Chapter 3: Decimals

1

CHAPTER 3: DECIMALS

Contents

CHAPTER 3: DECIMALS .......................................................................................................... 1

3.1 INTRODUCTION TO DECIMALS ..................................................................................... 2

A. WHAT IS A DECIMAL? ............................................................................................. 2

B. REPRESENTING DECIMAL FRACTIONS ............................................................... 5

C. PLACE VALUE AND DECIMALS ............................................................................. 5

EXERCISES ........................................................................................................................ 9

SECTION 3.2: OPERATIONS WITH DECIMALS ............................................................... 13

A. ADDING AND SUBTRACTING DECIMALS ........................................................... 13

B. MULTIPLYING DECIMALS ..................................................................................... 14

C. MULTIPLYING A DECIMAL BY A POWER OF TEN .............................................. 14

D. DIVIDING DECIMALS ............................................................................................. 15

EXERCISES ...................................................................................................................... 18

3.3 FRACTION AND DECIMAL CONNECTIONS ................................................................ 22

A. CONVERTING FRACTIONS TO DECIMALS.......................................................... 22

B. CONVERTING DECIMALS TO FRACTIONS ......................................................... 23

EXERCISES ......................................................................................................................... 26

Chapter 3: Decimals

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3.1 INTRODUCTION TO DECIMALS

A. WHAT IS A DECIMAL?

Decimals are a different way of representing fractions. In fact, each place value of a decimal represents a different fraction whose denominator is a power of ten.

Just like 234 can be written as

2 ∙ 100 + 3 ∙ 10 + 4 ∙ 1,

the decimal number 0.234 can be written as

2 ∙1

10+ 3 ∙

1

100+ 4 ∙

1

1000.

In this section we will develop the idea of a decimal by writing and representing them in numerous ways.

MEDIA LESSON Representing the Tenths Place using the Area Model (Duration 5:16)

View the video lesson, take notes and complete the problems below. The square below represents the unit. Using the tick marks, draw vertical lines to partition the unit into equal pieces.

a) How many equal pieces did you partition the square into? ________

b) If the square is the unit, what fraction number represents each piece? ________

c) If the square is the unit, what word name represents each piece? ________

d) Shade 3 of the equal parts with an orange highlighter. What fraction number represents

the shaded area? ________

e) What fraction number represents the area that is not shaded? ________

Chapter 3: Decimals

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MEDIA LESSON Representing the Hundredths Place using the Area Model (Duration 5:51)

View the video lesson, take notes and complete the problems below. The square below represents the unit. Using tick marks, draw vertical lines and horizontal lines to partition the unit into equal pieces.

a) How many pieces did you partition the square into? ___________ b) If the big square is the unit, what fraction number represents each small square piece? What

word name represents each piece? ___________

c) Shade 30 small squares with a yellow highlighter. What fraction number represents the

shaded area? _____________________________________

d) Compare this grid to the grid in the previous. What relationship do you see between the area

shaded orange on your first grid and the area shaded yellow on your second grid?

______________________________________________________________________

e) 1 orange is how many times as large as a yellow? ___________

f) 1 yellow is what part of an orange? ___________

Chapter 3: Decimals

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MEDIA LESSON Tenths and Hundredths Grids (Duration 9:46)

View the video lesson, take notes and complete the problems below. The big square represents the unit. Shade the following quantities on the grids below. Then write the quantities in terms of orange strips and yellow squares, the fraction word name and the fraction number name. 1) 6 out of 10 equal parts or 0.6

(Use orange strips as unit fraction)

a) Number of orange strips: _____________

b) Fraction Number: ___________________

c) Fraction word name: ________________

d) Equivalent number of yellow squares: ______

2) 40 out of 100 equal parts or 0.40 (Use yellow squares as unit fraction)

a) Number of yellow squares: _____________

b) Fraction Number: ___________________

c) Fraction word name: ________________

d) Equivalent number of orange strips: ______

3) 37 out of 100 equal parts or 0.37

(Use yellow squares as unit fraction)

a) Number of yellow squares: _____________

b) Fraction Number: ___________________

c) Fraction word name: ________________

d) Equivalent number of orange strips: ______

4) 5 out of 10 or 0.5 and 3 out of 100 or

0.03 equal parts (Use both orange strips and yellow squares)

a) Number of orange strips and yellow squares:

___________________________________

b) Fraction Number: ___________________

c) Fraction word name: ________________

d) Equivalent number of orange strips: ______

Chapter 3: Decimals

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B. REPRESENTING DECIMAL FRACTIONS

MEDIA LESSON Creating Benchmarks for Plotting Decimals (Duration 6:33)

View the video lesson, take notes and complete the problems below. The number line below is partitioned in fourths (or quarters). Use the given tick marks to approximate all of the decimals to the tenths place between -1 and 1 on the number line. Label the points underneath the number line.

C. PLACE VALUE AND DECIMALS

Recall that our number system is a base-10 number system. This means that 10 of a certain place value equals 1 of the next biggest place value.

1 one = 1

10 ten

1 ten = 1

10 hundred

1 hundred = 1

10 thousand

1 thousand = 1

10 ten thousand

Equivalently, we can say that 1 of a certain place value equals 1

10 of the next biggest place value.

1 tenth = 1

10 one

1 hundredth = 1

10 tenth

1 thousandth = 1

10 hundredth

The place value chart shows this relationship including the tenths, hundredths, and thousandths places.

Chapter 3: Decimals

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MEDIA LESSON Introduction to Decimals (Duration 8:03)

View the video lesson, take notes and complete the problems below. Decimal notation is used to write numbers according to place value in base -10. A decimal point is used to separate whole numbers from numbers less than 1.

It costs $2.89 per gallon of gas. The runner finished in 10.3 seconds.

The Down was up 124.6 points today.

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Ten T

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usand

ths

100,000 1,000 100 10 1 ● .1 .01 .001 .0001

1

10

1

100

1

1000

1

10,000

Read the following number using place values. Write the decimal in using fraction notation.

1. 6.3

2. 52.12

3. 1.125

Comparing using >, <, or =.

1) 5.219 5.22

2) 0.105 0.1050

3) 18.9 18.895

Order each list from least to greatest.

1) −5.8, 3.1, −3,04, 3.11, 3.009

2) 12.01, 12.0098, 12.011, 12.008

Chapter 3: Decimals

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Rounding Decimal Numbers

Procedure

1. Identify the round-off place digit.

2. If the digit to the right of the round-off place digit is:

Less than 5, do not change the round of place digit.

5 or more, increase the round-off place digit by 1.

3. In either case, drop all digits to the right of the round-off place digit.

Round each number to the indicated place value

1. 12.1258 to the hundredths

2. 5.1249 to the tenths

3. 16.098274 to the ten thousandths

YOU TRY:

a) Order the numbers from least to greatest: 2.8, 2.08, 2.88, 2.088, 2.008, 2.808, 0.28

.

.

.

.

.

.

.

b) Order from smallest to largest: 4.25, 0.425, 4.05, 4.2, 4.5

.

.

.

.

.

Chapter 3: Decimals

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c) Round 3.24 to the nearest tenth.

d) Round 0.073 to the nearest hundredth.

e) Round 5.076 to the nearest tenth.

f) Round 22.2965 to the nearest thousandth.

Chapter 3: Decimals

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EXERCISES

1) Use the give number lines to plot the following decimals.

a) Plot 0.3, 0.8, 0.5, 𝑎𝑛𝑑 0.9 on the number line below. Label the points underneath the number line.

b) Plot 1.8, 0.2, 1.1, 𝑎𝑛𝑑 2.7 on the number line below. Label the points underneath the number line.

2) Use the place value chart to order the numbers from least to greatest

2.8, 2.08, 2.88, 2.088, 2.008, 2.808, 0.28

_____________________________________________________

Ones . Decimals

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Chapter 3: Decimals

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3) Place the following numbers in order from smallest to largest.

0.2, 0.25, 0.74, 0.7, 0.40, 0.08

4) Order the signed decimals below using the symbols <, =, or >.

a) 0.45______0.54 b) 0.308______0.038

c) 0.32______0.99 d) 3.005______3.05

e) 0.33______0.3 f) 0.48______0.4800

5) Put the numbers in the place value chart. Use the place value chart as an aid to round the number to the indicated place value.

a) Round 8.53 to the nearest tenth.

b) Round 186.485 to the nearest whole number.

c) Round 5.283 to the nearest hundredth.

d) Round 139.081 to the nearest tenth.

e) Round 78.165 to two decimal places f) Round 8.53 to the ones place.

g) Round 186.485 to the nearest tenth.

Ones . Decimals

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Tenth

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Chapter 3: Decimals

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Round the results of the application problems so that it makes sense in the context of the

problem

5) Amy is buying ribbon for an art project. She estimates that she will need 3.34 meters of

ribbon. The store only sells ribbon by the tenth of a meter. How many meters should she

buy?

6) John is catering a luncheon and needs 12.37 pounds of sugar. If sugar is only sold in one

pound bags, how many bags should John buy?

7) Shelly is buying shoes online and computes that she has enough money to buy 2.78 pairs

of shoes. How many pairs of shoes can she buy?

8) Tia is making a work bench for her art studio. She measures the space and needs 8.24

meters of plywood. The store only sells plywood by the tenth of a meter. How many meters

should Tia buy?

9) Jamie is running a booth at the local fair. She computes that she needs to sell 86.25 snow

cones that day to make a profit. Since she can only sell a whole number of snow cones,

how many does she need to sell to make a profit?

Crystal is buying Halloween candy at the store. She has $20 and wants to buy as many bags

of candy as possible. She computes that she has enough to buy 6.91 bags of candy. How

many bags of candy can she buy?

Check your work with the answer key!

Chapter 3: Decimals

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Online Quiz

Log on to Canvas to take the section quiz

Directions: It is very useful to save your math exercise work and use it as a chapter test

review when you study for your chapter test and final.

1) Write each question on the screen down to for your record

2) Solve the problem step by step below each question

3) Double check your work to see whether your answer make sense

4) Enter your answer in the answer box in Canvas. Make sure you click on the “Preview”

button to make sure you enter the right format before you submit your answer. If you are

not sure how to enter your answer with the correct format, ask your instructor.

5) If you did not answer the question correctly, solve the question again from the beginning

below your 1st attempt. Sometimes, it is better to start a problem again from the beginning

and compare your steps with your 1st attempt to figure out your mistake.

6) Insert your work at the end of each section in your workbook so that you can use it to study

for your chapter test later.

Chapter 3: Decimals

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SECTION 3.2: OPERATIONS WITH DECIMALS

A. ADDING AND SUBTRACTING DECIMALS

MEDIA LESSON Add and subtract decimals (Duration 5:35)

View the video lesson, take notes and complete the problems below. Procedure

1. Write the numbers vertically and line up the decimal points. Add zeros to the right as

needed so each number has the same number of digits to the right of the decimal points.

2. Bring the decimal point down into the sum or difference.

3. Add or subtract as you normally would.

Example:

5.2 + 7 + 12.85 + 0.1248

75.98 + 1000.6 + 19 + 5.215

121.75 – 87.125

3 – 1.007

YOU TRY:

Perform the operations indicated below. Be sure to show your work.

a) 21.456 − 8.89

b) 437.9 + 52.438

c) 15.397 + 6.91

d) 0.56 − 0.24

e) 5.09 + 62.784

f) 32.456 + 7.98

Chapter 3: Decimals

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B. MULTIPLYING DECIMALS

MEDIA LESSON Multiply decimals (Duration 7:54)

View the video lesson, take notes and complete the problems below. Procedure

1. Multiply the numbers just as you would multiply whole numbers.

2. Find the sum of the decimal places in the factors.

3. Place the decimal point in the product so that the product has the same number of

decimal places as the sum of the decimal places. You may need to write zeros to the left

of the number.

Example: Multiply.

5.2 × 0.2

12.3 × 0.15

0.002 × 15.78

C. MULTIPLYING A DECIMAL BY A POWER OF TEN

To multiply a decimal by a power of 10, move the decimal point to the right the same number

of places as the number of zeros in the power of 10. It may be necessary to add zeros at the

end of the number.

Example: 5.378 × 100.

Why?

Multiply.

78.3 × 1,000

10,000 × 12.34985

Chapter 3: Decimals

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YOU TRY:

Multiply the numbers by the given powers of 10 by moving the decimal point the appropriate number of places. a) 1.126 ∙ 100 = __________

b) 0.049 ∙ 1000 = __________

c) 5.7 ∙ 10 = __________ d) 3.1415 ∙ 1000 = __________

Multiply the decimals.

e) 1.4 ∙ 3 =

f) 1.4 ∙ 0.3 =

g) 0.14 ∙ 0.3 =

h) 0.14 ∙ 0.03 =

D. DIVIDING DECIMALS

MEDIA LESSON Dividing decimals (Duration 9:11)

View the video lesson, take notes and complete the problems below.

i. Dividing a decimal by a whole number:

1. Place the decimal point in the answer directly above the decimal point in the dividend.

2. Divide as if there were no decimal point involve.

Example:

15.2 ÷ 4

42.2 ÷ 3

Chapter 3: Decimals

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ii. Dividing a decimal by a decimal

1. If the divisor is a decimal, change it to a whole number by moving the decimal point to

the right as many places as necessary.

2. Then move the decimal point in the dividend to the right the same number of places.

3. Place the decimal point in the answer directly above the decimal point in the dividend.

4. Divide until the remainder becomes zero or the remainder repeats itself, or the desired

number of decimal places it achieved.

→

67.08 ÷ 2.6

12 ÷ 0.36

iii. Dividing a decimal by a powers of ten

MEDIA LESSON Dividing decimal by Powers of Ten (Duration 5:09)

View the video lesson, take notes and complete the problems below. Divide the numbers by the given powers of 10 on your calculator then look for patterns to make a general strategy.

a) 4.23 ÷ 10 = __________

b) 3.7 ÷ 1000 = __________

c) 29.5 ÷ 100 = __________

d) 3.1415 ÷ 1000 = ______

e) 5.24 ÷ 10 = __________

f) 0.67 ÷ 100 = __________

g) Look for patterns in the examples above and complete the statement below.

To divide a decimal number by a power of 10, you move the decimal

place_______________________________________________________________________

___________________________________________________________________________

__________________________________________________________________________

Chapter 3: Decimals

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YOU TRY:

Divide the decimals.

a) 2.4 ÷ 0.8 =

b) 0.24 ÷ 0.8 =

c) 0.42 ÷ 0.07

d) 4.2 ÷ 0.7

Divide the numbers by the given powers of 10 by moving the decimal point the appropriate places.

e) 1.126 ÷ 100 = ________

f) 4.9 ÷ 1000 = _________

g) 5.7 ÷ 10 = __________

Chapter 3: Decimals

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EXERCISES

Perform the following operations with decimals. Show your work.

1) 43.136 + 21.823 2) 526.209 + 497.055

3) 37.528 − 23.106 4) 254.023 − 88.58

5) 279.381 − 102.16 6) 520 − 39.866

7) 2.1 ∙ 4 8) 2.1 ∙ 0.4

9) 0.21 ∙ 0.4 10) 0.05 ∙ 0.09

11) 0.42 ÷ 0.21 12) 4.2 ÷ 2.1

13) 0.42 ÷ 21 14) 24 ÷ 0.02

Chapter 3: Decimals

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Multiply or divide the numbers by the given powers of 10 by moving the decimal point the

appropriate number of places. Show your work.

15) 5.327 ∙ 100 = ____ 16) 1.002 ∙ 1000 = ___ 17) 3.14 ∙ 10 = ____

18) 32.81 ÷ 100 = ___ 19) 5 ÷ 1000 =____ 20) 53.91 ÷ 10 = ____

21) Sylvia just received her monthly water usage data from her local water department. For the

past 6 months, her water used (in thousands of gallons) was 19.9, 25.6, 28.8, 22.5, 20.3, and

19.2. What was her average usage during this time? (Round to the nearest tenth)

22) Glenn normally earns $8.50 per hour in a given 40-hour work-week. If he works overtime, he

earns time and a half pay per hour. During the month of October, he worked 40 hours, 50

hours, 45 hours, and 42 hours for the four weeks. How much did he earn total for October?

23) Dave is making a gazebo for his yard. He has a piece of wood that is 13 feet long and he

needs to cut it into pieces of length 1.5 inches. How many pieces of this size can he cut from

the 13 foot piece of wood?

Chapter 3: Decimals

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24) Callie ordered 4 items online. She is charged $2.37 per pound per shipping. The items

weighed 3.2 lbs., 4.6 lbs., 9.2 lbs. and 1.5 lbs. How much will be charged for shipping?

(Round to the nearest cent).

25) Mary’s son wants to go on the Gadget’s Go Coaster ride at Disneyland. The height

requirement for the ride is 35 inches. He is 29.3 inches tall now. How many inches more does

he need in order to get into the ride?

Check your work with the answer key!

Chapter 3: Decimals

21

Online Quiz

Log on to Canvas to take the section quiz

Directions: It is very useful to save your math exercise work and use it as a chapter test

review when you study for your chapter test and final.

1) Write each question on the screen down to for your record

2) Solve the problem step by step below each question

3) Double check your work to see whether your answer make sense

4) Enter your answer in the answer box in Canvas. Make sure you click on the “Preview”

button to make sure you enter the right format before you submit your answer. If you are

not sure how to enter your answer with the correct format, ask your instructor.

5) If you did not answer the question correctly, solve the question again from the beginning

below your 1st attempt. Sometimes, it is better to start a problem again from the beginning

and compare your steps with your 1st attempt to figure out your mistake.

6) Insert your work at the end of each section in your workbook so that you can use it to study

for your chapter test later.

Chapter 3: Decimals

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3.3 FRACTION AND DECIMAL CONNECTIONS

A. CONVERTING FRACTIONS TO DECIMALS

i. Method 1: Convert the denominator to a power of 10, then write the number using the

correct place value.

MEDIA LESSON Converting a Fraction to a Decimal (Power of 10) (Duration 5:50)

View the video lesson, take notes and complete the problems below.

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100,000 1,000 100 10 1 ● .1 .01 .001 .0001

1

10

1

100

1

1000

1

10,000

Example: Convert the fraction to the decimal by converting the denominator to a Power of 10.

a) 2

100

b) 15

1000

c) 7

25

d) 6

5

e) 303

250

f) 9

2000

ii. Method 2: Perform long division. Remember a fraction bar is a division symbol.

Chapter 3: Decimals

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𝑎

𝑏 = 𝑎 ÷ 𝑏 ⇒ 𝑏

MEDIA LESSON Converting a fraction to a decimal (long division) (Duration 6:05)

View the video lesson, take notes and complete the problems below. Convert the fractions to decimals.

a) 7

25 b)

2

5

c) 7

8

d) 3

11

B. CONVERTING DECIMALS TO FRACTIONS

MEDIA LESSON Converting decimals to fractions (Duration 5:10)

View the video lesson, take notes and complete the problems below.

When we rewrite a decimal as a simplified fraction, we will start by writing it as a fraction

based on its place value, a power of ten. Observe that 10’s prime factorization is 2 ∙ 5. So

any power of 10 is just a product of 2’s and 5’s. This will make the process of simplification

easier because we will only have to check the numerator for factors of 2’s and 5’s.

Complete the table below. Show all of your work for simplifying the fraction.

Decimal Fraction Simplified Fraction

a) 0.8

b) 0.65

c) 0.44

d) 0.002

Chapter 3: Decimals

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YOU TRY:

Convert the following fractions to decimals using the indicated method below. Show all of your work.

Fraction Powers of 10 method Long division method

a) 7

25

b) 3

5

c) 3

8

d) 11

20

Convert the following decimals to fractions and simplify your answer.

Decimal Fraction Simplified Fraction

e) 0.6

f) 0.85

g) 0.042

Chapter 3: Decimals

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h) 0.1275

Chapter 3: Decimals

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EXERCISES

Complete the table below. Show all of your work for simplifying the fraction.

Decimal Fraction Simplified Fraction

1) 0.8

2) 0.24

3) 0.95

4) 0.006

Write each fraction in decimal form. Round to the nearest thousandth as needed.

11) 33

100

Decimal: _________

12) 308

10

Decimal: _________

13) 81

10

Decimal: _________

14) 5

100

Decimal: _________

15) 400

100

Decimal: _________

16) 3

1000

Decimal: _________

Chapter 3: Decimals

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Convert the following fractions to decimals using the indicated method below. Show all

of your work.

Fraction Powers of 10 method if possible Long division method

17) 5

25

18) 7

5

19) 22

7

20) 5

20

Check your work with the answer key!

Chapter 3: Decimals

28

Online Quiz

Log on to Canvas to take the section quiz

Directions: It is very useful to save your math exercise work and use it as a chapter test

review when you study for your chapter test and final.

1) Write each question on the screen down to for your record

2) Solve the problem step by step below each question

3) Double check your work to see whether your answer make sense

4) Enter your answer in the answer box in Canvas. Make sure you click on the “Preview”

button to make sure you enter the right format before you submit your answer. If you are

not sure how to enter your answer with the correct format, ask your instructor.

5) If you did not answer the question correctly, solve the question again from the beginning

below your 1st attempt. Sometimes, it is better to start a problem again from the beginning

and compare your steps with your 1st attempt to figure out your mistake.

6) Insert your work at the end of each section in your workbook so that you can use it to study

for your chapter test later.