refer to Example 2 on page 57.
- Estimate and calculate. a) 2.7 × (-3.2) b) - 3.25 ÷ 2. c) - 5.5 × (-5.5) d) - 4.37 ÷ (-0.95)
- Calculate. Express your answer to the nearest thousandth, if necessary. a) - 2.4(-1.5) b) 8.6 ÷ 0. c) - 5.3(4.2) d) 19.5 ÷ (-16.2) e) 1.12(0.68) f) - 0.55 ÷ 0.
For help with #8 to #11, refer to Example 3 on page 58.
- Evaluate. a) - 2.1 × 3.2 + 4.3 × (-1.5) b) - 3.5(4.8 - 5.6) c) - 1.1[2.3 - (-0.5)]
Literacy Link In - 1.1[2.3 - (-0.5)], square brackets are used for grouping because - 0.5 is already in parentheses.
- Determine each value. a) (4.51 - 5.32)(5.17 - 6.57) b) 2.4 + 1.8 × 5.7 ÷ (-2.7) c) - 4.36 + 1.2[2.8 + (-3.5)] 10. In Regina, Saskatchewan, the average mid-afternoon temperature in January is - 12.6 °C. The average mid-afternoon temperature in July is 26.1 °C. How many degrees colder is Regina in January than in July?
- One January day in Penticton, British Columbia, the temperature read - 6.3 °C at 10:00 a.m. and 1.4 °C at 3:00 p.m. a) What was the change in temperature? b) What was the average rate of change in temperature?
Apply
- A pelican dives vertically from a height of 3.8 m above the water. It then catches a fish 2.3 m underwater. a) Write an expression using rational numbers to represent the length of the pelican’s dive. b) How long is the pelican’s dive?
- A submarine was cruising at a depth of 153 m. It then rose at 4.5 m/min for 15 min. a) What was the submarine’s depth at the end of this rise? b) If the submarine continues to rise at the same rate, how much longer will it take to reach the surface?
- Saida owned 125 shares of an oil company. One day, the value of each share dropped by $0.31. The next day, the value of each share rose by $0.18. What was the total change in the value of Saida’s shares?
Literacy Link A share is one unit of ownership in a corporation.
Practise
60 Chapter 2
- In dry air, the temperature decreases by about 0.65 °C for each 100-m increase in altitude. a) The temperature in Red Deer, Alberta, is 10 °C on a dry day. What is the temperature outside an aircraft 2.8 km above the city? b) The temperature outside an aircraft 1600 m above Red Deer is - 8.5 °C. What is the temperature in the city?
- Bella is more comfortable working with integers than with positive and negative decimal numbers. This is her way of understanding - 4.3 + 2.5. -4.3 is ____-43 10 or -43 tenths.
2.5 is 25 ___ 10 or 25 tenths. -43 tenths + 25 tenths is -18 tenths. -18 tenths is -18___ 10 or -1.8. So, -4.3 + 2.5 = -1.8. a) Use Bella’s method to determine 6.1 + (-3.9). b) How could you modify Bella’s method to determine 1.25 - 3.46?
- Two wooden poles measured 1.35 m and 0.83 m in length. To make a new pole, they were attached by overlapping the ends and tying them together. The length of the overlap was 12 cm. What was the total length of the new pole in metres? 1.35 m
0.83 m
12 cm
- Determine the mean of each set of numbers. Express your answer to the nearest hundredth, if necessary. a) 0, - 4.5, - 8.2, 0.4, - 7.6, 3.5, - 0. b) 6.3, - 2.2, 14.9, - 4.8, - 5.3, 1.
- A company made a profit of $8.6 million in its first year. It lost $5.9 million in its second year and lost another $6.3 million in its third year. a) What was the average profit or loss per year over the first three years? b) The company broke even over the first four years. What was its profit or loss in the fourth year?
- Research to find out the current price of gasoline in Calgary. It is 300 km from Calgary to Edmonton. How much less would it cost to drive this distance in a car with a fuel consumption of 5.9 L/100 km than in a car with a fuel consumption of 9.4 L/100 km? Give your answer in dollars, expressed to the nearest cent.
Web Link To find out prices of gas in Calgary, go to www.mathlinks9.ca and follow the links.
- Andrew drove his car 234 km from Dawson to Mayo in Yukon Territory in 3 h. Brian drove his truck along the same route at an average speed of 5 km/h greater than Andrew’s average speed. How much less time did Brian take, to the nearest minute?
- An aircraft was flying at an altitude of 2950 m. It descended for 3 min at 2.5 m/s and then descended for 2.5 min at 2.8 m/s. What was the plane’s altitude after the descent?
2.2 Problem Solving With Rational Numbers in Decimal Form 61
