Rational Numbers - msbayat.weebly.com

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Chapter

Rational NumbersThe degree of precision that you need for calculations depends on what you are doing. For example, the circumference of a circle is a bit more than three times the length of its diameter. If you are painting a circle with a diameter of 4 m at centre ice, you can say that the circumference of the circle is a bit more than 12 metres. If you want to be more precise, you can say that the circumference of a circle is about 3.14 times the length of its diameter. The symbol π, called pi, represents the ratio of the circumference of a circle to its diameter.

The value of π is actually a never-ending decimal. Pi has been calculated to over 13 trillion digits, and it cannot be expressed as a fraction. That means π is an irrational number. This chapter deals with rational numbers, that is, numbers that you can express as fractions.

When measuring quantities, different jobs require different degrees of precision. People in the construction trades measure using rational numbers, whether in fraction form or in decimal form. Tradespeople also need to be excellent at performing mathematical operations with rational numbers.

Big Idea

You can apply mathematical operations to rational numbers just like you do for whole numbers. The order in which you apply the operations is crucial. If you use the wrong order, you can get the wrong result.

Inquire and Explore

• What is the relationship between the different operations?• How can you use the order of operations to solve problems

involving rational numbers?• How can you use mental math to estimate reasonable answers?

1

2 MHR • Chapter 1 Rational Numbers

Chapter 1 Rational Numbers • MHR 3

Get Ready

Revisit Fractions

1. Make each set of fractions equivalent to the first one in the sequence.

a) 1 __ 2 = ◼ _ 4 = ◼ _ 8 = ◼ _ 16 = 16 _ ◼ = 30 _ ◼

b) 1 __ 4 = ◼ _ 8 = ◼ _ 16 = 8 _ ◼ = 16 _ ◼ = 30 _ ◼

2. Change the mixed numbers to improper fractions.a) 3 1 _ 4 b) 4 5 _ 7 c) −2 3 _ 4

3. Consider these improper fractions.

6 __ 4 10 __ 8 8 __ 5 20 __ 16

a) Which of the fractions above are equivalent fractions? How do you know?

b) Which fraction(s) are written in lowest terms?c) Write the other one(s) in lowest terms.d) Express each improper fraction as a mixed

number.

When adding or subtracting fractions, work with parts of the whole that are of equal size.

• Use diagrams:

2 _ 3 + 1 _ 6

= 4 _ 6 + 1 _ 6

= 5 _ 6

• Use a common denominator:

2 _ 3 − 5 _ 8

= 16 _ 24 − 15 _ 24

= 1 _ 24

1––31––3

1––31––6

1––61––6

1––61––6

1––6+

1––61––6

1––61––6

1––61––6

1––61––6

1––61––6

1––61––6 +

1––61––6

1––61––6

1––61––6

4. Add or subtract. Do as many as you can using mental math.

a) 1 __ 2 + 1 __ 4 b) 1 __ 2 + 1 __ 8 c) 1 __ 5 + 1 __ 10 d) 3 __ 4 + 1 __ 2

e) 1 __ 2 − 1 __ 4 f) 3 __ 5 − 1 __ 10 g) 1 __ 10 − 1 ___ 100 h) 3 __ 4 − 3 __ 8

To multiply two fractions, multiply the numerators and multiply the denominators.

1 _ 2 × 2 _ 3 = 1 × 2 _ 2 × 3

= 2 _ 6

= 1 _ 3 Reduce the fraction to lowest terms.

5. Multiply. State your answers in lowest terms.

a) 1 __ 2 × 1 __ 4 b) 3 __ 4 × 7 __ 6 c) 1 1 __ 2 × 1 __ 2 d) 3 3 __ 10 × 5 __ 2


Here are two ways to divide fractions.• Use a common denominator and divide

the numerators:

7 _ 10 ÷ 2 _ 5 = 7 _ 10 ÷ 4 _ 10

= 7 _ 4 or 1 3 _ 4

• Multiply by the reciprocal of the second fraction:

7 _ 10 ÷ 2 _ 5 = 7 _ 10 × 5 _ 2

= 35 _ 20 or 7 _ 4 or 1 3 _ 4

6. Divide. State your answers in lowest terms.

a) 1 __ 2 ÷ 1 __ 4 b) 3 __ 4 ÷ 5 __ 2 c) 7 __ 8 ÷ 7 __ 8 d) 2 1 __ 4 ÷ 1 1 __ 8

Revisit Decimals

You can estimate an answer to a multiplication or division question before you use your calculator. This helps you to know if you make a mistake when you use the calculator.

Estimate the answer to 19.76 × 31.18 .Round 19.76 × 31.18 to 20 × 30 = 600 .

19.76 31.18 616.1168

The answer should be close to 600 when you use a calculator.

A calculator display of 61 611.68 means that you missed entering the decimal for one of the numbers.

7. Add or subtract. Do as many as you can using mental math.a) 0.5 + 0.05 b) 2.6 + 4.44c) 7.55 − 3.56 d) 6.4 − 2.99

8. Estimate first, and then calculate. Compare your calculator answer to your estimate.a) 149.8 × 9.8 b) 2.7 × 101.4c) 40.96 ÷ 9.61 d) 26.4 ÷ 2.88

Connect Fractions and Decimals

9. Express each fraction as a decimal.

a) 4 __ 10 b) 4 ___ 100 c) 43 ___ 100 d) 43 ____ 1000

10. Express each decimal as a fraction.a) 0.7 b) 0.79 c) 0.07 d) 0.079

11. Express each improper fraction as a decimal.

a) 13 __ 10 b) 23 __ 10 c) 303 ___ 100 d) 330 ___ 100

12. Express each decimal as a mixed number in lowest terms.a) 3.900 b) 1.380 c) 2.05 d) 5.025

Squares

52 = 5 × 5

13. Evaluate. Do as many as you can using mental math.a) 32 b) 42 c) 72 d) 102

Get Ready • MHR 5

1.1 Introduction to Rational Numbers

>

1––3

3––42––3

1––3

1––41––4

1––4

> 2.2564––––25

Many lengths and areas have values that are rational numbers, such as 5 1 __ 4 . You can compare rational numbers using diagrams that represent the numbers. However, it is not always easy to accurately represent numbers using the area of a square. Do all squares with rational areas have side lengths that are rational?

Explore and Analyze

1. a) On grid paper, draw a 4 × 4 square as shown here.b) What is the area of the outlined square?c) What is the side length of the outlined square?d) A square root is a number that when multiplied by itself

takes you back to the given number. What is the square root of 16?

2. a) On grid paper, draw a 5 × 5 square.b) What is the area of the square?c) What is the square root of 25?

3. Model the square root of 20 using grid paper. Why is it more difficult to model this square root than the one in #2?

4. a) Shade in 1 __ 4 of the hundred grid as shown here.

b) What fraction of the side length of the hundred grid

gives an area of 1 __ 4 of the grid? In other words, what

is √ __

1 __ 4 ?

Focus On …In this lesson, I will learn to

• compare and order rational numbers

• determine the square or the square root of a rational number using estimation and technology

rational number

• a number that can be expressed as a fraction

• 7, −3, 4.5, 1 1 __ 2 , and 1 __ 3 are

rational numbers

• π and √ _

2 are irrational numbers because they cannot be expressed as a fraction


5. a) On your hundred grid, shade in a 7 __ 10 × 7 __ 10 square.

b) What fraction of the total area of the grid is shaded?

6. a) What conclusion(s) can you make about the square root of a number that is greater than 1?

b) What conclusion can you make about the square root of a number that is less than 1?

Develop Understanding

Example 1: Ordering Rational Numbers

Which rational number is greater, 7 __ 16 or 5 __ 8 ?

Solution

Method 1: Use a Common Denominator

A common denominator for 7 __ 16 and 5 __ 8 is 16.

5 __ 8 = 10 __ 16

7 __ 16 is less than 10 __ 16 .

So, 7 __ 16 < 5 __ 8 .

Method 2: Convert Fractions to Decimals

7 __ 16 = 7 ÷ 16 = 0.4375 5 __ 8 = 0.625

Therefore, 7 __ 16 < 5 __ 8 .

Method 3: Use a Benchmark

7 __ 16 is less than 5 __ 8 because 7 __ 16 is less than half and 5 __ 8 is greater than half.

Which rational number is smaller, 5 __ 9 or 3 __ 8 ?

Show You Know

Which of the three methods do you prefer?

1.1 Introduction to Rational Numbers • MHR 7

Example 2: Identify a Rational Number Between Two Given Rational Numbers

Identify a fraction between −0.6 and −0.7.

Solution

First, use a number line.

–0.7 –0.6

–1 0

A decimal number between −0.6 and −0.7 is −0.65.

Convert the decimal to a fraction.

−0.65 = − 65 ___ 100

A fraction between −0.6 and −0.7 is − 65 ___ 100 .

Identify a fraction between −2.4 and −2.5.

Show You Know

Example 3: Determine the Square of a Rational Number

A square trampoline has a side length of 2.6 m. Estimate and then calculate the area of the trampoline.

Solution

Estimate. 22 = 4 and 32 = 9 So, 2.62 is between 4 and 9.2.6 is closer to 3 than to 2, so 2.62 ≈ 7 .

An estimate for the area of the trampoline is 7 m2.

Calculate. 2.62 = 6.76 2.6 6.76

The area of a trampoline with a side length of 2.6 m is 6.76 m2.

You could change −0.6 and −0.7 into fraction form. What

would the number line look like?

What is another way to express − 65 ___ 100 as a fraction?

A = s2

s = 2.6 m


Estimate and then calculate the area of a square photo with a side length of 7.1 cm.

A = s2

s = 7.1 cm

Show You Know

Example 4: Determine the Square Root of a Rational Number

a) Estimate √ _____

0.73 .b) Use a calculator to determine √

_____ 0.73 . Round your answer to the nearest

thousandth.

Solution

a) Use the square root of a perfect square.

√ _

0.73 is equivalent to √ _

73 __ √ _

100 .

We know √ _

100 is 10.

√ _____

73 is about halfway between √ _____

64 and √ _____

81 .

9 8 7 10

64 8173

One reasonable estimate for √ _____

0.73 is about halfway between 0.8 and 0.9, which is about 0.85.

b) 0.73 0.854400375

So, √ _____

0.73 ≈ 0.854 , rounded to the nearest thousandth.

a) Estimate √ _____

0.34 . What perfect squares is 0.34 between? How do you know? How can this help you estimate an answer?

b) Use a calculator to determine √ _____

0.34 . Round to the nearest thousandth.

Show You Know

Numbers like 64 are called perfect squares because their

square roots are whole numbers.

0.64 is a perfect square, because its square root is a rational

number.

65 is not a perfect square, and neither is 0.65.

√ _

0.73 is not a rational number, so you need to round to

the nearest thousandth.


Connect and Reflect

Key Ideas

• Equivalent fractions represent the same rational number.

− 5 __ 2 , −5 ___ 2 , 10 ___ −4 , and −10 ___ 4 all represent −2 1 __ 2 or −2.5.

• One strategy for comparing and ordering rational numbers is to use a number line. ▪ On a horizontal number line, a larger rational number is to the right of a smaller

rational number. ▪ Opposite rational numbers are the same distance in opposite directions from zero.

0.75

0–1–2–3–4 +1 +2 +3 +4

–2 2–

opposites

1––41––4

1––2

• To compare fractions with the same denominator, compare the numerators.

−7 ___ 10 < −6 ___ 10 because −7 < −6 .

• One strategy for identifying a rational number between two given rational numbers is to use a number line. ▪ A rational number in fraction form between −0.3 and −0.1 is −1 ___ 5 .

–0.3 –0.1–0.2

–1 0

= – 1––5

• If the side length of a square models a number, the area of the square models the square of the number.

• If the area of a square models a number, the side length of the square models the square root of the number.

A = s2

s = A


Practise

For help with #1 refer to Example 1 on page 7.

1. Determine the greater fraction.

a) 6 __ 10 , 13 __ 20 b) 3 __ 7 , 8 __ 21

c) 7 __ 8 , 3 __ 4 d) 7 __ 10 , 2 __ 3

For help with #2 to #4, refer to Example 2 on page 8.

2. Identify a decimal number between each of the following pairs of rational numbers. Explain how you know your choice fits. Use a number line if that helps.

a) 3 __ 5 , 4 __ 5 b) − 1 __ 2 , − 5 __ 8

c) − 5 __ 6 , 1 d) − 17 __ 20 , − 4 __ 5

3. What is a possible decimal number between each of the following pairs of rational numbers?

a) 1 1 __ 2 , 1 7 __ 10 b) −2 2 __ 3 , −2 1 __ 3

c) 1 3 __ 5 , −1 7 __ 10 d) −3 1 ___ 100 , −3 1 __ 50

4. Identify a fraction between each of the following pairs of rational numbers.a) 0.2, 0.3 b) 0, −0.1c) −0.74, −0.76 d) −0.52, −0.53

For help with #5 and #6, refer to Example 3 on page 8.

5. Estimate and then calculate the square of each number.a) 3.1 b) 12.5c) 0.62 d) 0.29

6. Estimate and then calculate the area of each square, given its side length. Make a sketch on grid paper if that helps.a) 4.3 cm b) 0.38 kmc) 13.55 mm d) 2.91 m

For help with #7 to #10, refer to Example 4 on page 9.

7. Evaluate.a) √

____ 324 b) √

_____ 2.89

c) √ _______

0.0225 d) √ _____

2025

8. Calculate the side length of each square from its area.a) 169 m2 b) 0.16 mm2

9. Estimate each square root. Explain your reasoning. Then calculate it to the specified number of decimal places.a) √

___ 39 , to the nearest tenth

b) √ ____

4.5 , to the nearest hundredthc) √

_____ 0.87 , to the nearest thousandth

10. Given the area of each square, determine its side length. Express your answer to the nearest hundredth of a unit.a) 0.85 m2 b) 60 cm2

11. Use mental math to predict which rational number in each pair is greater. Explain how you know.

a) 0.7, 3 __ 4 b) 3 __ 8 , 0.3

c) 0.8, 7 __ 8 d) 6 __ 10 , 0. _

6

12. Use the strategy of your choice to verify the greater rational number in each pair in #11.

13. Sort the following fractions from least to greatest. − 5 __ 8 1 __ 3 13 __ 10 1 1 __ 4 4 __ 5 5 __ 3 3 __ 8

14. Sort the following decimals from least to greatest. 1.

_ 6 0.7 0.

_ 54 −0.07 1.301 −0.54 1.3 −0.45

15. Which two fractions in #13 have equivalent decimal values in #14?

The line over the digit indicates that it repeats forever.


Apply

16. a) Which temperature is warmer, 1 __ 4 °C or 0.3 °C? Explain.

b) Which temperature is warmer, −1 ___ 4 °C or −0.3 °C? Explain.

17. The table includes the melting points and the boiling points of six elements known as the noble gases.

Noble Gas Melting Point (°C) Boiling Point (°C)

Argon −189.2 −185.7

Helium −272.2 −268.6

Neon −248.67 −245.92

Krypton −156.6 −152.3

Radon −71.0 −61.8

Xenon −111.9 −107.1

a) Which noble gases have a melting point that is less than the melting point of argon?

b) Which noble gases have a boiling point that is greater than the boiling point of krypton?

c) Arrange the melting points in ascending order.d) Arrange the boiling points in descending order.

18. Dylan is replacing the strip of laminate that is glued to the vertical faces of a square tabletop. The tabletop has an area of 1.69 m2. What length of laminate does he need?

19. a) Rhys said that he ignored the fractions when he decided that

−2 1 __ 5 is smaller than −1 9 __ 10 . Explain his thinking.

b) Naomi said that she ignored the integer −1 when she decided that −1 1 __ 4 is greater than −1 2 __ 7 . Explain her thinking.

20. A bank offers an interest rate of 2 1 __ 4 % on deposits, while another bank offers 2.29%. Which bank offers the better interest rate? How do you know?

21. In your notebook, replace each ◼ with >, <, or = to make the statement true.

a) −9 ___ 6 ◼ 3 ___ −2 b) − 3 __ 5 ◼ −0. _

6

c) −1 3 __ 10 ◼ − ( −13 ___ −10 ) d) −3.25 ◼ −3 1 __ 5

e) − 8 __ 12 ◼ − 11 __ 15 f) −2 5 __ 6 ◼ −2 7 __ 8

A = 1.69 m2

laminate

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22. Competency Check The surface area of a cube is 100 cm2. Determine the edge length of the cube, to the nearest tenth of a centimetre.

23. The period, t, in seconds, of a pendulum is the time it takes for a complete swing back and forth. You can use the formula t = √

__ 4l to calculate the

period, where l is the length of the pendulum, in metres. Determine the period of a pendulum with each of the following lengths. Express each answer to the nearest hundredth of a second.a) 1.6 mb) 2.5 mc) 50 cm

Extend

24. A baseball diamond has a square area of about 8100 ft2. What is the straight-line distance between third base and first base? State your answer to the nearest inch. (Hint: There are 12 inches in 1 foot.)

secondbase

thirdbase

firstbase

25. In your notebook, replace each ◼ with an integer to make the statement true. In each case, is more than one answer possible? Explain.

a) ◼.5 < −1.9 b) ◼ ___ −4 = −2 1 __ 4

c) −3 ___ ◼ = − −15 ____ 5 d) −1.5◼2 > −1.512

e) − 3 __ 4 < −0.7◼ f) −5 1 __ 2 > 11 ___ ◼

g) −2 3 __ 5 = ◼ ___ 10 h) 8 __ ◼ < − 2 __ 3

26. Determine √ _______

√ _______

√ ______

65 536 .

27. Write a question similar to #26 that has an answer of 3.

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1.2 Rational Numbers in Decimal Form

Some things that are divided into smaller parts are written in fraction form, while other things tend to be written using decimal numbers. Decimals are just a different way to write fractions. In Fort Nelson, BC, the average mid-afternoon temperature in July is 16.7 °C. The average mid-afternoon temperature in January is −16.3 °C. Why are temperatures best expressed using decimals?

Explore and Analyze

1. Estimate, to the nearest degree, how much warmer Fort Nelson is in July than in January. Explain how you arrived at your answer.

2. You can think of a thermometer as a vertical number line. Sketch a thermometer in your notebook. Mark three temperatures on your thermometer: the July average, the January average, and 0 °C. Reflect on your estimate from #1. Are you satisfied with your estimate, or would you like to change it?

3. Many calculators distinguish between subtraction and a negative value. What keystrokes would you have to use on your calculator to determine the difference between 26.1 and −12.6? Is there more than one way to arrive at the answer of 38.7? Compare your method with those of your classmates.

4. In Kamloops, the January average temperature is −2.8 °C and the July average temperature is 21.5 °C. Is the range of average temperatures greater in Kamloops or Fort Nelson? How do you know?

Focus On …In this lesson, I will learn to…

• perform operations with decimal numbers

• use mental math to estimate reasonable answers

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5. Below are the average January and July temperatures in three Canadian cities. Can you match the temperatures to the city? Explain your reasoning. Which city has the greatest range of temperatures?a) Niagara Falls, ON

b) Yellowknife, NT

c) Winnipeg, MB

January: −18.3 °C July: 19.8 °CJanuary: −4.5 °C July: 22.3°CJanuary: −27.9 °C July: 16.5 °C


Example 1: Add or Subtract Rational Numbers in Decimal Form

Without using a calculator, explain whether the answer will be a positive or a negative value. Then, use a calculator to determine the actual answer.a) 2.65 + (−3.81) b) −5.96 − (−6.83)

Solution

a) 3.81 is greater than 2.65, so adding −3.81 to 2.65 will give an answer less than 0, close to −1.

Calculate: Here are two ways to input the keystrokes on a calculator:

2.65 3.81 −1.16

2.65 3.81 −1.16

b) Subtracting a negative number is equivalent to adding the opposite positive number. 6.83 is greater than 5.96, so adding 6.83 to −5.96 will give an answer greater than 0, close to +1.

Calculate: Here are two ways to input the keystrokes on a calculator: 5.96 6.83 0.87

5.96 6.83 0.87

Without using a calculator, explain whether the answer will be a positive or a negative value. Then, correctly use a calculator to determine the answer.a) −4.38 − (−5.12) b) 6.21 + (−6.84)

Show You Know

Your calculator may use different

keystrokes.

1.2 Rational Numbers in Decimal Form • MHR 15

Example 2: Multiply or Divide Rational Numbers in Decimal Form

Explain whether the sign of the answer will be positive or negative. Estimate the answer. Then, correctly use a calculator to determine the exact answer.a) 9.49 × 5.08 b) 2.4 ÷ −1.4 c) −2.5(2.5) d) −0.86 ÷ −0.42

Solution

a) A positive number multiplied by (or divided by) another positive number always gives an answer that is positive.

Estimate: 9.49 is very close to 9 1 __ 2 , and 5.08 is close to 5.

9 × 5 = 45 10 × 5 = 50

A reasonable estimate for the answer is between 45 and 50.

Calculate: 9.49 5.08 48.2092

b) A positive number divided by (or multiplied by) a negative number always gives an answer that is negative.

Estimate: 1.4 × 1 = 1.4 and 1.4 × 2 = 2.8

Since 2.8 is closer to 2.4, you know that 2.4 ÷ 1.4 will give an answer greater than 1 but less than 2.

Since the answer must be negative, a reasonable estimate is between −1 and −2.

Calculate. Here are two ways to input the keystrokes on a calculator: 2.4 1.4 1.71…

Then change the sign so the answer is −1.71…

2.4 1.4 −1.71…

c) The brackets show this is a multiplication problem. A negative number multiplied by (or divided by) a positive number always gives an answer that is negative.

Estimate: 2 × 2 = 4 and 3 × 3 = 9

Since the answer must be negative, a reasonable estimate is between −4 and −9.

How close is your estimate to your calculation?


Calculate. Here are two ways to input the keystrokes on a calculator: 2.5 2.5 6.25

Then change the sign so the answer is −6.25.

2.5 2.5 −6.25

d) A negative number multiplied by (or divided by) another negative number always gives an answer that is positive.

Estimate: Half of 0.86 is 0.43, so 0.86 ÷ 0.43 = 2 .

0.86 ÷ 0.42 (a smaller number than 0.43) must give an answer a little greater than 2.

Since the answer must be positive, a good estimate for the answer is a bit more than 2.

Calculate. Here are two ways to input the keystrokes on a calculator: 0.86 0.42 2.04…

0.86 0.42 2.04…

Explain whether the sign of the answer will be positive or negative. Estimate the answer. Then, correctly use a calculator to determine the exact answer.a) −2.52 × 7.07 b) 12.4 ÷ −2.9 c) −1.91 × −0.51

Show You Know

Connect and Reflect

Key Ideas

• Adding a negative rational number to another rational number is equivalent to subtracting the corresponding positive number.

• Subtracting a negative rational number is equivalent to adding the corresponding positive number.

• A negative rational number multiplied by or divided by another negative rational number always gives an answer that is positive.

• A positive rational number multiplied by or divided by a negative rational number always gives an answer that is negative.

Sign rules: (+) × (−) = (–) (−) × (−) = (+) (+) ÷ (−) = (−) (−) ÷ (−) = (+)


Practise


1. Without using a calculator, predict whether the answer will be a positive or a negative value. Then, use a calculator to determine the actual answer.a) 0.98 + (−2.91) b) 5.46 − 3.16 c) −4.23 + (−5.75) d) −1049 − (−6.83)

2. Calculate.a) 9.37 − 11.62 b) −0.512 + 2.385 c) 0.675 − (−0.061) d) −10.95 + (−1.99)


3. Decide whether the sign of the answer will be positive or negative. Estimate the answer. Then, use a calculator to determine the exact answer. Round your answers to 2 decimal places, if necessary.a) 2.7 × (−3.2) b) −3.25 ÷ 2.5 c) −5.5 × (−5.5) d) −4.37 ÷ (−.095)

4. Decide whether the sign of the answer will be positive or negative. Then, calculate.a) −2.4(−1.5) b) 8.6 ÷ 0.9 c) −5.3(4.2) d) 19.5 ÷ (−16.2) e) 1.12(0.68) f) −0.55 ÷ 0.66

Apply

5. A pelican dives vertically from a height of 3.8 m above the water. It then catches a fish 2.3 m underwater.a) Estimate the length of the pelican’s dive from the top of the dive to

the fish.b) Write an expression using rational numbers to represent the length

of the pelican’s dive.c) How long is the pelican’s dive?d) Interpret your answer to part c) as a positive number and a negative number.

6. A submarine is cruising at a depth of 153 m. It then rises at 4.5 m/min for 15 min.a) What is 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?

7. Saida owns 125 shares of an oil company. The value of each share drops by $0.31 on Monday. On Tuesday, the value of each share rises by $0.18. What is the total change in the value of Saida’s shares?

8. Copy and complete each statement.a) ◼ + 1.8 = −3.5 b) −13.3 − ◼ = −8.9 c) ◼ × (−4.5) = −9.45 d) −18.5 ÷ ◼ = 7.4

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9. Competency Check In dry air, the temperature decreases by about 0.65 °C for each 100-m increase in altitude.a) The temperature in a city is 10 °C on a dry day. What is the temperature

2.8 km above the city?b) The temperature 1600 m above the city is −8.5 °C. What is the

temperature in the city? Explain your reasoning to a partner.

10. Two wooden poles measure 1.35 m and 0.83 m in length. A worker makes a new pole by overlapping the ends and tying them together. The length of the overlap is 12 cm. What is the total length of the new pole, in metres? Explain your thinking.

1.35 m

0.83 m12 cm

11. One week in October in Iqaluit, Nunavut, the daily high temperatures are −4.6 °C, −0.5 °C, 1.2 °C, 2.3 °C, −1.1 °C, 1.5 °C, and −3.1 °C. Without using a calculator, do you think the mean daily high temperature for the week is above or below 0 °C? Explain your reasoning.

12. Create a word problem that involves operations with rational numbers in decimal form. Make sure you can solve your problem. Then, have a classmate solve your problem.

Extend

13. Competency Check Four points have the following coordinates:A(0.75, 0.81) B(0.75, −0.65) C(−1.22, −0.65) D(−1.22, 0.81)a) What is the perimeter of quadrilateral ABCD, to the nearest hundredth

of a unit?b) What is the area of quadrilateral ABCD to the nearest tenth of a square unit?

14. Evaluate each expression.a) 3.6 + 2y , y = −0.05 b) (m − 1.8)(m + 1.8) , m = 1.7

c) 4.5 ___ q − q ___ 4.5 , q = −3.6

15. Add one pair of brackets to the left side of each equation to make it a true statement.a) 3.5 × 4.1 − 3.5 − 2.8 = −0.7 b) 2.5 + (−4.1) + (−2.3) × (−1.1) = 4.29 c) −5.5 − (−6.5) ÷ 2.4 + (−1.1) = −0.5

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1.3 Rational Numbers in Fraction Form

Explore and Analyze

Canada officially adopted metric measurement in the 1970s, but many building materials, foods, and other goods are still sold using imperial measures. Imperial measures frequently use fractional measures. For example, many recipes use fractions. Why would you need to use operations on fractions in recipes?

A cake recipe calls for 1 __ 4 cup of milk.

1. If you double the recipe, you could measure 1 __ 4 cup of milk twice. How else could you add the correct amount?

2. a) If you quadruple the recipe, how much milk would you need to add?

b) How could you add the amount from part a) to the other ingredients?c) Which is the most efficient way?

3. Suppose you work at a bakery and you want to make enough batter for 10 cakes.a) Would you measure 1 __ 4 cup of milk 10 times? Explain why or why not.

b) How much milk would you need?c) How would you measure the right amount?

4. The recipe also calls for 1 __ 2 cup of flour. If you have 4 1 __ 2 cups of flour, could you make 10 cakes? Explain your reasoning.

Focus On…In this lesson, I will learn to

• apply multiple strategies to add and subtract rational numbers in fraction form

• apply multiple strategies to multiply and divide rational numbers in fraction form

• solve problems using rational numbers in fraction form

To quadruple means to multiply by 4.



Example 1: Add or Subtract Rational Numbers in Fraction Form

Calculate.

a) 2 __ 5 − (− 1 __ 10 ) b) 3 2 __ 3 + (−1 3 __ 4 )

Solution

a) 2 __ 5 − (− 1 __ 10 )

= 4 __ 10 + 1 __ 10

= 5 __ 10

= 1 __ 2

b) 3 2 __ 3 + (−1 3 __ 4 )

Method 1: Subtract Whole Numbers and Fractions SeparatelySubtract whole numbers: 3 − 1 = 2 Subtract fractions: 2 _ 3 − 3 _ 4

= 8 _ 12 − 9 _ 12

= − 1 _ 12

Add the whole number and fraction together: 2 + (− 1 _ 12 )

= 24 _ 12 − 1 _ 12

= 23 _ 12

= 1 11 _ 12

Method 2: Express Mixed Numbers as Improper Fractions

44 __ 12 − 21 __ 12

= 23 __ 12

= 1 11 __ 12

Calculate.a) − 3 __ 4 − 1 __ 5 b) −2 1 __ 2 + 1 9 __ 10

Show You Know

To subtract a negative number, add the opposite positive number.

Find a common denominator: 2 __ 5 = 4 __ 10

To add a negative number, subtract the corresponding positive number.

Find a common denominator: 2 _ 3 = 8 _ 12 and 3 _ 4 = 9 _ 12

Find a common denominator: 11 __ 3 = 44 __ 12 and 7 __ 4 = 21 __ 12

1.3 Rational Numbers in Fraction Form • MHR 21

Example 2: Multiply or Divide Rational Numbers in Fraction Form

Solve and simplify if necessary.a) 3 __ 4 × (− 2 __ 3 )

b) −1 1 __ 2 ÷ (−2 3 __ 4 )

Solution

a) Multiply the numerators and multiply the denominators.

3 __ 4 × (− 2 __ 3 ) = 3 __ 4 × ( −2 ___ 3 )

= 3 × (−2) _______ 4 × 3

= −6 ___ 12

= −1 ___ 2 or − 1 __ 2

b) Method 1: Multiply by the Reciprocal

−3 ___ 2 ÷ ( −11 ___ 4 ) = −3 ___ 2 × 4 ___ −11

= −12 ___ −22

= 6 __ 11

Method 2: Use a Common Denominator

Write the fractions with a common denominator and divide the numerators.

−1 1 __ 2 ÷ (−2 3 __ 4 ) = − 3 __ 2 ÷ (− 11 __ 4 )

= −6 ___ 4 ÷ ( −11 ___ 4 )

= −6 ___ −11

= 6 __ 11

Solve and simplify if necessary.

a) − 2 __ 5 (− 1 __ 6 ) When there is no sign between a number and a bracket, it means multiply.

b) −2 1 __ 8 ÷ 1 1 __ 4

Show You Know

Remember that you can write a negative fraction

as −1 ____ 2

or (−1)

____ 2

or − ( 1 __ 2

) or 1 _ (−2)

You could remove the common factors of 3 and 2 from the numerator and denominator before

multiplying:

3 1 _ 4

2 × ( −2

−1 _ 3

1 ) = −1 _ 2 or − 1 _ 2

You could remove the common factor of 2 from the numerator and denominator before multiplying:

−3 _ 2 1 × 4

2 _ 11 = 6 _ 11

−6 _ 4 ÷ ( −11 _ 4 ) = −6 ÷ (−11) __ 4 ÷ 4

= −6 ÷ (−11) __ 1

= 6 _ 11


Example 3: Apply Rational Numbers

A “two by four,” or 2 × 4, is a standard building material used throughout Canada. The piece of lumber measures 2 inches by 4 inches and comes in a variety of lengths. After the wood has been dried and planed smooth, it actually measures 1 1 __ 2 inches by

3 1 __ 2 inches , but it is still called a “two by four”! Layering pieces of wood together gives additional strength. This process is called laminating.

You are building a work bench by laminating 2 × 4s together so the top surface is 24 inches wide and 3 1 __ 2 inches thick. How many 2 × 4s will you need?

Solution

How many boards of width 1 1 __ 2 inches do you need to make 24 inches?

1 1 __ 2 × x = 24

x = 24 ÷ 1 1 __ 2

Method 1: Convert the Mixed Number to a Decimal

x = 24 ÷ 1 1 __ 2 = 24 ÷ 1.5 = 16

Method 2: Convert the Mixed Number to an Improper Fraction

x = 24 ÷ 1 1 __ 2

= 24 ÷ 3 __ 2

= 24 8

× 2 _ 3 1

= 8 × 2 = 16

You will need 16 laminated 1 1 __ 2 -inch boards to make a work bench with a width of 24 inches.

A soup recipe calls for 1 1 __ 4 cups of broth. How many batches of soup could you

make with 10 cups of broth?

Show You Know


Connect and Reflect

Key Ideas

• You can add, subtract, multiply, or divide rational numbers expressed as fractions using the same rules as for rational numbers expressed as decimals.

• When working with rational numbers, it is often helpful to estimate first by using close whole numbers.

• When adding and subtracting, it is sometimes easier to change mixed numbers to improper fractions.

Practise


1. Calculate.

a) 3 __ 10 + 1 __ 5

b) 2 1 __ 3 + (−1 1 __ 4 )

c) − 5 __ 12 − 5 __ 12

d) −2 1 __ 2 − (−3 1 __ 3 )

e) − 5 __ 6 + 1 __ 3

f) 3 __ 8 − (− 1 __ 4 )

2. Calculate.

a) 2 __ 3 − 3 __ 4

b) − 2 __ 9 + (− 1 __ 3 )

c) − 1 __ 4 + (− 3 __ 5 )

d) − 3 __ 4 − (− 5 __ 8 )

e) 1 1 __ 2 − 2 1 __ 4

f) 1 2 __ 5 + (−1 3 __ 4 )


3. Calculate.

a) 4 __ 5 ÷ 5 __ 6

b) (3 1 __ 3 ) (1 3 __ 4 )

c) 1 __ 8 × (− 2 __ 5 )

d) − 9 __ 10 ÷ (− 4 __ 5 )

e) − 3 __ 8 × 5 1 __ 3

f) 1 __ 10 ÷ (− 3 __ 8 )

4. Calculate.

a) − 3 __ 4 × (− 1 __ 9 )

b) 1 1 __ 3 ÷ 1 1 __ 4

c) − 3 __ 8 ÷ 7 __ 10

d) −2 1 __ 8 ÷ 1 1 __ 4

e) ( 7 __ 9 ) (− 6 __ 11 )

f) −1 1 __ 2 ÷ (−2 1 __ 2 )


Apply

For help with this section, refer to Example 3 on page 23.

5. Lori owes her mother $39. Lori pays back 1 __ 3 of this debt and then pays back 1 __ 4 of the remaining debt. How much does Lori still owe her mother?

6. A recipe calls for 2 __ 3 cup of butter. If the recipe is quadrupled, express the amount of butter needed asa) an improper fraction b) a mixed number

7. A carpenter has 64 feet of baseboard. He installs 1 __ 2 of the baseboard in one room. He installs 3 __ 4 of the remaining amount of baseboard in another room. How much baseboard does he have left?

8. The table uses positive numbers to show how many hours the time in a location is ahead of the time in London, England. Negative numbers show how many hours the time is behind the time in London.

Location Time Zone

Alice Springs, Australia +9 1 __ 2

Brandon, Manitoba −6

Chatham Islands, New Zealand +12 3 __ 4

Istanbul, Turkey +2

Kathmandu, Nepal +5 3 __ 4

London, England 0

Mumbai, India +5 1 __ 2

St. John’s, Newfoundland and Labrador −3 1 __ 2

Tokyo, Japan +9

Victoria, British Columbia −8

a) How many hours is the time in St. John’s ahead of the time in Brandon?b) How many hours is the time in Victoria behind the time in Mumbai?c) On any given day, which location will experience sunrise first, Istanbul or

St. John’s? Explain why.d) In which location is the time exactly halfway between the times in

Istanbul and Alice Springs?e) If the time is 3:30 p.m. in London, what time is it in Victoria?

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9. Predict the next three numbers in each pattern.

a) −1 1 __ 2 , − 7 __ 8 , − 1 __ 4 , 3 __ 8 , …

b) 1 1 __ 3 , − 2 __ 3 , 1 __ 3 , − 1 __ 6 , 1 __ 12 , …

10. Draw a semicircle in your notebook. Imagine travelling counterclockwise around the outside of the semicircle. Indicate the location of each of the following fractional distances.

a) 1 __ 2 b) 1 __ 4 c) 1 __ 3 d) 1 __ 6

e) 3 __ 4 f) 2 __ 3 g) 5 __ 6 h) 2 __ 5

11. Taj has three scoops for measuring flour. The largest scoop holds 2 1 __ 2 times as

much as the smallest one. The middle scoop holds 1 3 __ 4 times as much as the smallest one. Describe two ways in which Taj could measure each of the following quantities if he can only use full scoops.

a) 3 1 __ 4 times as much as the smallest scoop holds

b) 1 __ 2 as much as the smallest scoop holds

12. Competency Check a) Write a subtraction statement involving two negative fractions or negative

mixed numbers so that the difference is − 4 __ 3 .

b) Write an addition, a multiplication, and a division statement with the same answer.

c) Compare your statements with a classmate’s. How are they alike? How are they different?

13. In a magic square, the sum of each row, column, and diagonal is the same. Copy and complete this magic square.

− 1 __ 2

− 5 __ 6

1 __ 2

−1 1 __ 6

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Extend

14. Competency Check Suppose the sum of two rational numbers is less than each of the rational numbers and that neither of these numbers are integers.a) Which of the following statements could be true?

A Both numbers are positive.B Both numbers are negative.C One number is positive and the other is negative.

b) Give an example of a pair of numbers that either supports or refutes each statement. Explain your thinking.

15. Refer to #10. Draw a line segment from the centre of the circle to the right edge of the semicircle. Label this 0°, as shown below.

0°

For each mark that you placed on the semicircle, draw a line segment to the centre of the circle.

0°

1––4

Copy and complete the table that is started below.

Distance Travelled Around the Semicircle 1 __ 2 1 __ 4

Angle Formed

16. For what value(s) of x does x − 1 __ x = 1 1 __ 2 ?

17. You multiply a fraction by − 1 __ 2 , then add 3 __ 4 , then divide by − 1 __ 4 , and get an

answer of −3 3 __ 4 . What was the original fraction?

18. Can the sum of two rational numbers be less than both of the rational numbers? Explain using examples in fraction form.

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1.4 Order of Operations With Rational Numbers

In Canada, contestants need to answer a skill testing question in order to collect their prize. This is because Canadian laws ban games of chance, with the exception of lotteries and those held at casinos. Companies figured out that if contestants need to answer a question to get their prize, it is no longer a game of chance. For some reason, these skill testing questions are almost always math questions! Why do calculations with three or more operations need to be done in a specific order?

Explore and Analyze

1. Answer the following question using the order of operations. 2 + 8 × (5 − 22) ÷ 2 − 6

2. Will you get the same answer if you ignore the order of operations, and just solve #1 by making each calculation from left to right as you encounter it? Explain why or why not.

3. a) Write a skill testing question that includes rational numbers.

b) Solve the question using the order of operations.c) Solve the question ignoring the order of operations.

Did you get the same answer? Explain why or why not.

4. Exchange questions with a partner and solve their question using the correct order of operations.

Focus On …In this lesson, I will learn to

• use the order of operations to solve problems involving rational numbers

Use the same order of operations for rational numbers that you use with whole numbers

and integers.



Example: Order of Operations With Rational Numbers

Solve.

a) 12 ÷ 4 × √ ______

3 __ 4 + 1 1 __ 2 + ( 5 __ 8 − 1 __ 16 ) − 1 3 __ 8

b) −0.7 + [2.2(1.58 − 3.12)] + √ _____________

12.5 + (−3.5)

Solution

a) Follow the order of operations.

12 ÷ 4 × √ ______

3 __ 4 + 1 1 __ 2 + ( 5 __ 8 − 1 __ 16 ) − 1 3 __ 8

= 12 ÷ 4 × √ ___

2 1 __ 4 + 9 __ 16 − 1 3 __ 8 BRACKETS Treat what is under the square root symbol as if it is inside brackets. Also do the subtraction in the brackets.

= 12 ÷ 4 × 1 1 _ 2 + 9 __ 16 − 1 3 __ 8 EXPONENTS Treat the square root symbol as if it is an exponent.

= 4 1 __ 2 + 9 __ 16 − 1 3 __ 8 DIVIDE and/or MULTIPLY Do these operations in the order that they appear.

= 9 __ 2 + 9 __ 16 − 11 __ 8 ADD and/or SUBTRACT Write fractions with a common denominator and do these operations in the order that they appear.

= 72 __ 16 + 9 __ 16 − 22 __ 16

= 59 __ 16

= 3 11 __ 16

b) −0.7 + [2.2(1.58 − 3.12)] + √ _____________

12.5 + (−3.5) = −0.7 + [2.2 × (−1.54)] + √

__ 9 BRACKETS

= −0.7 + [−3.388] + 3 BRACKETS and EXPONENTS

= −1.088 ADD

Solve using the order of operations.a) 15 ÷ (−2.5) + √

_____ 6.25 − 32

b) [1 1 __ 2 + ( 3 __ 4 − 1 __ 2 ) − √ __

1 __ 4 ] × 4

Show You Know

√ ___

2 1 __ 4 = √ __

9 __ 4 = √

__ 9 __

√ __

4 = 3 __ 2 = 1 1 __ 2

12 ÷ 4 × 1 1 _ 2 = 3 × 3 _ 2 = 9 _ 2 = 4 1 _ 2

Find a common denominator for 2, 16, and 8.

All of the multiplication was

done inside the brackets.

1.4 Order of Operations With Rational Numbers • MHR 29

Connect and Reflect

Key Ideas

• Use the same order of operations on rational numbers as you use with whole numbers and integers. ▪ Calculate what is inside the BRACKETS first. Treat what is under the square root symbol

as if it is inside brackets. ▪ Solve EXPONENTS next. Treat the square root symbol as if it is an exponent. ▪ DIVIDE or MULTIPLY next, in the order that they appear from left to right. ▪ Finally, ADD or SUBTRACT in the order that they appear from left to right. ▪ The acronym BEDMAS will help you remember the order.

Practise

For help with #1 and #2, refer to the Example on page 29.

1. Solve.

a) 10 [ ( 1 __ 2 + 1 __ 4 ) + 2 ( 1 __ 8 ) ] ÷ 2

b) √ ______________

(0. 6 2 ) + (0. 8 2 )

c) ( 1 __ 5 − 3 __ 5 ) × √ ______

6 × 3 __ 2 + ( √ ___

36 ÷ √ __

5 2 )

d) [−1.5 + √ _____

0.25 − (−0.75)] × 24

2. Solve. Express your final answer as a fraction in lowest terms or as a decimal rounded to the nearest hundredth.

a) (2 1 __ 3 + 1 __ 2 − 1 __ 12 ) − 2 ( 1 __ 3 + 1 __ 4 ) − √ __

1 __ 9

b) −4.9 (−12 + √ _____

0.81 ) − (2.2) 2 × 1.59

c) [ ( −7 ___ 10 − 1 __ 2 ) + 4 ( 1 __ 5 + 1 __ 4 ) ] 2

d) 1 − √ ____________________

6.35 − 5(1.44 − 0.78)

Apply

3. Convert each fraction in #2c) to a decimal and redo the question. Did you get the same answer? Explain why.

4. Competency Check Ben works at a local coffee shop. On New Year’s Day, which is a statutory holiday, Ben was paid time and a half. This means that he was paid 1 1 __ 2 times his regular $15/h pay rate. He worked from 7:00 a.m. to 2:30 p.m. How much did Ben earn?

5. Nicki decides to bake three different desserts. The recipes call for 2 1 __ 2 cups,

3 __ 4 of a cup, and 3 cups of flour, respectively.

a) How much flour does Nicki need to make the three desserts?

b) If she triples the recipe that calls for 3 __ 4 of a cup, what is the total amount of flour needed for the three desserts?

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6. A company makes sheets of plywood. A sheet made up of 8 layers is 3 __ 4 inches thick. How thick is a sheet of plywood made up of 6 layers?

7. Leslie evaluated −2.2 + 4.6 × (−0.5) as −1.2. Zack evaluated the same expression as −4.5. Who is correct? Explain your answer.

Extend

8. Some artists’ panels are laminated, 7-layer boards. Many have a centre core made of five layers of material and two thin wood veneer outer panels.

Design a 7-layer board that is a total of 1 __ 2 -inch thick and has a 7 __ 8 -inch centre core. Make a sketch and include the thickness of all layers.

9. In her last basketball game, Jacqui made 3 __ 7 of her free throws. In the game before that, she made 5 __ 8 of her free throws. Jacqui believes that she made 8 __ 15 of her total free throws. Is Jacqui right or wrong? Explain your reasoning.

10. Simplify √ _________________________________

14.58 − 3.5 (1.14 + 2.69) + (−9.44) . Explain why it is not possible to find a final answer.

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1.4 Order of Operations With Rational Numbers • MHR 31

1. a) When you multiply two fractions, the answer is 4 __ 7 . What are two possible fractions?

b) When you multiply two fractions, the answer is close to 1. What are two possible fractions?

2. If a U.S. dollar is worth $1.30 in Canadian dollars, what is a Canadian dollar worth in U.S. dollars? Explain your reasoning.

3. Use the four numbers 3, 4, 5, and 7 to create two fractions that give the biggest and smallest answers when multiplied and divided.

4. First Peoples often used Bentwood boxes to carry materials during canoe trips. The sides of the boxes are made from a single piece of wood that is steamed so it bends to make the edges. The bottom is made from a separate piece of wood that is pegged to the sides.a) Draw three different Bentwood boxes that could each

hold 24 litres of berries. Label the dimensions of each box. What is the length of the piece of wood needed to make the sides of each box? Assume the box is 20 cm tall.

b) Imagine you have a piece of wood that is 36 units long. What different sizes of boxes could you design using the entire piece of wood? Include the bottom of the box, but not the top. What are the dimensions of the largest box you could make with 20 cm sides?

5. The table shows the number of points you get for shading different areas on a 10 × 10 grid.Shade a 10 × 10 grid to try to obtain the greatest number of points given the following restrictions:• Squares cannot overlap.• You may only use the squares listed in the chart.• You may use the same size square a maximum of 3 times.

Area of Square Points

4 10

9 20

16 30

25 45

36 50

49 65

64 80

81 100

1 __ 4 1 __ 8

1 __ 9 1 __ 5

Rich Problems


Learning Goals

Inquire and Explore: What is the relationship between the different operations?How can you use the order of operations to solve problems involving rational numbers?How can you use mental math to estimate reasonable answers?

After this section, I can

1.1 ▪ compare and order rational numbers ▪ determine the square or the square root of a rational number using estimation and technology

1.2 ▪ perform operations with decimal numbers ▪ use mental math to estimate reasonable answers

1.3 ▪ apply multiple strategies to add and subtract rational numbers in fraction form ▪ apply multiple strategies to multiply and divide rational numbers in fraction form ▪ solve problems using rational numbers in fraction form

1.4 ▪ use the order of operations to solve problems involving rational numbers

1.1 Introduction to Rational Numbers, pages 6–13

1. In your notebook, replace each ◼ with >, <, or = to make the statement true.

a) −9 ___ 6 ◼ 3 ___ −2 b) −0.86 ◼ −0.84

c) − 3 __ 5 ◼ −0. _

6 d) −1 3 __ 10 ◼ − ( −13 ___ −10 )

e) − 8 __ 12 ◼ − 11 __ 15 f) −2 5 __ 6 ◼ −2 7 __ 8

2. Determine whether each rational number is a perfect square. Explain your reasoning.

a) 64 _ 121 b) 7 _ 4

c) 0.49 d) 1.6

3. Estimate √ _

220 to one decimal place. Describe your method.

4. On a number line, would you find −3 5 __ 11 to the left or to the right of −3.4545? Explain how you know.

5. Find two fractions in lowest terms between 0 and −1 that have 5 as the numerator.

6. Canada’s Heather Moyse and Kaillie Humphries won the gold medal in the two-women bobsleigh competition at the Sochi Winter Olympics. Their total time after 4 runs was 3 minutes 50.61 seconds. They beat the second-place finishers from the United States by 10 ___ 100 of a second. What was the American team’s total time?

7. a) Which temperature is colder, 3 __ 4 °C or 0.7 °C? Show your thinking.

b) Which temperature is colder, −3 ___ 4 °C or −0.7 °C? Show your thinking.

Chapter 1 Review

Chapter 1 Review • MHR 33

8. Suppose a 1-L can of paint covers 11 m2.a) How many cans of paint would you need to

paint a ceiling that is 5.2 m by 5.2 m? Show your work.

b) Determine the maximum dimensions of a square ceiling you could paint with 4 L of paint. Express your answer to the nearest tenth of a metre.

1.2 Rational Numbers in Decimal Form, pages 14–19

9. Calculate.a) −5.68 + 4.73 b) −0.85 − (−2.34) c) 1.8(−4.5) d) −3.77 ÷ (−2.9)

10. Evaluate. Express your answer to the nearest tenth, if necessary. a) 5.3 ÷ (−8.4) b) −0.25 ÷ (−0.031) c) −5.3 + 2.4[7.8 + (−8.3)] d) 4.2 − 5.6 ÷ (−2.8) − 0.9

11. One evening in Prince George, BC, the temperature decreased from 2.4 °C to −3.2 °C in 3.5 h. What was the average rate of change in temperature per hour?

12. Over a 4-year period, a company lost an average of $1.2 million per year. The company’s total losses by the end of 5 years were $3.5 million. What was the company’s profit or loss in the fifth year?

1.3 Rational Numbers in Fraction Form, pages 20–27

13. Simplify.

a) 2 __ 3 − 4 __ 5 b) − 3 __ 8 + (− 3 __ 4 )

c) −3 3 __ 5 + 1 7 __ 10 d) 2 1 __ 3 − (−2 1 __ 4 )

14. Evaluate.

a) − 1 __ 2 (− 8 __ 9 ) b) − 5 __ 6 ÷ 7 __ 8

c) 2 3 __ 4 × (−4 2 __ 3 ) d) −4 7 __ 8 ÷ (−2 3 __ 4 )

15. How many hours are there in 2 1 __ 2 weeks ? Show your work using decimals and fractions.

16. The area of Manitoba is about 1 1 __ 5 times the total area of the four Atlantic provinces. The area of Yukon Territory is about 3 __ 4 the area of Manitoba. Express the area of Yukon Territory as a fraction of the total area of the Atlantic provinces.

17. Without doing any calculations, compare the values of the following two quotients. Explain your reasoning.

96 7 _ 8 ÷ 7 3 _ 4

−96 7 _ 8 ÷ (−7 3 _ 4 )

1.4 Order of Operations With Rational Numbers, pages 28–31

18. Calculate.

a) ( 1 __ 7 + 1 __ 3 ) ÷ ( 1 __ 3 − 1 __ 7 ) + 1 1 __ 4

b) √ __________________________

4 + 0. 5 2 × 12 + (20 × 0.1)

c) 9.7 + 4.9 − 20.5 × 5.2

d) (2 2 _ 3 ) ( 2 _ 5 ) + (− 2 _ 5 ) ÷ 3 _ 8

e) 1.3 × 2.5 + 5.6 × (−2.5) ÷ 1.4

f) √ ______

3.4 + 2.3 − (0.4 × 5.5) + 5.5

19. A Canadian quarter is made from nickel, copper, and steel. The quarter is 11 ___ 500 nickel ,

19 ___ 500 copper , and 47 __ 50 steel .

a) Estimate the sum of the three fractions. Justify your answer.

b) Test your estimation by calculating the sum of the three fractions.


c) How many times as great is the mass of the steel as the combined mass of the nickel and copper?

d) The mass of a Canadian quarter is 4.4 g. In a roll of 40 quarters, how much greater is the mass of copper than the mass of nickel?

20. Axel, Bree, and Caitlin were comparing −1 1 __ 2 and −1 1 __ 4 .

a) Axel first wrote the two mixed numbers as improper fractions. Describe the rest of his method.

b) Bree first wrote each mixed number as a decimal. Describe the rest of her method.

c) Caitlin first ignored the integers and wrote − 1 __ 2 and − 1 __ 4 with a common denominator. Describe the rest of her method.

d) Which method do you prefer? Explain.

21. The membership fee at a gym is $51.25 per month. If you pay for one year in advance, there is a discount of $5 per month. How much would it cost for one year if you paid all at once?

22. Maya had $200.78 in her bank account. Over the next two months she made four deposits of $63.75. She took out the following amounts over the same time period: $47.10, $25.91, $102.00, and $58.43. a) How much money is in her account at the

end of the two months?b) If Maya puts 1 _ 4 of her remaining money in a

savings account, how much will that be?c) If Maya gives 1 _ 2 of the money left in her

account to a charity, how much will that be?

Connect the Concepts

23. Find the area of a rectangle with length 7.8 cm and width 2.03 cm.

24. A construction company can dig 5.9 m3 per hour. A new tool will allow the company to dig 8 times as fast. How fast will the company be able to dig with the new tool?

25. You need to buy a refrigerator for your apartment. The chart shows the details for two different sizes of fridge. Fridge B costs 3 times as much as Fridge A. Which fridge would you buy? Justify your answer.

Area of Base (m2)

Height (m)

Price ($)

Fridge A 1 __ 2 1 100

Fridge B 3 __ 4 1.5 300

26. The temperature at 2:30 a.m. was −10.6 °C. At noon, the temperature had risen 16.8°. At 5 p.m., the temperature had dropped 8.7°.a) Without using a calculator, do you think

the temperature at 5 p.m. was above or below 0°?

b) What was the temperature at 5 p.m.?

27. The chart shows the cost of school supplies.

Item Cost ($)

Notebook 3.39

Package of pencils 2.79

Three-ring binder 5.99

Gillian buys 6 notebooks, 2 packages of pencils, and 1 three-ring binder.a) Write an expression to find the total amount

Gillian spent on school supplies.b) Calculate the after-tax cost of Gillian’s

purchases.

Chapter 1 Review • MHR 35

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