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Rational NumbersA

Multi-Part Lesson

12-1PART B C

Main IdeaCompare and order fractions, decimals, and percents.

NGSSSMA.7.A.5.1 Express rational numbers as

terminating or repeating decimals.

Compare and Order Rational NumbersINSECTS The most common types of insects are shown in the table.

1. Which of the insects is the longest?

2. Is the fire ant or housefly longer? Explain.

3. Are more than or less than half of the insects shown longer than 0.6 inch?

4. Order the lengths from shortest to longest.

Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers.

Compare Decimals

Replace the � with <, >, or = to make a true statement.

-1.2 � 0.8

Graph the decimals on a number line.

1.4-1.4 1.210.80.60.40.2-1.2 -1 -0.8-0.6-0.4-0.2 0

Since -1.2 is to the left of 0.8, -1.2 < 0.8.

-2.40 � -2.45

Graph the decimals on a number line.

-2.50 -2.48 -2.46 -2.44 -2.42 -2.40

Since -2.40 is to the right of -2.45, -2.40 > -2.45.

a. 3.1 � -3.7 b. -4.5 � -4.49

Insect Length(in.)

Bumble bee 3 _ 4

Cricket 1 _ 1

Fire ant 1 _ 3

Firefly 3 _ 4

Housefly 1 _ 4

Japanese beetle 1 _ 2

Mosquito 5 _ 8

Compare Fractions

Replace the � with <, >, or = to make a true statement.

-

7 _

8 �

6 _

8

Since the denominators are the same, compare the numerators by graphing on a number line.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 76 8

Since -7 < 6, - 7 _

8 <

6 _

8 .

-

3 _

8 � -

5 _

16

Rename the fractions using the least common denominator.

- 3 _

8 = -

3 × 2 _

8 × 2 = -

6 _

16 -

5 _

16 = -

5 × 1 _

16 × 1 = - 5 _

16

Since -6 < -5, - 6 _

16 < -

5 _

16 and -

3 _

8 < -

5 _

16 .

Check

616-

516-

416-

316-

216-

116- 0

�

c. 9 _

16 � -

12 _

16 d. -

7 _

10 � -

4 _

5

Compare Decimals and Fractions

Replace the � with <, >, or = to make a true statement.

-0.51 � -

8 _

15

Rename - 8 _

15 as a decimal. Then graph both decimals on a

number line.

-0.6 -0.55

-0.53 -0.51

-5 -0.5 -0.45 -0.4 -

8 _ 15

= -0.5 − 3

Since -0.51 is to the right of -0.5 −

3 on the number line,

-0.51 > - 8 _ 15

.

Replace the � with <, >, or = to make a true statement.

e. -3 5 _ 8 � -3.625 f.

3 _

7 � 0.413

Least Common Multiple (LCM)Find the LCM of 8 and 16.8: 8, 16, 2416: 16, 32, 48The LCM is 16.

To order rational numbers, first write them in the same form and then order the numbers.

Order Rational Numbers

SCIENCE Mr. Plum’s science class is Difference from

Average Plant Growth

Student Difference(in.)

Ricky 3 1 _ 4

Debbie -2.2

Suni 1.7

Leonora -1 7 _ 10

growing plants under different conditions. The average plant growth during a week was 5.5 inches. The table shows the difference from the average for some students’ plants. Order the differences from least to greatest.

Express each number as a decimal.

Ricky’s plant: 3 1 _ 4 = 3.25 Debbie’s plant: -2.2

Suni’s plant: 1.7 Leonora’s plant: -1 7 _ 10

= -1.7

Graph each number on a number line.

-3.5-3 -2-2.5

-2.2-1.7 1.7 3.25

-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5

From least to greatest, the differences are -2.2, -1 7 _ 10

, 1.7, and 3 1 _ 4 .

g. OCEANS The average depths of the three deepest oceans are -2.46 miles, -2.88 miles, and -2.44 miles. Order the depths from least deep to deepest.

Real-World LinkThe longest species of seaweed measures 197 feet. This kelp is the fastest growing species, growing more than 13 inches per day.

Replace the � with <, >, or = to make a true statement.Examples 1–5(pp. 677–678)

1. -7.8 � -7.82 2. 9.7 � -10.3 3. 5 _ 8 � -

3 _

8

4. - 7 _

12 � 5 _

12 5. -6.7 � -6 7 _

10 6. -

5 _

6 � -0.94

Example 6(p. 679)

7. MONEY Stephen recorded the following amounts in his checkbook: -$6.50, $7.00, -$6.75, and $7.25. Order these amounts from least to greatest.

Order the following numbers from least to greatest.

8. -3 1 _ 3 , 3.3, -3 3 _

4 , 3.5 99 2.8, -2 3 _

4 , 3 1 _

8 , -2.

− 2

= Step-by-Step Solutions begin on page R31. Extra Practice is on page EP32.

Replace the � with <, >, or = to make a true statement.Examples 1–5(pp. 677–678)

10. -2.5 � -2.9 11. -18.6 � -18.06 12. -4.08 � -4.7

13. - 4 _ 9 � - 5 _

9 14. 5 _

4 � - 1 _

4 15. - 3 _

7 � - 2 _

5

16. -6 1 _ 3 � -6.375 17. - 3 _

5 � -0.6 18. -9 2 _

7 � -9.3

Example 6(p. 679)

19. STOCK MARKET The change in four stocks during Company Change

A -4 1 _ 2

B 5.6

C -2 3 _ 8

D 1.35

a day are shown. Order the changes from least to greatest.

20. IN-LINE SKATING The average amount of time Brent spent in-line skating for one week was 34 minutes. During the next week, the difference between the average time and actual time spent skating was

4.2 minutes, -5 1 _ 3 minutes, -2 1 _

2 minutes, and 3.75 minutes.

Order these differences from least to greatest.

Replace the � with <, >, or = to make a true statement.

B

21. -0. −

4 � -0. −−

45 22. -2. −−

21 � -2. −−−

212 23. -6. −

4 � -6 4 _ 9

Order the following numbers from least to greatest.

24. 2. −

1 , -2.1, 2 1 _ 11

, -2 25. 1 _ 8 , -0.02

− 5 , 0.2, - 1 _

7

26. 2 _ 3 , 60%, 0.65, 4 _

5 27. 125%, 1 3 _

4 , 1.2

− 5 , 1 1 _

5

28. SNACKS The bar graph shows the

Loss

Prof

it

$0.00$2.00

-$2.00

$4.00

-$8.00-$6.00-$4.00

$6.00$8.00

Day321 4

Profit or Lossprofit or loss of the after-school snack stand.

a. Write each profit as a positive number and each loss as a negative number.

b. Order the numbers from least to greatest.

2929 MODEL ROCKETS For a technology Trials

Difference BetweenAverage Height and

Actual Height (ft)

1 -10.8

2 11.4

3 9.0

4 -9.7

competition, Julienne constructed a model rocket. The average height the rocket can reach is 545 feet. The table shows the differences between the average height and the actual height reached. Order the differences from least to greatest.

C 30. REASONING Determine whether the following statement is always, sometimes, or never true. Give examples to justify your answer.

If x and y are both greater than zero and x > y, then -x < -y.

31. NUMBER SENSE Determine whether the fractions - 4 _

5 , -

4 _

6 , -

4 _

7 , and -

4 _

8

are arranged in order from least to greatest. Explain.

32. Explain why -0.33 is greater than -0. −−

33 .

Write each fraction as a decimal. (Lesson 12-1B)

37. 5 _ 9 38. -

1 _

8 39. 2 _

3

40. Find the value of x ÷ y if x = -15 and y = -3. (Lesson 11-3E)

41. BUSINESS The table shows the results of a company’s Month Profit Loss

April $15,000

May $19,000

June $12,000

operations over a period of three months. Find the company’s overall profit or loss over this time period. (Lesson 11-2B)

PracticeMA.6.A.5.2, MA.7.A.5.1

NGSSS

33. Which point shows the approximate

location of - 1 _

3 ?

-0.4 -0.3 -0.1-0.2 0 0.1 0.2 0.3 0.4

A. point A

B. point B

C. point C

D. point D

34. GRIDDED RESPONSE Which of the following numbers is the greatest?

-0.73, 0.32, -0.21, 0.19

35. SHORT RESPONSE Student Council’s goal was to raise $50 each week for 4 weeks to have enough money for the school dance. The table shows the difference between the goal and the actual amount raised. Order these amounts from least to greatest.

Week 1 2 3 4

Difference ($) 5.50 -6.25 7.80 -2.45

36. Which of the following numbers is less

than - 2 _

3 ?

F. 0.6 H. 0. −

6

G. 1 _ 3 I. -

5 _

6