CHAPTER
2
Rational Numbers
GET READY
52
Math Link
54
2.1 Warm Up
55
2.1 Comparing and Ordering Rational Numbers
56
2.2 Warm Up
64
2.2 Problem Solving With Rational Numbers in
Decimal Form
65
2.3 Warm Up
74
2.3 Problem Solving With Rational Numbers in
Fraction Form
75
2.4 Warm Up
88
2.4 Determining Square Roots of Rational
Numbers
89
Graphic Organizer
100
Chapter 2 Review
101
Key Word Builder
105
Chapter 2 Practice Test
106
Math Link: Wrap It Up!
109
Challenge
110
Answers
112
Name: _____________________________________________________
Date: ______________
Get Ready
Working With Decimal Numbers
Use estimation to place the decimal point in the answer.
50.1 × 2.1 = 1 0 5 2 1
Estimate: 50 × 2 = 100
Answer: 105.21
1.
Place the decimal so the
answer is close to 100.
Use estimation to place the decimal point in the answer.
a) 49.8 ÷ 0.98 = 5 0 8 1 6
b) 2.7 × 100.9 = 2 7 2 4 3
=
50 ÷
Understanding Fractions
The shaded part of the diagram shows
Which is larger,
×4
1 4
=
2 8
4
1
or
or 0.5.
8
2
1
3
or ? Make the denominators the same to compare fractions.
2
8
4>3
4
3
1
3
is greater than , so
> .
8
8
2 8
×4
2.
a) Write the fraction for the shaded
parts of each diagram.
b) Compare the fractions from part a).
Use > or <.
Adding or Subtracting Fractions
To add or subtract fractions, make the denominators the same.
●
52
use diagrams
●
1
1
+
2
4
+
=
2 1
+
4 4
+
=
3
4
MHR ● Chapter 2: Rational Numbers
use a common denominator
3 1
−
4 2
=
3 2
−
4 4
=
1
4
1 2
=
2 4
Name: _____________________________________________________
3.
Date: ______________
Solve.
a)
1
+
5
1 3
+
2 8
b)
+
3
Multiplying and Dividing Fractions
To multiply fractions, multiply the numerators and the denominators.
1 3
×
2 4
1× 3
=
2× 4
3
=
8
numerator × numerator
denominator × denominator
To divide fractions,
● find a common denominator and divide
the numerators
●
7 2
÷
10 5
7 2
÷
10 5
=
7
4
÷
10 10
7
=
4
4.
multiply by the reciprocal
=
7 5
×
10 2
Multiply by the reciprocal.
=
35
20
Multiply the numerators
Multiply the denominators
=
7
4
Write in lowest terms.
Find a common denominator.
Divide the numerators.
The reciprocal of
1 2
is .
2
1
Solve. Write your answer in lowest terms.
a)
2
1
×
3
3
b)
2
1
÷
3
3
Get Ready ● MHR 53
Name: _____________________________________________________
Date: ______________
Math Link
back
new
Problem Solving With Games
Use the gameboard to answer the questions.
1.
How many small squares are there in total on the gameboard?
What fraction of the board is 1 small square?
2.
What fraction of the board squares have Xs?
3.
What fraction of the board squares have Xs and Os?
4.
Play the game with a partner. What is the minimum number of squares you need to use to win?
Write your answer as a fraction of the total squares played.
5.
How could you play this game with 3 players? Describe how to play your new game.
Does your new game change the fraction of the total squares needed to win?
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
54
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
2.1 Warm Up
1.
Circle the larger number.
a) 1.1
2.
1.2
b) –10
–1
Change the fractions to decimal numbers.
1
2
a)
b)
=
3
5
÷
=
3.
0.3
a)
=
4.
Tens Ones . Tenths Hundredths
Change the decimal numbers to fractions.
The 3 is in the tenths place
so the denominator is 10.
b) 0.85
10
a) Plot – 0.5, –1, 0.3, and 1.2 on the number line.
-1
0
+1
b) Write the numbers in ascending order (smallest to largest):
5.
Make equivalent fractions.
÷
×
a)
2
=
3
b)
8
=
12
×
6.
÷
Write the opposite of each number.
Opposite numbers have the same
numeral but different signs.
a) −5 →
b) 3.4 →
c)
3
→
4
d) −
2
→
5
2.1 Warm Up ● MHR 55
Name: _____________________________________________________
Date: ______________
2.1 Comparing and Ordering Rational Numbers
Link the Ideas
Working Example 1: Compare and Order Rational Numbers
rational number
●
●
●
a
, where a and b are integers and b ≠ 0
b
any number that can be written as a fraction
−4
7
1 3
0
examples: – 4 or
, 3.5 or , − , 1 , 0 or
1
2
2 4
7
a number written as
Compare and order the rational numbers. Show your work.
–1.2,
4 7
7
, , 0.5 , −
5 8
8
Solution
Method 1: Use Estimation
What numbers are smaller than 0?
and −
Is –1.2 or −
7
are smaller than 0.
8
7
closer to 0?
8
Look at the number line.
-1.2
-1
0
7
-_
8
is closer to 0.
_4
5
So, 0.5,
4
7
, and will be to the right of 0.
5
8
0
+1
0.5
_7
8
0.5 is about
8
2
.
is closest to 1, so it is the greatest number.
7
4
7
Using estimation, the order from least to greatest is –1.2, − , 0.5, , and .
8
5
8
56
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Method 2: Use Decimals
Write all the numbers in the same form. Change the fractions to decimal numbers.
−1.2
4
=
5
4÷5
7
=
8
-1
,
, and
7
= − 0.
8
+1
0
The numbers in ascending order are
7
−1.2, − ,
8
−
Label the number line using
the original form of the numbers.
Plot each number on the number line.
-2
0.5 = 0.555…
Ascending means
from least to greatest.
.
The numbers in descending order (largest to smallest) are
.
Compare and order the rational numbers.
Write them in ascending and descending order.
Show your work.
– 0.6,
3 3
3
, , −
4 8
2
-2
-1
0
1
Ascending order:
Descending order:
2.1 Comparing and Ordering Rational Numbers ● MHR 57
Name: _____________________________________________________
Date: ______________
Working Example 2: Compare Rational Numbers
Which fraction is greater, −
3
2
or − ?
4
3
Solution
Method 1: Use Equivalent Fractions
Write the fractions as equivalent fractions with a common denominator.
A common denominator for 4 and 3 is
×3
−
3
=−
4
Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
.
×
−
12
×3
2
=−
3
12
×
The denominators are the same, so you can compare the numerators.
−
9
−9
=
12
12
−
8
−8
=
12
12
9 = _3
− __
−
12
4
> means greater than.
−1
8 = _2
− __
−
12
3
0
–8 > −9, so − 2 is the greater fraction.
3
Method 2: Use Decimals
Change both fractions to decimals.
3
2
−
−
4
3
= –3 ÷ 4
= –2 ÷ 3
=–
= − 0.6
Compare the place values.
– 0.6 > – 0.75 because – 0.6 is closer to 0
So, −
58
is the greater fraction.
MHR ● Chapter 2: Rational Numbers
-0.75
-1
-0.6
-0.8 -0.7 -0.6 -0.5
0
Name: _____________________________________________________
a) Which is smaller, −
7
3
or − ?
10
5
Date: ______________
b) Which is greater, – 0.25 or 0.0001?
0
0
Working Example 3: Identify a Rational Number Between Two Given Rational Numbers
Find a fraction between – 0.6 and – 0.7.
Solution
Use the number line to find a decimal number between – 0.6 and – 0.7.
−0.7
−0.6
−1
0
A decimal number between – 0.60 and – 0.70 is – 0.65.
Change the decimal to a fraction.
The 5 is in the hundredths place,
so the denominator is 100.
– 0.65 as fraction is –
100
.
A fraction between – 0.6 and – 0.7 is –
. Another fraction between − 0.6 and − 0.7 is −
100
a) Find a fraction between 0.56 and 0.58.
0
.
b) Find a fraction between –2.4 and –2.5.
-2
2.1 Comparing and Ordering Rational Numbers ● MHR 59
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Use a number line to show that −
-5
2.
-4
-3
-2
-1
0
1
1
is greater than –3.
2
2
3
4
5
Is Dominic correct?
Circle YES or NO. Give 1 reason for your answer.
________________________________________
________________________________________
________________________________________
Practise
3.
Match each rational number to a point on the number line.
A
−3
−2
B C
−1
D
0
+1
+2
+3
3
2
a)
b) – 0.7
c) –2 1
d) –1
5
4.
Plot
-2
60
8
1
5
, – 0.4, 2 , – on the number line.
9
10 3
-1
MHR ● Chapter 2: Rational Numbers
1
3
Change the fractions
to decimal numbers.
0
1
2
Name: _____________________________________________________
5.
3
9 1
Write – , 1. 8 , , – , and –1 in descending order.
8
5 2
6.
Write each fraction as an equivalent fraction.
a) −
7.
b)
Write each number
in decimal form.
10
=
6
Which rational number in each pair is greater? Show your thinking.
a)
8.
2
=
5
Date: ______________
1
, –1
5
1
3
b) − , −
2
5
Write a decimal number between each pair of rational numbers.
a)
3 4
,
5 5
1
5
b) − , −
2
8
Change each fraction
to a decimal number.
2.1 Comparing and Ordering Rational Numbers ● MHR 61
Name: _____________________________________________________
9.
Change each
decimal to a fraction.
Write a fraction between each pair of rational numbers.
a) 0.2, 0.3
Date: ______________
b) – 0.52, – 0.53
Apply
10. Rewrite each amount as a positive or negative number. Example: “losing 2 dollars” = –2
a) a temperature increase of 8.2 °C =
b) growth of 2.9 cm =
c) 3.5 m below sea level =
d) earning $32.50 =
11. The table shows the average early-morning temperature for 7 communities in May.
Average Early-Morning
Temperature (°C)
Community
Churchill, Manitoba
–5.1
Regina, Saskatchewan
3.9
Edmonton, Alberta
5.4
Penticton, British Columbia
6.1
Yellowknife, Northwest Territories
– 0.1
Whitehorse, Yukon Territory
0.6
Resolute, Nunavut
–14.1
a) Write the temperatures in descending order.
,
,
,
,
highest to lowest
,
,
.
b) Which community has an average temperature between the values for Whitehorse and
Churchill?
62
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
12. Write >, <, or = to make each statement true.
a)
−9
6
c) –3.25
3
−2
–3
b) –
1
5
d)
−
3
5
4
7
Date: ______________
> means greater than.
< means less than.
– 0. 6
To compare fractions, change them
to decimals or equivalent fractions
with a common denominator.
−
2
3
Math Link
Play this card game with a partner.
●
Remove the jokers, kings, queens, jacks, and 10s from the deck.
●
Divide the cards between you and your partner.
●
The numbered cards are decimals.
Red is positive and black is negative.
Example: A black 5 is – 0.5. A red 4 is 0.4.
●
The red aces are +1. The black aces are –1.
●
Both players lay a card face up at the same time.
The greatest value wins and the winner keeps both cards.
●
If there is a tie, both players lay 2 more cards face down and then a card face up.
Whichever card is greater wins all cards from that turn.
●
The player who ends up with all the cards is the winner.
4
5
5
4
represents − 0.5
represents 0.4
2.1 Math Link ● MHR 63
Name: _____________________________________________________
Date: ______________
2.2 Warm Up
1.
Solve.
a) 5 + (–3) =
b) (–10) + (–2) =
(– 4) – 2
c)
=(
d) 7 – (–5)
)+(
=
e) 3 × (–2) =
g) (–8) ÷ (4) =
2.
)
Add the opposite.
+×+=+
-×-=+
+×-=-
-×+=-
+÷+=+
-÷- =+
+÷-=-
-÷+ =-
f) (–5) × (–1) =
h) (–15) ÷ (–3) =
Estimate and calculate.
a) 1.99 + 3.25
b) 0.57 – 0.14
Estimate:
Calculate:
+
Estimate:
Calculate:
1.99
+3.25
=
c) 3.1 × 6.5
d) 9.6 ÷ 3.2
Estimate:
3.
Calculate:
Use the order of operations to solve.
b) (11 + 3) – 10 ÷ 2
8 – 4 ÷ (–2)
a)
=8−(
=8+
)
Add the opposite.
=
64
Estimate:
MHR ● Chapter 2: Rational Numbers
Calculate:
Name: _____________________________________________________
Date: ______________
2.2 Problem Solving With Rational Numbers in
Decimal Form
Link the Ideas
Working Example 1: Add and Subtract Rational Numbers in Decimal Form
Estimate and calculate.
a) 2.65 + (–3.81)
Solution
Estimate.
2.65 + (–3.81)
≈ 3 + (– 4)
≈
Calculate.
Method 1: Use Paper and Pencil
Adding the opposite is the same as subtracting.
2.65 + (–3.81) = 2.65 – 3.81
3.81
−2.65
When the signs are opposite, subtract the
smaller number from the larger number and
take the sign of the larger number.
The answer 1.16 must be negative since 3.81 is larger than 2.65.
Is the answer close to the estimate?
So, 2.65 + (–3.81) = −
Method 2: Use a Calculator
C 2.65 + 3.81 +
−
= −1.16
2.2 Problem Solving With Rational Numbers in Decimal Form ● MHR 65
Name: _____________________________________________________
b) –5.96 – (– 6.83)
Solution
Estimate.
–5.96 – (– 6.83)
≈ – 6 – (–7)
≈ –6 +
≈
Calculate.
Method 1: Use Paper and Pencil
Subtracting a negative is the same as adding the opposite.
–5.96 – (– 6.83) = –5.96 + 6.83
Find 6.83 + (–5.96).
6.83
−5.96
6.83 + (–5.96)
= 6.83 – 5.96
=
Method 2: Use a Calculator
C 5.96 +
-
- 6.83 +
-
= 0.87
Estimate and calculate.
a) 1.52 + (– 4.38)
b) –1.25 – 3.55
Estimate:
Estimate:
+
=
Calculate:
66
MHR ● Chapter 2: Rational Numbers
Calculate:
Date: ______________
Name: _____________________________________________________
Date: ______________
Working Example 2: Multiply and Divide Rational Numbers in Decimal Form
Estimate and calculate.
a) 0.45 × (–1.2)
b) –2.3 ÷ (– 0.25)
Solution
Solution
Estimate.
Estimate.
0.45 × (–1.2)
–2.3 ÷ (– 0.25)
≈ 0.5 ×
≈
Ask yourself, what is half of –1?
≈
Calculate.
Method 1: Use Paper and Pencil
÷ (− 0.2)
≈
+÷+=+
-÷- =+
Calculate.
+÷-=-
-÷+ =-
C 2.3 +
- ÷ 0.25 +
- = 9.2
Multiply the decimal numbers.
0.45
× 1.2
90
450
+×+=+
-×-=+
+×-=-
-×+=-
Use the sign rules to find the sign of the answer.
0.45 × (–1.2) =
Method 2: Use a Calculator
C 0.45 × 1.2 +
-
= − 0.54
Estimate and calculate.
a) − 0.6 × (– 6.1)
Estimate:
≈
b) (–2.4) ÷ (2.0)
Calculate:
Estimate:
Calculate:
×
≈
2.2 Problem Solving With Rational Numbers in Decimal Form ● MHR 67
Name: _____________________________________________________
Date: ______________
Working Example 3: Apply Operations With Rational Numbers in Decimal Form
The temperature at Blood Reserve in 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?
Solution
The temperature changes are
negative because they decreased.
The time periods are 3.5 and 1.5 hours.
The temperature decreases are –1.2 and –
.
Method 1: Calculate in Steps
Step 1: Temperature decrease in first 3.5 h = 3.5 × (–1.2)
= – 4.2
Step 2: Temperature decrease in last 1.5 h = 1.5 × (– 0.9)
=
Step 3: Add to find the total temperature decrease: – 4.2 + (
The total decrease in temperature was
)=–
°C.
Method 2: Evaluate One Expression
The expression shows the total temperature decrease: 3.5 × (–1.2) + 1.5 × (
Evaluate using the order of operations.
3.5 × (–1.2) + 1.5 × (
= (–
) + (–
Literacy Link
Order of Operations
• Perform operations inside
brackets first
• Multiply and divide in
order from left to right
• Add and subtract in order
from left to right
)
)
=
You can also use a calculator.
C 3.5 × 1.2 +
-
+ 1.5 × 0.9 +
-
The total decrease in temperature was
68
).
MHR ● Chapter 2: Rational Numbers
= −5.55
°C.
Name: _____________________________________________________
Date: ______________
b) What was the average rate of decrease in temperature?
Solution
Average rate of decrease in temperature =
=
total decrease in temperature
total number of hours
− 5.55
5
total number of hours = 3.5 + 1.5 = 5
=
The average rate of decrease in temperature was
°C/h.
A student did an experiment where the temperature went from 20.8 °C to 50.4 °C in 10 minutes.
How much did the temperature change per minute?
Total temperature change =
−
=
Total number of minutes =
Average temperature change per minute =
temperature change
total number of minutes
=
=
The average temperature change was
°C/min.
2.2 Problem Solving With Rational Numbers in Decimal Form ● MHR 69
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
a) Do you think – 0.32 + 6.5 will give a positive or negative answer? Give 1 reason for your answer.
_____________________________________________________________________________
_____________________________________________________________________________
b) Evaluate – 0.32 + 6.5.
2.
a) The products of these 2 expressions are
.
(the same or different)
2.54 × (− 4.22)
−2.54 × 4.22
b) Give 1 reason for your answer.
_____________________________________________________________________________
Practise
3.
Estimate and calculate.
a) 0.9 + (− 0.2)
b) 0.34 + (−1.22)
Estimate:
Estimate:
≈
+
≈
Calculate:
70
MHR ● Chapter 2: Rational Numbers
Calculate:
Name: _____________________________________________________
4.
Date: ______________
Estimate and calculate.
a) 5.46 − 3.16
b) −1.49 − (− 6.83)
Estimate:
≈
Estimate:
−
≈
≈
−(
)
≈
+(
) Add the opposite.
≈
Calculate:
5.
Estimate and calculate.
a) 2.7 × (−3.2)
6.
Calculate:
Use the sign rules.
b) −5.5 × (−5.5)
Estimate:
Estimate:
Calculate:
Calculate:
Estimate and calculate.
a) (− 40.4) ÷ (– 4.04)
b) –3.25 ÷ 2.5
Estimate:
Estimate:
Calculate:
Calculate:
2.2 Problem Solving With Rational Numbers in Decimal Form ● MHR 71
Name: _____________________________________________________
7.
Date: ______________
Evaluate. Use the order of operations.
a) −2.1 × 3.2 + 4.3 × (−1.5)
=(
)+(
b) −1.1[2.3 − (− 0.5)]
= −1.1 × [2.3 +
)
When there is more
than 1 set of brackets,
use square brackets.
] Add the opposite.
= −1.1 ×
=
=
Apply
8.
The temperature in Kelowna went from −2.2 °C to −11.0 °C in 4 h.
How many degrees did the temperature drop per hour?
Temperature change = (
)−(
)
=
Total time =
Average temperature drop =
temperature change
total number of hours
Sentence: ________________________________________________________________________
9.
A pelican dives vertically from a height of 3.8 m above the water.
It then catches a fish 2.3 m underwater.
Sketch a diagram of the situation.
a) Write an expression using rational numbers to show the length of the pelican’s dive.
Distance down to the water =
Distance from the top of the water to the fish =
Expression:
b) How far did the pelican dive?
Solve the expression.
Sentence: _____________________________________________________________________
72
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
10. A submarine was cruising at a depth of 304.5 m. It then rose at 10.5 m per minute.
How many minutes did it take to reach the surface?
Sentence: ________________________________________________________________________
11. A company made a profit of $8.6 million in its first year.
It lost $5.9 million in its second year. It lost another $6.3 million in its third year.
a) After 3 years, did the company make or lose money? Show your calculations.
Sentence: _____________________________________________________________________
b) What was the average amount of money the company made or lost per year?
Average =
sum of numbers
number of years
Sentence: _____________________________________________________________________
Math Link
Play this game with a partner. You will need 2 dice and 1 coin.
• Roll 2 dice, one at a time. The numbers on the dice create a decimal number.
Example: rolling 6, then 5 means 6.5.
• Toss the coin. Tossing heads means the rational number is positive.
Tossing tails means the rational number is negative.
• Roll the dice and toss the coin again to get your second
number.
• Add the 2 numbers.
• The person with the sum closest to 0 wins 2 points.
(+1.2)
+
(−5.6)
=?
If there is a tie, each person wins 1 point.
• The first player to reach 10 points wins.
0
2.2 Math Link ● MHR 73
Name: _____________________________________________________
Date: ______________
2.3 Warm Up
1.
Write the fractions in lowest terms.
÷
a)
12
=
15
b)
10
45
÷
2.
3.
Change 4
2
to an improper fraction.
3
Example: 3
1 3× 2 +1
=
2
2
7
=
2
or
3
1 2 2 2 1
= + + +
2 2 2 2 2
7
=
2
Add or subtract. Write your answers in lowest terms.
2 1
+
5 2
a)
=
10
+
1
5
b) 2 − 1
3
6
10
+
=
=
Find a common denominator.
10
10
4.
Multiply or divide. Write your answers in lowest terms.
2 1
3
1
a) 3 × 1
b) 2 ÷ 3
Change to improper fractions.
5
2
4
2
74
MHR ● Chapter 2: Rational Numbers
Write as improper fractions.
Name: _____________________________________________________
Date: ______________
2.3 Problem Solving With Rational Numbers in
Fraction Form
Link the Ideas
Working Example 1: Add and Subtract Rational Numbers in Fraction Form
Estimate and calculate.
a)
2 ⎛ 1⎞
– −
5 ⎜⎝ 10 ⎟⎠
Solution
Estimate:
2
1
1
is close to ; ⎛⎜ − ⎞⎟ is close to 0
5
2 ⎝ 10 ⎠
1
–0=
2
Calculate.
Multiples of 5: 5, 10, 15.
Multiples of 10: 10, 20, 30.
2 ⎛ 1⎞
Find a common denominator.
− −
5 ⎜⎝ 10 ⎟⎠
Subtracting −
4
1
− ⎛⎜ − ⎞⎟
=
10 ⎝ 10 ⎠
=
4 − (−1)
10
=
4 +1
10
=
=
10
2
1 is the same as adding the opposite of − 1 .
10
10
4
––
10
Add the opposite.
0
1
––
10
1
÷5
Write in lowest terms.
5
=
10
2
÷5
Is the calculated answer close to the estimate? Circle YES or NO.
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 75
Name: _____________________________________________________
b) 3
Date: ______________
2 ⎛ 3⎞
+ −1
3 ⎜⎝ 4 ⎟⎠
Solution
Estimate.
3
2
≈4
3
–1
4+(
3
≈
4
)=
Calculate.
Method 1: Rewrite the Mixed Numbers as Improper Fractions
3
2
3
=
3 3 3 2
+ + +
3 3 3 3
=
11
3
=–
⎛
11 ⎜⎜
+ −
3 ⎝
=
3
4
4
3
= ⎛⎜ − ⎞⎟ + ⎛⎜ − ⎞⎟
⎝ 4⎠ ⎝ 4⎠
–1
⎞
⎟
⎟
⎠
4
( )
4
Multiples of 3: 3, 6, 9, 12.
Multiples of 4: 4, 8, 12, 16.
Find a common denominator.
44
21
+ −
12
12
⎛
44 + ⎜ −
⎝
=
=
12
Add the numerators.
Write as a mixed number.
12
= _________
76
⎞
⎟
⎠
12
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Method 2: Add the Integers and Add the Fractions
2 ⎛ 3⎞
+ −1
3 ⎜⎝ 4 ⎟⎠
2
3
=3+
+ (−1) + ⎛⎜ − ⎞⎟
3
⎝ 4⎠
3
= 3 + (–1) +
Separate the whole numbers and fractions.
2 ⎛ 3⎞
+ −
3 ⎜⎝ 4 ⎟⎠
Combine like terms.
=
+
2 ⎛ 3⎞
+ −
3 ⎜⎝ 4 ⎟⎠
Add the whole numbers.
=
+
8 ⎛ 9⎞
+ −
12 ⎜⎝ 12 ⎟⎠
Find a common denominator.
=
⎛
8 + ⎜−
⎝
+
=
12
⎛
⎜
+ ⎜−
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
⎛
12 ⎜⎜
=1
+ −
12 ⎜
⎜
⎝
⎛
12 + ⎜
⎝
=1
12
⎞
⎟
⎠
Add the numerators.
⎞
⎟
⎟
⎟
⎟
⎠
⎞
⎟
⎠
Add the numerators.
=1
Is the estimated answer close to the calculated answer? Circle YES or NO.
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 77
Name: _____________________________________________________
Date: ______________
Estimate and calculate. Write your answers in lowest terms.
a) −
3 2
−
4 5
Estimate:
−
Calculate:
3
≈
4
−
2
≈
5
= −
−
=
b) 2
3 ⎛ 2⎞
− +
4 ⎜⎝ 5 ⎟⎠
=
Find a common denominator.
=
Add the opposite.
=
Write as a mixed number.
3 ⎛ 1⎞
– −
4 ⎜⎝ 2 ⎟⎠
Estimate:
78
3 2
−
4 5
MHR ● Chapter 2: Rational Numbers
Calculate:
Name: _____________________________________________________
Date: ______________
Working Example 2: Multiply and Divide Rational Numbers in Fraction Form
Solve.
3 ⎛ 2⎞
× −
4 ⎜⎝ 3 ⎟⎠
a)
Solution
3 ⎛ 2⎞
× −
4 ⎜⎝ 3 ⎟⎠
3 × ( −2 )
=
4×3
=
=
3
≈1
4
2
1
− ≈−
3
2
Multiply the numerators
Multiply the denominators
( 21 ) = − 21
So, 1 × −
−
−
or −
b) –
You can remove the common factors
of 3 and 2 from the numerator and
denominator before multiplying.
3 1 ⎛ −2 −1 ⎞ −1
1
×⎜
⎟ = 2 or − 2
42 ⎝ 3 1 ⎠
Write in lowest terms.
5 1
÷
8 3
Solution
Method 1: Use a Common Denominator
–
5 1
÷
8 3
Method 2: Multiply by the Reciprocal
–
=–
=
−15
÷
24
=
−15
8
= −1
24
5
×
8
3
Find a common denominator.
Divide the numerators.
=–
=–
8
5 1
÷
8 3
Write as a mixed number.
8
7
8
Write as a mixed number.
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 79
Name: _____________________________________________________
–1
c)
Date: ______________
1 ⎛ 3⎞
÷ −2
2 ⎜⎝ 4 ⎟⎠
Solution
Method 1: Use a Common Denominator
–1
=–
=
Method 2: Multiply by the Reciprocal
1 ⎛ 3⎞
÷ −2
2 ⎜⎝ 4 ⎟⎠
3 ⎛ 11 ⎞
÷ −
2 ⎜⎝ 4 ⎟⎠
–1
Write as improper fractions.
− 6 ⎛ −11 ⎞
÷⎜
⎟
4
⎝ 4 ⎠
−6
=
−11
Divide the numerators.
Dividing 2 negatives = positive
=
=
1 ⎛ 3⎞
÷ −2
2 ⎜⎝ 4 ⎟⎠
−3 ⎛ −11 ⎞
÷⎜
⎟
2
⎝ 4 ⎠
⎛
⎜
−3 ⎜
=
×
2
⎝
=
Write in lowest terms.
=
−11
80
2 ⎛ 1⎞
× −
5 ⎜⎝ 6 ⎟⎠
MHR ● Chapter 2: Rational Numbers
⎞
⎟
⎟
⎠
Multiply by the reciprocal.
−22
11
Calculate.
a) –
Write as improper fractions.
b) –2
1
1
÷1
8
4
Write in lowest terms.
Name: _____________________________________________________
Date: ______________
Working Example 3: Apply Operations With Rational Numbers in Fraction Form
At the start of the week, Maka had $30.
1
1
1
She spent of the money on bus fares, another shopping, and on snacks.
5
2
4
How much does she have left?
Solution
Write 30 instead of $30.
Use negative fractions to show each of the amounts spent.
Bus fares: –
1
5
Shopping: −
2
Snacks: −
4
Calculate each dollar amount.
Bus fares:
1
– × 30
5
=–
1 30
×
5
1
= −
30
Shopping:
1
– × 30
2
Snacks:
1
− × 30
4
1
=– ×
2
1
= − ×
4
=–
=
1
2
=
=–
= −
1
4
2
or –7.5
Add the amounts.
(− 6) + (−15) + (−7.5) = $
Maka spent
.
Find how much Maka has left.
$30 + (−28.5) =
Maka has $
When the number is negative,
add to find the difference.
left.
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 81
Name: _____________________________________________________
Date: ______________
1
1
1
of it on a movie, on a round of golf, and
on a snack.
5
2
10
How much does he have left?
Stefano had $50. He spent
Movie:
–
Golf:
Snack:
1
×
5
)+(
Total amount spent: (
)+(
)=
Find how much he has left.
Stefano has
left.
Check Your Understanding
Communicate the Ideas
1.
−3 3
÷ using a common
4
8
denominator and dividing the numerators.
a) Calculate
b) Calculate
−3 3
÷ by multiplying by the
4
8
reciprocal.
c) Which method do you prefer? Give 1 reason for your answer.
_____________________________________________________________________________
82
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Practise
2.
Estimate and calculate.
3 ⎛ 1⎞
– −
a)
8 ⎜⎝ 4 ⎟⎠
Estimate:
Calculate:
3
≈
8
−
–
1
≈
4
3 ⎛ 1⎞
– −
8 ⎜⎝ 4 ⎟⎠
⎛
3 ⎜
= – ⎜−
8 ⎜
⎜
⎝
=
=
3+
⎞
⎟
⎟ Find a common denominator.
⎟
⎟
⎠
Add the opposite.
=
b) 1
2 ⎛ 3⎞
+ −1
5 ⎜⎝ 4 ⎟⎠
Estimate:
Calculate:
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 83
Name: _____________________________________________________
3.
Date: ______________
Estimate and calculate.
3
1
a) − × ⎛⎜ − ⎞⎟
4 ⎝ 9⎠
Estimate:
b) 3
1
3
×1
3
4
Estimate:
c)
Calculate:
Change to improper
fractions.
1 ⎛ 3⎞
÷ −
10 ⎜⎝ 8 ⎟⎠
Estimate:
d) –
Calculate:
3
1
÷3
8
3
Estimate:
84
Calculate:
MHR ● Chapter 2: Rational Numbers
1
to an
3
improper fraction.
Change 3
Calculate:
Name: _____________________________________________________
Date: ______________
Apply
4.
Virginia made 75 sandwiches for a party.
1
1
1
1
She made ham and cheese, roast beef,
salmon,
tuna, and the rest chicken salad.
3
3
15
15
How many sandwiches were chicken salad?
Ham and cheese:
Roast beef:
Salmon:
Tuna:
1
× 75
3
1 75
= ×
3 1
=
=
Number of sandwiches that are not chicken salad =
+
+
+
=
Number of chicken salad sandwiches = total sandwiches – non-chicken sandwiches
=
–
=
Sentence: _______________________________________________________________________
5.
A vegetarian pizza is cut into 8 equal pieces. A Hawaiian pizza
is cut into 6 equal pieces. Li ate 2 slices of the vegetarian pizza
and 1 slice of the Hawaiian pizza.
a) How much pizza did Li eat?
Vegetarian pizza
Hawaiian pizza
Sentence: ____________________________________________________________________
b) How much pizza was left over?
Sentence: ____________________________________________________________________
2.3 Problem Solving With Rational Numbers in Fraction Form ● MHR 85
Name: _____________________________________________________
6.
A science experiment tracked temperature changes. The table shows the results.
Complete the table. The first row is done for you.
Start Temp (°C)
End Temp (°C)
a) 100
−
1
2
b) –100
−
1
2
c) –1
d)
86
Date: ______________
3
5
9
10
5
−2
5
MHR ● Chapter 2: Rational Numbers
Change in Temp (°C)
(End Temp – Start Temp)
1
− 100
2
1 100
=− −
2
1
1 200
=− −
2
2
−1 − 200
=
2
−201
=
2
1
= −100 °C
2
−
°C
Name: _____________________________________________________
7.
Date: ______________
Paul had $120 to spend on school supplies.
1
1
1
He spent on software, on paper, on pens and pencils, and the rest on other supplies.
2
4
5
How much did he spend on other supplies?
Sentence: _______________________________________________________________________
Math Link
Play this game with a partner or in a small group. You will need a deck of playing cards.
●
Aces represent 1 or −1.
●
Each player gets 4 cards. Use 2 of the 4 cards to make a fraction.
●
The player whose fraction is furthest from 0 gets 1 point.
●
The first player to get 10 points wins.
represents −7
7
represents 1
3
7
Red cards represent positive integers.
Black cards represent negative integers.
8
3
●
7
8
Remove the jokers and face cards from the deck.
7
●
represent −7, 1, −3, and 8
7
The fraction furthest from 0 is − .
8
2.3 Math Link ● MHR 87
Name: _____________________________________________________
Date: ______________
2.4 Warm Up
1.
Evaluate.
a)
22
=2×2
b) 102
=
c) (1.2)2
2.
d) (0.5)2
Find the square root of each number. Example:
a)
9 =
2×2=
4 = 2, because 2 × 2 = 4
b)
16
d)
100
Too low.
3×3=
c)
3.
4.
88
25
Use a calculator to find the square root of each number.
a)
1.21 =
b)
0.09 =
c)
1.44 =
d)
2.25 =
Round each number to the place value in brackets.
a) 3.555 (tenth) ≈
b) 0.785 (hundredth) ≈
c) 289.99 (whole number) ≈
d) 0.729316 (thousandth) ≈
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
2.4 Determining Square Roots of
Rational Numbers
Link the Ideas
Working Example 1: Determine a Rational Number From Its Square Root
A square trampoline has a side length of 2.6 m.
Estimate and calculate the area of the trampoline.
Solution
A s2
s 2.6 m
Estimate.
Calculate.
A = s2
≈ 22
A = s2
≈ 32
≈2×2
≈
≈
≈
C 2.6 x2 6.76
The area of the trampoline is 6.76 m2.
×
Is the calculation close
to your estimate?
So, 2.62 is between 4 and 9.
22
4
2.62
5
6
7
32
8
9
2.6 is closer to 3 than 2, so 2.62 ≈ 7.
An estimate for the area of the trampoline is 7 m2.
A square photo has a side length of 4.4 mm. Estimate and calculate the area of the square.
Estimate:
A = s2
16
25
Calculate:
A = s2
2.4 Determining Square Roots of Rational Numbers ● MHR 89
Name: _____________________________________________________
Date: ______________
Working Example 2: Determine Whether a Rational Number Is a Perfect Square
perfect square
a number that is the product of 2 identical numbers
● examples: 0.5 × 0.5 = 0.25
3
3
9
× =
4 16
4
●
Determine whether each number is a perfect square.
25
49
a)
Solution
If both the numerator and denominator are perfect squares,
the fraction is also a perfect square.
25 is a perfect square because 5 × 5 =
49 is a perfect square because 7 × 7 =
How does this diagram
represent the situation?
25
5 5
= ×
49
7 7
25
So,
is a perfect square.
49
25
A = __
49
b) 0.25
Solution
0.25 =
Perfect Squares:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
5
s= _
7
25
100
25 = 5 ×
The 5 is in the hundredths place,
so the fraction is out of 100.
100 = 10 ×
Since 25 and 100 are perfect squares,
c)
25
is also a perfect square. So, 0.25 is a perfect square.
100
0.4
Solution
0.4 =
4
4 is a perfect square, but 10 is not a perfect square. So, 0.4 is not a perfect square.
90
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Is each of the numbers a perfect square? Show your work.
a)
36
64
36
64
b) 0.41
(is or is not)
a perfect square.
0.41
(is or is not)
a perfect square.
Working Example 3: Determine the Square Root of a Perfect Square
Evaluate 1.44 .
Solution
A 1.44
s √A
Find a positive number that, when multiplied by itself, equals 1.44.
Method 1: Use Guess and Check
1.0 × 1.0 =
Too low.
1.1 × 1.1 =
Too low.
1.3 ×
=
Too high.
1.2 ×
=
Correct!
So, 1.44 = 1.2
2.4 Determining Square Roots of Rational Numbers ● MHR 91
Name: _____________________________________________________
Method 2: Use Fraction Form
1.44 =
=
=
144
The second 4 is in the hundredths
place, so the denominator is 100.
144
12
=
Write as a decimal.
So, 1.44 = 1.2
Check: C 1.44 x 1.2
Method 3: Use Inspection
1.2 × 1.2 =
So, 1.44 = 1.2
Think: 12 × 12 = 144, so
1.2 × 1.2 = 1.44
Evaluate.
a)
92
0.16
MHR ● Chapter 2: Rational Numbers
b)
1.21
Date: ______________
Name: _____________________________________________________
Date: ______________
Working Example 4: Determine a Square Root of a Non-Perfect Square
non-perfect square
● a number that cannot be written as the product of 2 equal factors
● example: 16 is a perfect square because 4 × 4 = 16
17 is not a perfect square; you cannot multiply a number by itself to get 17
a) Estimate
0.73 .
Solution
Think about the perfect squares that are close to 73: 82 = 64 and 92 =
Use the square root of a perfect square on each side of 0.73 .
So,
0.73 is about halfway between
0.64 and
0.81 .
A reasonable estimate is about halfway between 0.8 and
So, 0.73 ≈ 0.85.
b) Calculate
.
√0.64
0.7
√0.73
0.8
√0.81
0.9
1
, which is about 0.85.
0.73 . Round your answer to the nearest thousandth.
Solution
C 0.73 x 0.854400375
Check:
Square your answer to check.
So,
0.8542 =
0.73 ≈ 0.854, rounded to the nearest thousandth.
0.8542 is close to 0.73.
a) Estimate
0.34 .
b) Calculate 0.34 .
Round your answer to the nearest thousandth.
0.5 0.6
0
+1
2.4 Determining Square Roots of Rational Numbers ● MHR 93
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Max says the square root of 6.4 is 3.2.
Lynda says the square root of 6.4 is 0.8.
a) Who is correct? Circle MAX or LYNDA or NEITHER.
b) Explain your reasoning.
_____________________________________________________________________________
2.
Use estimation. Circle the square root that has a value between 5.0 and 5.5.
19.9
35.7
25.4
Give 1 reason for your choice.
________________________________________________________________________________
________________________________________________________________________________
Practise
3.
Use the diagram to find a rational number with a square root between 3 and 4.
94
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
4.
Date: ______________
Estimate and calculate the number that has the given square root.
a) 3.1
Estimate:
Calculate:
(3)2 =
(3.1)2 =
(4)2 =
(3.1)2 ≈
b) 12.5
Estimate:
Calculate:
c) 0.62
Estimate:
5.
Calculate:
Which of the numbers is a perfect square? Show your work.
a)
1
4
b)
1
5
9
a perfect square.
(is or is not)
4
a perfect square.
(is or is not)
So,
1
4
a perfect square.
(is or is not)
c) 0.36 =
36
d) 0.9
2.4 Determining Square Roots of Rational Numbers ● MHR 95
Name: _____________________________________________________
6.
Evaluate.
a)
Use inspection or
guess and check.
196
102 =
Date: ______________
1.21
b)
1.22 =
Too low.
132 =
142 =
196 =
0.25
c)
7.
b) 0.16 mm2
The length of 1 side is
m.
The length of 1 side is
mm.
Estimate each square root.
Then, calculate the square root and round it to the specified number of decimal places.
a)
39, to the nearest tenth
b)
Square root of perfect squares on
either side of 39:
36 =
96
A = s2
Calculate the side length of each square from its area.
a) 169 m2
8.
0.64
d)
=
The closer square root is
.
A reasonable estimate is
.
Check with a calculator:
.
MHR ● Chapter 2: Rational Numbers
4.5, to the nearest hundredth
Name: _____________________________________________________
Date: ______________
Apply
9.
A square tabletop has an area of 1.69 m2. What is the length of 1 side?
s
1.69 m2
A = s2
= s2
=s
=s
Sentence: ________________________________________________________________________
10. a) A 1-L can of paint will cover an area of 9 m2.
What is the side length of the largest square area the paint will cover?
Sentence: _____________________________________________________________________
b) What area will a 2-L can of the same paint cover?
Sentence: _____________________________________________________________________
c) What is the side length of the area in part b)? Round your answer to 1 decimal place.
Sentence: _____________________________________________________________________
d) Nadia uses 2 coats of the same paint for an area that is 5 m by 5 m.
How many litres of paint will she use if the same amount of paint is used for each coat?
Sentence: _____________________________________________________________________
2.4 Determining Square Roots of Rational Numbers ● MHR 97
Name: _____________________________________________________
Date: ______________
11. It costs $80 to build 1 metre of fence.
How much does it cost to build a fence around a square with an area of 225 m2?
Round your answer to the nearest dollar (whole number).
225 m2
Length of side:
Perimeter:
Total cost:
Sentence: ________________________________________________________________________
12. Leon’s rectangular living room is 8.2 m by 4.5 m.
2
A square rug covers of the area of the floor.
5
What is the side length of the rug, to the nearest tenth of a metre
(1 decimal place)?
8.2 m
4.5 m
Area of living room:
Rug area =
2
of area of living room:
5
Side length of rug:
Find the square root
of the rug area.
Sentence: ________________________________________________________________________
98
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Math Link
Sudoku, a Japanese logic puzzle, uses a 9 × 9 square grid.
The grid has nine 3 × 3 sections.
a) The smallest squares have a side length of 1.1 cm.
What is the length of 1 side of the whole grid? Show your thinking.
Sentence: ________________________________________________________________________
b) What is the area of the whole grid?
Sentence: ________________________________________________________________________
c)
A different size Sudoku grid has an area of 182.25 cm2.
What is the length of 1 side of the grid?
Sentence: ________________________________________________________________________
d) What are the dimensions of each 3 by 3 section in part c)?
Dimensions means the
lengths of the sides.
Sentence: ________________________________________________________________________
2.4 Math Link ● MHR 99
Name: _____________________________________________________
Date: ______________
Graphic Organizer
Define each term in the first box, then give some examples in the second box.
Definition:
Examples:
___________________________
___________________________
___________________________
___________________________
___________________________
Rational number
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
Perfect square
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
___________________________
Non-perfect
square
___________________________
___________________________
___________________________
___________________________
___________________________
100
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Chapter 2 Review
For #1 to #3, use the clues to unscramble the letters.
1.
STISPOPOE
+7 and −7 are
2.
SEUAQR OOTR
asks the question, “one of 2 equal factors equals this...” (2 words)
3.
CREFPET QUESAR
the product of 2 equal rational factors; example: 7 × 7 (2 words)
4.
TALINARO BRUNME
the quotient of 2 integers, where the divisor is not zero; example:
5
6
2.1 Comparing and Ordering Rational Numbers, pages 56–63
5.
> means greater than.
< means less than.
Write >, <, or = to make each statement true.
a)
1
2
3
6
b) − 0.86
− 0.84
×
-1 -0.95 -0.9 -8.5 -0.80 -0.75
1
=
2
6
×
c) −
3
4
− 0.75
Change the fraction
to a decimal.
d)
3
4
3
8
Chapter 2 Review ● MHR 101
Name: _____________________________________________________
Date: ______________
2.2 Problem Solving With Rational Numbers in Decimal Form, pages 65–73
6.
7.
Calculate.
a) –5.68 + 4.73
b) – 0.85 – (–2.34)
c) 1.8(– 4.5)
d) –3.77 ÷ (–2.9)
Evaluate. Round your answer to the nearest tenth, if necessary.
5.3 ÷ 2[7.8 + (−8.3)]
a)
= 5.3 ÷ 2 (
=
)
×(
b) 4.2 – 5.6 ÷ (–2.8)
Brackets first.
)
=
8.
One evening in Dauphin, Manitoba, the temperature decreased from 2.4 °C to –3.2 °C.
How much did the temperature change?
–3.2 –
Sentence: ________________________________________________________________________
9.
A company lost an average of $1.2 million per year.
How much did the company lose in 4 years?
The company lost
102
MHR ● Chapter 2: Rational Numbers
in 4 years.
Name: _____________________________________________________
Date: ______________
2.3 Problem Solving With Rational Numbers in Fraction Form, pages 75–87
10. Add or subtract.
a)
2 4
−
3 5
b) –
3 ⎛ 1⎞
+ −
5 ⎜⎝ 5 ⎟⎠
d) 2
1 ⎛ 1⎞
– −2
3 ⎜⎝ 4 ⎟⎠
← Find a common denominator
← Add the opposite
← Solve →
c) –1
1 ⎛ 1⎞
+ −
2 ⎜⎝ 4 ⎟⎠
← Write as improper fractions →
← Find a common denominator →
← Solve →
11. Multiply or divide.
a)
1 ⎛ 1⎞
× −
2 ⎜⎝ 3 ⎟⎠
b) –
5 7
÷
6 8
2
1
÷−
5
10
d) 2
3 ⎛
2
× − 4 ⎞⎟
3⎠
4 ⎜⎝
c) –1
Chapter 2 Review ● MHR 103
Name: _____________________________________________________
12. How many hours are there in 2
1
weeks?
2
Date: ______________
24 hours = 1 day
7 days = 1 week
Sentence: _______________________________________________________________________
2.4 Determining Square Roots of Rational Numbers, pages 89–99
13. Circle each rational number that is a perfect square. Show your work.
a)
3
4
b)
c) 0.49
16
4
d) 22
14. Estimate
220 to 1 decimal place. Check your answer with a calculator. Show your work.
Estimate:
Calculate:
220 is between perfect squares
and
.
15. A 1-L can of paint covers 11 m2. How many cans of paint would you need to paint a ceiling that is
5.2 m by 5.2 m? Show your work.
Area of the ceiling = 5.2 ×
Sentence: ________________________________________________________________________
104
MHR ● Chapter 2: Rational Numbers
Name: _____________________________________________________
Date: ______________
Key Word Builder
Fill in the blanks. Then, find each term in the word search.
1.
The rational number 2.4 is a
number.
2.
The bottom number in a fraction is the
3.
An
4.
A
5.
A(n)
6.
A
7.
The
is the top number in a fraction.
8.
The
of 4 is – 4.
9.
4 is an example of a
.
is an educated guess.
has a numerator and a denominator.
fraction has a numerator that is greater than the denominator.
number has a whole number and a proper fraction.
.
10. The answer when you multiply is called the
.
1 2
11. –2, 3.5, − , 4 , and 0 are all
2 5
.
12. A
written as the product of 2 equal rational numbers.
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is a rational number that cannot be
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Key Word Builder ● MHR 105
Name: _____________________________________________________
Date: ______________
Chapter 2 Practice Test
For #1 to #6, circle the best answer.
1.
Which fraction does not equal
A −
C
2.
4
?
−6
8
12
B
−12
−18
⎛ −6 ⎞
D –⎜
⎟
⎝ −9 ⎠
Which value is greater than –1.5?
0
A –1.6
C –
3.
6.
3
2
−1
5
C –
5.
B –1.2
D –
5
2
B –
1
6
Which fraction is between – 0.4 and – 0.6?
A
4.
−2
3
1
2
D
−1
3
Which value equals –3.78 – (–2.95)?
A – 6.73
B – 0.83
C 0.83
D 6.73
Which value is the best estimate for 1.6 ?
A 2.6
B 1.3
C 0.8
D 0.4
Which rational number is a non-perfect square?
A
B 0.16
1
25
D
C 0.9
121
4
Complete the statements in #6 and #7.
7.
A square has an area of 1.44 m2. The length of 1 side of the square is
So, the perimeter of the square is
106
MHR ● Chapter 2: Rational Numbers
m.
m.
Name: _____________________________________________________
8.
5
On a number line, –3 11 is to the
Date: ______________
of –3.
(right or left)
Short Answer
9.
Find a fraction in lowest terms that is between 0 and –1 and has 5 as the denominator.
-1
0
10. Calculate. Write your answers in lowest terms.
1
1
1
a) 1 – 2
b) – + ⎛⎜ − ⎞⎟
2
3 ⎝ 6⎠
c) − 2
3 ⎛ 1⎞
× −1
4 ⎜⎝ 2 ⎟⎠
d)
5 ⎛ 11 ⎞
÷ −
6 ⎜⎝ 12 ⎟⎠
Chapter 2 Practice Test ● MHR 107
Name: _____________________________________________________
Date: ______________
11. Canada’s Donovan Bailey won the gold medal in the 100-m sprint at the Summer Olympics.
His time was 9.84 s.
5
He beat Frankie Fredericks of Namibia by
of a second. What was Fredericks’s time?
100
First change
5
to a decimal.
100
Sentence: ________________________________________________________________________
12. Is 31.36 a perfect square? Show how you know.
Sentence: ________________________________________________________________________
13. Calculate.
a) the square of 6.1
108
MHR ● Chapter 2: Rational Numbers
b)
1369
Name: _____________________________________________________
Date: ______________
Math Link: Wrap It Up!
Both positive and negative rational numbers
•
Dice, coins, playing cards, or other materials to make numbers
2
8
7
A
1
9
6
2
3
5
4
6
•
6 4
A
Design a game that you can play with a partner or a small group.
The game must include:
+, ―, ×, ÷
• Calculations using 1 or 2 operations
a) Write the rules of your game. How is the winner decided?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
b) Give 4 different examples of calculations used in your game.
c) Play the game with a partner or a small group.
Math Link: Wrap It Up! ● MHR 109
Name: _____________________________________________________
Date: ______________
Challenge
Reaction Time
Sometimes drivers need to be able to stop quickly.
You be the driver!
How quickly do you think you could react to an object on the road?
Materials
• 30-cm ruler
Calculate your reaction time.
1.
Do this experiment with a partner.
● Your partner holds a 30-cm ruler vertically (↕) in front of you.
The zero should be at the bottom.
●
Put your thumb and index finger on each side of the ruler so that
the zero mark is just above your thumb. Do not touch the ruler.
●
Your partner drops the ruler without warning you.
Catch it as quickly as you can.
●
Read the measurement above your thumb, rounded to the
nearest tenth of a centimetre. This is your reaction distance.
●
Do the experiment 5 times. Record each distance in the table.
●
Switch roles and record your partner’s reaction distances.
●
Find the average reaction distance for you and your partner.
●
Change each average reaction distance (d ) to metres.
Your Reaction Distance
Partner’s Reaction Distance
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Average Reaction Distance (d )
sum of 5 reaction distances
d=
sum of 5 reaction distances
5
=
5
d=
Average Reaction Distance in
metres (d )
110
d=
=
MHR ● Chapter 2: Rational Numbers
cm ÷ 100
m
d=
d=
sum of 5 reaction distances
5
Name: _____________________________________________________
2.
Date: ______________
d
to calculate your and your partner’s average reaction time.
4.9
d is the reaction distance in metres and t is the time in seconds
Use the formula t =
●
Player 1:
t=
Player 2:
d
4.9
t=
← Formula →
← Substitute average reaction distance →
← Divide →
4.9
t=
← Find the square root →
t =
s
3. a) Imagine you are driving a car at 40 km/h.
Suppose you need a reaction time of 0.75 s to step on the brake.
How far would the car travel before you step on the brake?
Round your answer to the nearest tenth.
40 km =
1h=
m
40 km/h = ? m/s
1 h = 60 min
1 min = 60 s
m
40 km
→
1h
s
s
÷ 3600
m
=
s
m
1s
÷ 3600
Distance travelled in 0.75 s = m/s × 0.75
=
× 0.75
=
The car would travel
m before you step on the brake.
b) What might influence your reaction time and your stopping distance?
_____________________________________________________________________________
Challenge ● MHR 111
Answers
Get Ready, pages 52–53
6. Answers will vary. Examples: a) – 4 b) 20 ; 5
12 3
10
1
1
7. a)
b) –
5
2
1. a) 50.816 b) 272.43
2. a) 3 , 3 b) 3 > 3
12 4
4 12
8. Answers may vary. Example: a) 0.7 b) – 0.6
3. a) 5 or 1 b) 7
2
10
8
9. Answers may vary. Example: a) 25 or 1 b) – 527
100
1000
4
4. a) 2 b) 2
9
Apply
10. a) +8.2 b) +2.9 c) –3.5 d) +32.50
Math Link
1. 9, 1
9
2. 4
9
3. 8
9
4. 3 or 1
3
9
11. a) 6.1, 5.4, 3.9, 0.6, – 0.1, –5.1, –14.1 b) Yellowknife
12. a) = b) > c) < d) >
2.2 Warm Up, page 64
1. a) 2 b) –12 c) – 6 d) 12 e) – 6 f) 5 g) –2 h) 5
2. Estimates will vary. a) Estimate: 5; Calculate: 5.24 b) Estimate: 0.4;
Calculate: 0.43 c) Estimate: 18; Calculate: 20.15 d) Estimate: 3;
Calculate: 3
5. Answers may vary. Example: Use X, Y, and O for symbols. The fraction
of total squares needed to win does not change.
3. a) 10 b) 9
2.1 Warm Up, page 55
2.2 Problem Solving With Rational Numbers in Decimal Form,
pages 65–73
1. a) 1.2 b) –1
Working Example 1: Show You Know
Estimates will vary. a) Estimate: – 3; Calculate: –2.86 b) Estimate: – 4;
Calculate: – 4.8
2. a) 0.5 b) 0.6
3. a) 3 b) 85 or 17
10
100
20
Working Example 2: Show You Know
−0.5
4. a)
0.3
−1
1.2
+1
0
b) –1, –0.5, 0.3, 1.2
Estimates will vary. a) Estimate: 3; Calculate: 3.66 b) Estimate: –1;
Calculate: –1.2
Working Example 3: Show You Know
2.96 °C/min
5. Answers may vary. Example: a) 4 b) 4 or 2
6
6
3
3
2
6. a) +5 b) –3.4 c) –
d)
4
5
Communicate the Ideas
1. Answers will vary. Examples: a) Positive because 6.5 is greater than
– 0.32 b) 6.18
2.1 Comparing and Ordering Rational Numbers, pages 56–63
Working Example 1: Show You Know
Ascending order: – 3 , – 0.6, 3 , 3 ; Descending order: 3 , 3 , – 0.6, – 3
2
8 4
4 8
2
2. a) the same b) Answers will vary. Example: A negative times a positive
equals a negative. It does not matter whether the negative number is first
or second.
Practise
Working Example 2: Show You Know
3. Estimates will vary. a) Estimate: 1.0; Calculate: 0.7 b) Estimate: –1;
Calculate: – 0.88
a) – 7 b) 0.0001
10
4. Estimates will vary. a) Estimate: 2; Calculate: 2.3; b) Estimate: 5.5;
Calculate: 5.34
Working Example 3: Show You Know
5. Estimates will vary. a) Estimate: – 9; Calculate: –8.64 b) Estimate: 30;
Calculate: 30.25
Answers may vary. Example: a) 57 b) –2 45 or – 245
100
100
100
Communicate the Ideas
1.
7. a) –13.17 b) –3.08
1
−_
2
−3
6. Estimates will vary. a) Estimate: 10; Calculate: 10 b) Estimate: –1.5;
Calculate: –1.3
Apply
8. The temperature dropped 2.2 °C per hour.
−5
−4
−3
−2
−1
0
1
2
3
4
5
2. NO. Answers may vary. Example: When numbers are negative, the bigger
the number, the smaller the value.
Practise
3. a) D b) C c) A d) B
4.
5
−_
3
−2
_8
9
−0.4
−1
5. 1. 8 , 9 , – 3 ,
5
8
0
1
1
2 __
10
2
− 1 , –1
2
112 MHR ● Chapter 2: Rational Numbers
9. a) 3.8 + 2.3 b) The pelican’s dive is 6.1 m.
10. It took 29 min to reach the surface.
11. a) The company lost money. b) On average, the company lost
$1.2 million per year.
2.3 Warm Up, page 74
1. a) 4 b) 2
5
9
14
2.
3
3. a) 9 b) 1
10
2
1
4. a) 5
b) 11
10
14
2.3 Problem Solving With Rational Numbers in Fraction Form,
pages 75–87
Practise
Working Example 1: Show You Know
Estimates will vary. a) Estimate: –1 1 ; Calculate: –1 3 b) Estimate: 3;
20
2
1
Calculate: 3
4
4. Estimates will vary. a) Estimate: 9; Calculate: 9.61 b) Estimate: 150;
Calculate: 156.25 c) Estimate: 0.4; Calculate: 0.3844
5. a) is b) is not c) is d) is not
Working Example 2: Show You Know
7. a) 13 b) 0.4
3. Answers will vary: Example: 10
6. a) 14 b) 1.1 c) 0.5 d) 0.8
8. Estimates will vary. a) Estimate: 6.1; Calculate: 6.2 b) Estimate: 2.1;
Calculate: 2.1
a) 1 b) –1 7
10
15
Apply
Working Example 3: Show You Know
9. The length of one side is 1.3 m.
$10
Communicate the Ideas
1. a) –2 b) –2 c) Answers will vary. Example: I like multiplying by the
reciprocal because cancelling is possible.
Practise
2. Estimates will vary. a) Estimate: 1 ; Calculate: 5 b) Estimate: – 1 ;
8
2
2
7
Calculate: –
20
3. Estimates will vary. a) Estimate: 0; Calculate: 1 b) Estimate: 6;
12
5
4
Calculate: 5
c) Estimate: 0; Calculate: –
d) Estimate: 0;
6
15
Calculate: – 9
80
Apply
4. There are 15 chicken salad sandwiches.
5. a) Li at 10 of a pizza. b) There was 1 7 of a pizza left.
24
12
6.
Change in Temp (°C)
Start Temp (°C) End Temp (°C) (End Temp – Start Temp)
b) –100
−1
2
c) –1 3
5
d) 9
10
5
−2
5
99 1
2
63
5
–1 3
10
10. a) The side length is 3 m. b) A 2-L can will cover 18 m2. c) The side
length is 4.2 m. d) She will use 5.6 L of paint.
11. It costs $4800.
12. The side length of the rug is 3.8 m.
Math Link
a) The length of one side is 9.9 cm. b) The area of the whole grid is
98.01 cm2. c) The length of one side is 13.5 cm.
d) The dimensions are 4.5 × 4.5.
Graphic Organizer, page 100
Answers will vary. Example:
Rational number:
Definition: A number that can be written as a fraction.
Examples: 1 , 2 1 , – 51 , 1.2, – 0.5
25
4 2
Perfect square:
Definition: The product of two identical numbers
Examples: 3 × 3 = 9, 3 × 3 = 9
4 4 16
Non-perfect square:
Definition: A number that cannot be written as the product of two identical
numbers.
Examples: 10, –7,
−
4 , 0.18
5
Chapter 2 Review, pages 101–104
1. opposites 2. square root 3. perfect square 4. rational number
7. Paul spent $6 on other supplies.
5. a) = b) < c) = d) >
2.4 Warm Up, page 88
6. a) –0.95 b) 1.49 c) –8.1 d) 1.3
1. a) 4 b) 100 c) 1.44 d) 0.25
7. a) –1.325 b) 6.2
2. a) 3 b) 4 c) 5 d) 10
8. The temperature changed –5.6 °C.
3. a) 1.1 b) 0.3 c) 1.2 d) 1.5
9. $4.8 million
4. a) 3.6 b) 0.79 c) 290 d) 0.729
10. a) – 2 b) – 4 c) –1 3 d) 4 7
4
12
15
5
2.4 Determining Square Roots of Rational Numbers, pages 89–99
Working Example 1: Show You Know
Estimate will vary. Example: 20; Calculate: 19.36 mn 2
Working Example 2: Show You Know
a) is b) is not
Working Example 3: Show You Know
a) 0.4 b) 1.1
Working Example 4: Show You Know
11. a) – 1 b) – 20 c) 14 d) –12 5
21
6
6
12. There are 420 hours in 2 1 weeks.
2
13. a) no b) yes c) yes d) no
14. Estimate will vary. Example: 14; Calculate: 14.8
15. You would need 3 cans of paint.
a) 0.6 b) 0.583
Communicate the Ideas
1. a) NEITHER
b) Answers will vary. Example: Max divided by 2 instead of finding the
square root. Lynda squared 0.8 and misplaced the decimal.
2. 25.4 ; 52 is 25, so the square must be slightly more than 25.
Answers ● MHR 113
Key Word Builder, page 105
Math Link: Wrap It Up!, page 109
1. decimal 2. denominator 3. estimate 4. fraction 5. improper 6. mixed
7. numerator 8. opposite 9. perfect square 10. product 11. rational
numbers 12. non-perfect square
a) Answers will vary. Example: Use 1 red die and 1 black die to play. Red is
a positive integer, black is a negative integer. Roll the two dice and add
the scores. The first player to 20 wins.
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Chapter 2 Practice Test, pages 106–108
1. C 2. B 3. C 4. B 5. B 6. C
7. 1.2; 4.8
8. left
9. – 1 or – 2 or – 3 or – 4
5
5
5
5
1
1
1
b) –
c) 4
d) – 10
10. a) –
11
2
2
8
11. Fredericks’s time was 9.89 s.
12. Yes. Its square root is 5.6.
13. a) 37.21 b) 37
114 MHR ● Chapter 2: Rational Numbers
b) Answers will vary. Examples: +2 + (–5) = –3; (–3) + (+3) = 0; +6 + (+5)
= +11; (+6) + (+6) = +12. So the score is –3 + 0 + 11 + 12 = 20.
Challenge, pages 110–111
Answers will vary. Example:
1.
Your Reaction Distance
10
11
9
8
7
sum
of
5
reaction
distances
d=
5
d=9
9
d=
cm ÷ 100
= 0.09
Partner’s Reaction Distance
12
13
10
8
7
sum
of
5
reaction
distances
d=
5
d = 10
0.01 m
m
2. Player 1: 0.136 s; Player 2: 0.143 s
3. a) 8.3 m b) Answers will vary. Example: Being tired might influence
your reaction time.