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NSSAL i Draft

©2012 C. D. Pilmer

Decimal Numbers and

Percent Unit

(Pilot Materials)

NSSAL

(Draft)

C. David Pilmer

2013

(Last Updated: November, 2013)

NSSAL ii Draft

©2012 C. D. Pilmer

This resource is the intellectual property of the Adult Education Division of the Nova Scotia

Department of Labour and Advanced Education.

The following are permitted to use and reproduce this resource for classroom purposes.

Nova Scotia instructors delivering the Nova Scotia Adult Learning Program

Canadian public school teachers delivering public school curriculum

Canadian non-profit tuition-free adult basic education programs

The following are not permitted to use or reproduce this resource without the written

authorization of the Adult Education Division of the Nova Scotia Department of Labour and

Advanced Education.

Upgrading programs at post-secondary institutions (exception NSCC's ACC)

Core programs at post-secondary institutions

Public or private schools outside of Canada

Basic adult education programs outside of Canada

Individuals, not including teachers or instructors, are permitted to use this resource for their own

learning. They are not permitted to make multiple copies of the resource for distribution. Nor

are they permitted to use this resource under the direction of a teacher or instructor at a learning

institution.

Acknowledgments

The Adult Education Division would like to thank Dr. Genevieve Boulet (MSVU) for reviewing

this resource and providing valuable feedback.

The Adult Education Division would also like to thank the following ALP instructors for piloting

this resource and offering suggestions during its development.

Eileen Burchill (IT Campus)

Lynn Cuzner (Marconi Campus)

Carissa Dulong (Truro Campus)

Krys Galvin (Truro Campus)

Barbara Gillis (Burridge Campus)

Nancy Harvey (Akerley Campus)

Barbara Leck (Pictou Campus)

Suzette Lowe (Lunenburg Campus)

Shelly Meisner (IT Campus)

Alice Veenema (Kingstec Campus)

NSSAL iii Draft

©2012 C. D. Pilmer

Table of Contents

Introduction (for Learners) ……………………………………………………………….. iv

Prerequisite Knowledge …………………………………………………………………… v

Introduction (for Instructors) ……………………………………………………………… vi

Introduction to Decimal Numbers ………………………………………………………… 1

Comparing Decimals ……………………………………………………………………… 6

Equivalent Fractions and Decimals ……………………………………………………….. 13

Introduction to Percent ……………………………………………………………………. 17

Comparing Fractions, Decimals and Percentages ………………………………………… 25

Adding and Subtracting Decimal Numbers ………………………………………………. 29

Multiplying Decimal Numbers ……………………………………………………………. 36

Dividing Decimal Numbers ……………………………………………………………….. 42

Estimation Questions Involving Percentages ………………………..……………………. 53

Calculator Questions ………………………………………………………………………. 57

Appendix …………………………………………………………………………………… 61

Connect Four Fraction Decimal Equivalency Game ………………………………….. 62

Connect Four Fraction Percent Equivalency Game …………………………………… 63

Connect Four Percentage Game ………………………………………………………. 64

Answers ………………………………………………………………………………... 65

NSSAL iv Draft

©2012 C. D. Pilmer

Introduction (for Learners)

Welcome to the Decimal Numbers and Percent Unit. Like the Fractions Unit, we will initially

spending a bit of time understanding what a decimal and percent are and how to order decimals

and percentages from smallest to largest. This understanding is very important before we try to

introduce operations (i.e. addition, subtraction, multiplication, and division) with decimals and

percents.

Prerequisite Knowledge

This unit was written under the assumption that learners understand the concepts covered in the

Level III Whole Number Operations Unit and Level III Fractions Unit. We will be revisiting

many of the concepts addressed in those units.

The expectation for this unit is that learners are comfortable with:

The addition, subtraction, and multiplication of multi-digit numbers.

e.g. 198 35 e.g. 928 294 e.g. 251 58

Divide a multi-digit number by a single digit number.

e.g. 8456 7

Ordering fractions from smallest to largest without using a calculator.

e.g. Order 6

210

, 1

18

, 1

16,

12

7,

12

12,

3

16,

42

5, and

8

9 from smallest to largest.

Introduction (for Instructors)

This unit is similar to the Fraction Unit in that learners initially spend a significant amount of

time understanding the magnitude of decimals and percentages, before ever completing

operations with decimals and percentages. We actually tap into much of the information and

understanding that the learners acquired in the Fractions Unit; hence, it is a prerequisite for this

unit. Please ensure that learners arrange fractions, decimals and percentages from smallest to

largest before they proceed to the sections involving operations.

NSSAL 1 Draft

©2012 C. D. Pilmer

Introduction to Decimal Numbers

We see decimal numbers everywhere.

Money: "Yoshi deposited $210.75 into his bank account."

Measurement: "The distance from my house to work is 14.3 kilometres."

"The container holds 1.57 litres of fluid."

"The package weighs 6.8 kilograms."

"The property is 0.85 acres in size."

"The winning time in the 100 metre dash was 9.72 seconds."

"When we started the experiment, the fluid was at 18.2oC."

Probability: "The probability of obtaining a head when flipping a fair coin is 0.5."

Statistics: "The mean weight (i.e. average weight) of males in the class is 85.2 kg."

Decimals are just another way of writing fractions, and vice versa.

For example, the fraction 1

10 ("one-tenth"), which is represented by the

area model on the right, can also be written in its decimal form as 0.1

("zero decimal one").

For example, the fraction 1

100 ("one-hundredth"), which is represented

by the area model on the right, can also be written in its decimal form as

0.01 ("zero decimal zero one").

Fractional Form Decimal Form

7

10 ("seven tenths") 0.7 ("zero decimal seven")

81

100 ("eighty-one hundredths') 0.81 ("zero decimal eight one")

3

100 ("three hundredths") 0.03 ("zero decimal zero three")

417

1000 ("four hundred seventeen thousandths") 0.417 ("zero decimal four one seven")

79

1000 ("seventy-nine thousandths") 0.079 ("zero decimal zero seven nine")

NSSAL 2 Draft

©2012 C. D. Pilmer

Please note that many math resources and math teachers will say that decimal numbers should be

read in the same manner as their fractional counterparts.

e.g. 0.7 should be read as "seven tenths."

e.g. 0.81 should be read as "eighty-one hundredths."

e.g. 0.417 should be read as "four hundred seventeen thousandths."

The rationale for this approach from people in the education community is that it forces learners

to understand place value, and therefore conveys a deeper level of understanding. However,

mathematicians disagree with this approach stating that decimals like 0.7 should be read as "zero

decimal seven" because it clearly conveys to the listener that we are dealing with a decimal,

rather than its equivalent fraction. In this resource, we are going to follow the practices of the

mathematicians.

Place Value and Decimals

Millions

Period

Thousands

Period

Ones

Period

Hundre

d M

illi

ons

Ten

Mil

lions

Mil

lions

Hundre

d T

housa

nds

Ten

Thousa

nds

Thousa

nds

Hundre

ds

Ten

s

Ones

Ten

ths

Hundre

dth

s

Thousa

ndth

s

Ten

-Thousa

ndth

s

Some decimals are larger than 1.

28.93 "twenty-eight decimal nine three"

Fractional Form: 93

28100

("twenty-eight and ninety-three hundredths")

Expanded Form: 20 8 0.9 0.03

or

1 1

2 10 8 1 9 310 100

4319.2 "four thousand, three hundred nine decimal two"

Fractional Form: 2

430910

("four thousand three hundred nine and two tenths")

Expanded Form: 4000 300 9 0.2

or

1

4 1000 3 100 9 1 210

NSSAL 3 Draft

©2012 C. D. Pilmer

7.065 "seven decimal zero six five"

Fractional Form: 65

71000

("seven and sixty-five thousandths')

Expanded Form: 7 0.06 0.005

or

1 1

7 1 6 5100 1000

25.304 "twenty-five decimal three zero four"

Fractional Form: 304

251000

("twenty-five and three hundred four thousandths")

Expanded Form: 20 5 0.3 0.004

or

1 1

2 10 5 1 3 410 1000

Questions

1. What decimal numbers are represented by each of these area models?

(a)

Answer: ________

(b)

Answer: ________

(c)

Answer: ________

(d)

Answer: ________

(e)

Answer: ________

(d)

Answer: ________

NSSAL 4 Draft

©2012 C. D. Pilmer

2. Circle the adjoining numbers that are equivalent decimals and fractions.

92

10 0.05

5

10 0.5

46

1000 0.46

81

10

0.07

7

100 0.087

87

100 0.046

81

100 1.8

527

1000 0.0527

37

1000 0.037

67

100 0.13

8

10

0.009

9

100 0.09

37

100 6.7

13

1000 1.26

643

1000 2.07

72

10 0.27

93

100 1.3

261

100

3.64

72

1000 2.7

93

1000 3.09

56

1000 0.126

6043

1000 3.604

4063

1000 0.39 3

9

10 0.056

56

100

3. Express each fraction in its decimal form.

(a) 256

1000 (b)

6

100 (c)

93

10

(d) 97

11000

(e) 7

21000

(f) 58

13100

4. Express each decimal in its fractional form. Do not put the fraction in its simplest form.

(a) 0.95 = (b) 0.4 = (c) 4.508 =

(d) 1.08 = (e) 2.003= (f) 6.059 =

5. Write the decimal equivalent to each of the following.

(a) thirty-five and six tenths _____________

(b) seven and nine hundredths _____________

(c) fifty-eight thousandths _____________

(d) one thousand and fifteen hundredths _____________

(e) two hundred six and three hundred nine thousandths _____________

(f) seventy and one tenth _____________

NSSAL 5 Draft

©2012 C. D. Pilmer

(g) five and thirty-seven thousandths _____________

(h) four hundred and twenty-nine thousandths _____________

6. Write each decimal as a fraction, using both numerals and words. A completed example has

been provided.

Fraction

Decimal Numerals Words

e.g. 2.04

42

100 two and four hundredths

(a) 32.8

(b) 0.472

(c) 13.067

(d) 7.59

(e) 327.09

7. Write each decimal number in both expanded forms.

(a) 42.8

(b) 9.31

(c) 302.429

(d) 18.034

(e) 4209.07

NSSAL 6 Draft

©2012 C. D. Pilmer

Comparing Decimals

In this section we will be comparing decimal numbers, and in a few instances comparing decimal

numbers to fractions. You will have to remember the strategies we used in the Fraction Unit that

you completed earlier.

There are two techniques that we would like you to learn for comparing decimals.

1. Benchmarks

When we were working with fractions, we used the benchmarks 0, 1

2, and 1 to gauge the

size of fractions. We will do the same for decimals.

Examples of Decimals that

are Close to Zero


are Close to One Half


are Close to One

0.1

1i.e.

10

0.08 8

i.e. 100

0.017 17

i.e. 1000

0.6 6

i.e. 10

0.46 46

i.e. 100

0.519 519

i.e. 1000

0.9 9

i.e. 10

0.94 94

i.e. 100

1.004 4

i.e. 11000

Example 1

Order the numbers 0.53, 1.1 and 0.091 from smallest to largest.

Answer:

0.53 53

or 100

is close to one half.

1.1 1

or 110

is close to one.

0.091 91

or 1000

is close to zero.

Order from Smallest to Largest: 0.091, 0.53, 1.1

Example 2

Order the numbers 0.952, 0.489, 9

8 , 0.07, and

9

16 from smallest to largest.

Answer:

The question involves decimals and fractions, but we can use the benchmark strategy

with all of these numbers.

NSSAL 7 Draft

©2012 C. D. Pilmer

0.952 952

or 1000

is close to one (slightly less than one).

0.489 489

or 1000

is close to one half (slightly less than one half).

9

8 is close to one (slightly more than one).

0.07 7

or 100

is close to zero.

9

16 is close to one half (slightly more than one half).

Order from Smallest to Largest: 0.07, 0.489, 9

16, 0.952,

9

8

2. Comparing Digits

Start on the left of both numbers and compare corresponding digits. If the digit of one

number is larger, then this is the larger decimal number. If the digits are the same, move one

place to the right and repeat the procedure. In some cases, you might want to add additional

zeros to the decimal number for comparison purposes (e.g. 0.54 = 0.540).

Example 3

Which is larger?

(a) 1.6 or 1.4

(b) 0.576 or 0.582

(c) 2.95 or 2.9

Answers:

(a) Step 1: Start on the left and compare the unit digits

Same

1.6 1.4

Step 2: Move one place to the right (to the tenths) and compare the digits

Different

1.6 1.4

Step 3: Since the 6 is bigger than the 4, we can conclude that 1.6 is larger than 1.4

NSSAL 8 Draft

©2012 C. D. Pilmer

(b) Step 1: Unit digits are the same.

Same

0.576 0.582

Step 2: Tenths digits are the same.

Same

0.576 0.582

Step 3: Hundredths digits are different.

Different

0.576 0.582

Step 4: Since the 8 is bigger than the 7, we can conclude that 0.582 is larger than

0.576.

(c) Step 1: Add a zero to 2.9 such that both numbers have three digits

Step 2: Unit digits are the same.

Same

2.95 2.90

Step 3: Tenths digits are the same.

Same

2.95 2.90

Step 4: Hundredths digits are different.

Different

2.95 2.90

Step 5: Since the 5 is bigger than the 0, we can conclude that 2.95 is larger than 2.9.

NSSAL 9 Draft

©2012 C. D. Pilmer

Questions

Do not use a calculator for any of these questions.

1. For each of the following decimal numbers, indicate whether it is closer to 0, 1

2, or 1.

Decimal Closest

to:

Decimal Closest

to:

Decimal Closest

to:

(a) 0.6 (b) 0.1 (c) 0.9

(d) 0.08 (e) 0.45 (f) 1.01

(g) 0.502 (h) 0.89 (i) 0.12

(j) 0.901 (k) 0.005 (l) 0.486

(m) 0.892 (n) 0.59 (o) 0.092

2. In each case, you are given two numbers. Circle the larger number. You will have to use the

benchmark strategy because every question deals with both decimals and fractions.

(a) 0.89 1

12 (b) 0.56

7

8

(c) 4

8 0.1 (d)

3

32 0.907

(e) 0.451 3

3 (f) 0.879

5

12

(g) 13

12 0.009 (h)

7

16 0.58

3. In each case, you are given two numbers. Circle the larger number.

(a) 0.7 0.3 (b) 0.47 0.52

(c) 0.198 0.192 (d) 1.24 1.04

(e) 2.04 2.4 (f) 0.09 0.078

(g) 3.1 3.098 (h) 0.57 0.507

NSSAL 10 Draft

©2012 C. D. Pilmer

(i) 5.61 5.618 (j) 2.09 1.98

(k) 7.029 7.08 (l) 12.899 12.988

(m) 0.4 0.409 (n) 31.29 31.3005

(o) 3.01 2.999 (p) 7.5 7.809

(q) 15.35 15.2 (r) 0.75 0.739

(s) 15

16 0.44 (t)

1

8 0.81

(u) 0.51 1

10 (v) 2.04

52

6

(w) 1

42

4.7 (x) 11

312

3.009

4. Place the following numbers by the appropriate arrow on the number line below.

1.4, 1.9, 0.6, 2.7, 0.1, 3.1, 0.8, 2.2, 1.3, 2.5


0.54, 2.89, 2.4, 0.097, 1.46, 1.039, 2.62, 1.75, 3.05, 1.95

0 2 1 3

0 2 1 3

NSSAL 11 Draft

©2012 C. D. Pilmer


2.78, 7

10 , 3.3, 1.57,

12

16, 0.07,

43

100,

3

8, 1.2, 2.44

7. Order the following numbers from smallest to largest.

(a) 0.9, 1.3, 0.4, 1.6, 0.2

_________, _________, _________, _________, _________

(b) 0.59, 1.23, 0.08, 0.55, 1.14

_________, _________, _________, _________, _________

(c) 0.8, 0.09, 0.52, 1.01, 0.83, 1.1

_________, _________, _________, _________, _________, _________

(d) 0.26, 0.19, 1, 0.98, 0.3, 0.2

_________, _________, _________, _________, _________, _________

(e) 0.2, 0.08, 0.72, 0.006, 0.24, 0.209

_________, _________, _________, _________, _________, _________

(f) 0.64, 0.7, 0.05, 0.78, 0.619, 0.092, 0.4

_________, _________, _________, _________, _________, _________, _________

(g) 0.542, 0.9, 7

100, 0.85,

1

2, 0.862, 0.3

_________, _________, _________, _________, _________, _________, _________

0 2 1 3

NSSAL 12 Draft

©2012 C. D. Pilmer

(h) 43

11000

, 0.16, 8

8, 0.201, 0.6, 1.3, 0.649

_________, _________, _________, _________, _________, _________, _________

(i) 0.48, 1.002, 1

16, 0.509,

51

8, 0.4, 1.1

_________, _________, _________, _________, _________, _________, _________

NSSAL 13 Draft

©2012 C. D. Pilmer

Equivalent Fractions and Decimals

We are aware that decimals are just another way of expressing fractions, and vice versa.

However, there are some common equivalent fractions and decimals that we should all know off-

the-top-of-our-heads.

Fractions with a denominator of 2

1

0.52

21

2

3 11 1.5

2 2

42

2

Notice the resulting sequence: 0.5, 1, 1.5, 2,…

(As we go up by 1

2 with the fractions, the corresponding decimals go up by 0.5)


10.333...

3

0.3

20.666...

3

0.6

31

3

4 11 1.333...

3 3

1.3

Many of these result in repeating decimals. The line above a digit indicates that digit

repeats indefinitely.

Notice the resulting sequence: 0.3 , 0.6 , 1, 1.3 ,…

(As we go up by 1

3 with the fractions, the corresponding decimals go up by 0.333… or

0.3 .)


1

0.254

20.5

4

30.75

4

41

4

Notice the resulting sequence: 0.25, 0.5, 0.75, 1, 1.25,…

(As we go up by 1



1

0.25

20.4

5

30.6

5

40.8

5

Notice the resulting sequence: 0.2, 0.4, 0.6, 0.8, 1,…

(As we go up by 1


NSSAL 14 Draft

©2012 C. D. Pilmer

Example 1:

Order the numbers 3

5, 0.58,

2

3, 0.46, 1.3,

1

20,

51

9 0.962 from smallest to largest.

Answer:

It is important not to convert all the fractions to their decimals equivalents. Such conversions

work in some cases, but not in all. Do not forget to use benchmarks.

3

5 can be converted to 0.6, its decimal equivalent.

2

3 can be converted to 0.666.. or 0.6 , its decimal equivalent.

1

20 is close to the benchmark 0.

5

19

is close to (and slightly more than) the benchmark 1

12

.

Appropriate Order: 1

20, 0.46, 0.58,

3

5,

2

3, 0.962, 1.3,

51

9

Questions

Do not use a calculator to complete any of these questions.

1. For each of the following decimals, state the equivalent fraction or mixed number.

(a) 0.75 = (b) 3.5 = (c) 1.8 =

(d) 2.3 = (e) 7.2 = (f) 5.6 =

2. For each of the following decimals, state the equivalent fraction or mixed number.

(a) 1

94 (b)

2

5 (c)

16

3

(d) 3

74 (e)

19

2 (f)

28

3

3. In each case, you are given two numbers. Circle the larger number.

(a) 3

5 0.65 (b)

3

4 0.875

(c) 1

13

1.09 (d) 1

22

2.6

(e) 2

43

4.59 (f) 4

65

6.78

NSSAL 15 Draft

©2012 C. D. Pilmer

(g) 1

24

2.304 (h) 1

75

7.19


3

14

, 2.069, 2

5, 0.9,

13

5 1.44,

72

8,

1

3, 3.37, 2.539,

1

25


2

13

, 1.87, 3.12, 95

100, 0.098,

9

16, 2.2,

13

4, 2.43,

21

5, 2.71

6. Order the following numbers from smallest to largest.

(a) 3

25

, 1

14

, 1.6, 0.09, 93

100, 2.4,

111

12,

5

8

_________, _________, _________, _________, _________, _________, _________, _________

(b) 1

13

, 9

210

, 0.587, 3

14

,1

216

, 1.58, 2.7, 41

100

_________, _________, _________, _________, _________, _________, _________, _________

0 2 1 3

0 2 1 3

NSSAL 16 Draft

©2012 C. D. Pilmer

(c) 1

24

, 9

16, 2.011, 0.892,

11

10, 2.6,

191

20,

4

5

_________, _________, _________, _________, _________, _________, _________, ________

7. Open-ended Questions (i.e. more than one acceptable answer)

Your Answer

(a) Provide a decimal number that is between 3.4 and 3.5.

(b) Provide a mixed number that is between 2.5 and 2.8.

(c) Provide a decimal number that is between 1

15

and 1

13

(d) Provide a decimal number that is between 2

3 and

9

10

(e) Provide a mixed number that is between 3.1 and 3.3.

(f) Provide a decimal number that is between 4

25

and 3.01

(Have your instructor check your answers to question 7.)

8. With a classmate, friend, or family member, play at least two rounds of the Connect Four

Fraction Decimal Equivalency Game found in the appendix of this resource. Record in the

chart below whom you played and who won.

Opponent Winner

Round #1

Round #2

NSSAL 17 Draft

©2012 C. D. Pilmer

Introduction to Percent

We see percentages everywhere.

The union negotiated a 2% wage increase for this year.

The dress is marked 30% off.

Approximately 70% of the class is female.

Babe Ruth, who played professional baseball from 1914 to 1935, hit a homerun 11.76%

of the time at bat.

Candice left a tip of 20% for the exceptional service she received at the restaurant.

The mortgage rate on Lei's condominium is 5.25% per annum.

The word percent comes from the Latin phrase per centum, which means

"per 100." For example, when one says 13%, it means 13 per 100 and

can be represented by the fraction 13

100, or by the decimal 0.13 . The

area model for this particular percentage is shown on the right; 13 of the

100 equal parts are shaded.

Percentages are just another way of expressing fractions or decimals; they all mean the same

thing but look slightly different.

Percent Fraction

(or Mixed Number)

Decimal

7%

7

100 0.07

23%

23

100 0.23

89%

89

100 0.89

109%

109 91

100 100 1.09

16.7%

16.7 167

100 1000 0.167

0.1%

0.1 1

100 1000 0.001

25%

25 1

100 4 0.25

80%

80 4

100 5 0.80

150%

150 50 11 1

100 100 2 1.50

NSSAL 18 Draft

©2012 C. D. Pilmer

Converting Percentages to Decimals

Simply remove the percent sign and slide the decimal point two places to the left (i.e. divide by

100).

Example 1

Convert the following percentages to decimals.

(a) 68% (b) 135% (c) 15.9%

Answers:

(a) 68% 68. 0.68 Therefore: 68% = 0.68

(b) 135% 135. 1.35 Therefore: 135% = 1.35

(c) 15.9% 15.9 0.159 Therefore: 15.9% = 0.159

Converting Decimals to Percentages

Simply slide the decimal point two places to the right (i.e. multiply by 100) and add the percent

sign.

Example 2

Convert the following percentages to decimals.

(a) 0.52 (b) 2.68 (c) 0.743

Answers:

(a) 0.52 52. 52% Therefore: 0.52 = 52%

(b) 2.68 268. 268% Therefore: 2.68 = 268%

(c) 0.743 74.3 74.3% Therefore: 0.743 = 74.3%

Remove percent sign Slide decimal point two places to the left

Remove percent sign

Remove percent sign

Slide decimal point two places to the left

Slide decimal point

two places to the left

Add a percent sign Slide decimal point

two places to the right

Slide decimal point two places to the right

Slide decimal point two places to the right

Add a percent sign

Add a percent sign

NSSAL 19 Draft

©2012 C. D. Pilmer

Converting Percentages to Fractions

Simply drop the percent sign, express as fraction with a denominator of 100, and simplify the

fraction if necessary.

Example 3

Convert the following percentages to fractions (or mixed numbers).

(a) 43% (b) 65% (c) 108%

(d) 7.9% (e) 0.6% (f) 216.4%

Answers:

With questions (d) through (f), we initially have fractions with decimals in them. We do not

leave the number in this form. If we multiply the numerator and denominator by 10, we can

rectify this problem.

(a) 43% 43

100

(b) 65% 65 65 5 13

100 100 5 20

(c) 108% = 108 8 8 4 2

1 1 1100 100 100 4 25

(d) 7.9% 7.9 7.9 10 79

100 100 10 1000

(e) 0.6% 0.6 0.6 10 6 6 2 3

100 100 10 1000 1000 2 500

(f) 216.4% 216.4 16.4 16.4 10 164 164 4 41

2 2 2 2 2100 1000 100 10 1000 1000 4 250

Equivalent Fractions, Decimals, and Percentages

In the previous section, we examined equivalent fractions and decimals; we are going to expand

on this slightly by also including equivalent percentages.


1

0.5 50%2

21 100%

2

31.5 150%

2

42 200%

2


1

0.3 33.3%3

20.6 66.6%

3

31 100%

3

41.3 133.3%

3


1

0.25 25%4

20.5 50%

4

30.75 75%

4

41 100%

4

NSSAL 20 Draft

©2012 C. D. Pilmer


1

0.2 20%5

20.4 40%

5

30.6 60%

5

40.8 80%

5

Questions

Calculators are not permitted for any of these questions.

1. For each of the area models below, supply the corresponding percent, decimal, and fraction.

(a) (b) (c)

Percent: Percent: Percent:

Decimal: Decimal: Decimal:

Fraction: Fraction: Fraction:

2. When you are downloading program or application for your digital device, you will often see

a bar on your screen indicating what portion of that program or application has been

downloaded at that instant. Below, you have been supplied with download bars with shaded

portions. In each case, estimate the percentage of the program or application that has been

downloaded at that time (i.e. There is a range of acceptable answers.).

(a) Percent: _______

(b) Percent: _______

(c) Percent: _______

(d) Percent: _______

(e) Percent: _______

NSSAL 21 Draft

©2012 C. D. Pilmer

3. Convert the following percentages to decimals. No work needs to be shown.

(a) 79% = _______ (b) 16% = _______

(c) 9% = _______ (d) 145% = _______

(e) 29.4% = _______ (f) 7% = _______

(g) 208% = _______ (h) 81.7% = _______

(i) 4.5% = _______ (j) 0.8% = _______

4. Convert the following decimals to percentages. No work needs to be shown.

(a) 0.19 = _______ (b) 0.48 = _______

(c) 1.73 = _______ (d) 0.692 = _______

(e) 0.06 = _______ (f) 2.09 = _______

(g) 0.073 = _______ (h) 1.548 = _______

(i) 0.002 = _______ (j) 1.7 = _______

5. Convert the following percentages to fractions (or mixed number). In some cases, the

fraction will have to be simplified.

(a) 39%

(b) 91%

(c) 16%

(d) 129%

(e) 235%

(f) 5.1%

(g) 4.6%

(h) 48.2%

(i) 0.4%

(j) 320.6%

NSSAL 22 Draft

©2012 C. D. Pilmer

6. Convert the following fractions or mixed numbers to percentages. No work needs to be

shown; it all comes down to remembering equivalent fractions and percentages.

(a) 3

4 (b)

21

5

(c) 1

32 (d)

12

3

(e) 4

5 (f)

13

4

(g) 2

13 (h)

12

5

7. Complete the following table of equivalent fractions, decimals and percentages.

Percent Fraction Decimal Percent Fraction Decimal

(a) 83

100 (b) 67%

(c) 3

45

(d) 0.39

(e) 5% (f) 3

14

(g) 0.719 (h) 216.3%

8. Of the three percentages supplied, which one makes the most sense in the context of the

given situation.

Situation Available Percentages

(a) It was a fantastic sale item. The price of the item had

been reduced by _______.

120% 3% 40%

(b) When Jacob renewed his mortgage, he was pleased that

the rate of interest had dropped by _______ per annum.

13% 28% 1%

(c) With a few more men in the course than woman, we

were not surprised when we were told that _______ of

the class was comprised of women.

45% 62% 18%

(d) Maxine's science teacher was very pleased with Maxine's

performance on the test. Her mark was _______, which

was the highest mark in the class.

72% 96% 58%

(e) Montez was satisfied with the service at the restaurant

and therefore left a tip of _______ for the waiter.

15% 45% 2%

NSSAL 23 Draft

©2012 C. D. Pilmer

9. Below you have been supplied with diagrams of cylindrical containers filled with fluid.

Match each of the numbers below with the most appropriate diagram. Place your answers in

the boxes below each diagram. This is an estimation activity; no calculations are required.

Do not assume that equivalent fractions, decimals, and percentages will be going in the same

box. For example 82%, 0.81 and 4

5 might all go in the same box even though they are not

equivalent; they are, however, very close to each other.

2.16 140% 0.8

1

4 0.91

52

8 51%

7

10 1.93

7

9 0.52 69% 194%

12

6 81%

7

8 0.24 260%

0.72 2.57 218%

31

7 89%

191

20 26% 1.43

1

2

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

NSSAL 24 Draft

©2012 C. D. Pilmer

10. Circle the two adjoining numbers that are equivalent decimals, fractions, or percentages.

93

100 9.3%

17

1000 1.7% 0.053 1.387 138.7%

93% 0.8 0.17 5.3%

53

100

3871

100 0.275

8

100 0.8% 0.008 0.28 0.53 2.75%

32

4

13.1% 1.31

131

10 280% 5.03%

503

1000 275%

131

1000 0.131% 2.75

42

5 47% 3.3

1

3

34.7% 0.347 0.275% 4.4

24

3 4.6 25%

3

47

100 347% 0.3%

3

10 30%

3

100 3

11. Use the numbers in the chart below to correctly complete the following statement.

There are _____ people in the Sampson family. Of those, ____ are female. That means that

percentage of females in this family is _____%, which can also be represented by the fraction

_____. The percentage of males in this family is _____%, which can also be represented by

the fraction _____.

75 6

3

4

1

4 8 25


Fraction Percent Equivalency Game found in the appendix of this resource. Record in the

chart below whom you played and who won.

Opponent Winner

Round #1

Round #2

NSSAL 25 Draft

©2012 C. D. Pilmer

Comparing Fractions, Decimals and Percentages

In this section we will be comparing fractions, decimals and percentages for the purpose of

ordering them from smallest to largest. Most learners will tend to leave the decimals as

decimals, convert percentages to decimals mentally, and use the benchmarks 0, 1

2, and 1 for

fractions. This strategy works well in most, but not all, cases.

Example

Order the following from smallest to largest.

107%, 8

9, 1.546, 9.8%,

1

100,

7

16, 0.51,

3

4

Answer:

107% = 1.07 (slightly larger than the benchmark 1)

8

9 is slightly less than the benchmark 1.

1.546 is slightly larger than 1.5 or 1

12

.

9.8% = 0.098 (close to the benchmark 0)

1

100 = 0.01 (close to the benchmark 0, and smaller than 9.8% or 0.098)

7

16 is slightly less than the benchmark

1

2.

0.51 is slightly more than the benchmark 1

2.

3

4 = 0.75, which is half way between the benchmarks

1

2 and 1.

Proper Order: 1

100, 9.8%,

7

16, 0.51,

3

4,

8

9, 107%, 1.546

Questions


1. For each of the following, indicate whether it is closer to 0, 1

2, or 1.

Closest

to:

Closest

to:

Closest

to:

(a) 98% (b) 11% (c) 45%

(d) 0.02 (e) 0.899 (f) 0.6

NSSAL 26 Draft

©2012 C. D. Pilmer

Closest

to:

Closest

to:

Closest

to:

(g) 8

9 (h)

11

20 (i)

1

16

(j) 0.3% (k) 1.05 (l) 102%

(m) 13

24 (n) 56.2% (o)

16

15

2. In each case, circle the larger number.

(a) 39% 83% (b) 14.7% 14.2%

(c) 136% 98% (d) 3.1% 2.99%

(e) 9% 0.48 (f) 83% 0.78

(g) 2.3% 0.005 (h) 1

12 0.65

(i) 0.99 105% (j) 1.45 89%

(k) 7

8 29% (l) 47%

9

16

(m) 135% 14

15 (n)

1

10 2%

(o) 0.198 7

16 (p) 93.5%

3

4

(q) 2

3 30% (r)

11

3 215%

(s) 0.546 21.5% 4

5 (t) 7.59% 0.48

1

4

(u) 5

8 0.19 81.2% (v) 8.3%

1

100 0.009

3. Place the following by the appropriate arrow on the number line below.

200%, 0.93, 5

28

, 54.7%, 1.099, 1

316

, 125%, 1

20, 0.422,

12

4, 180%

0 2 1 3

NSSAL 27 Draft

©2012 C. D. Pilmer

4. Place the following by the appropriate arrow on the number line below.

19

120

, 2.85, 9.7%, 3

24

, 0.713, 155%, 3.2, 3

6, 1.389, 96%, 215%

5. Order the following from smallest to largest.

(a) 32%, 124%, 0.7%, 91.2%, 5.8%

_________, _________, _________, _________, _________

(b) 0.82, 14%, 0.1, 64%, 0.745

_________, _________, _________, _________, _________

(c) 123%, 1.45, 8.2%, 0.61, 57.2%

_________, _________, _________, _________, _________

(d) 9

10, 0.792, 3.8%, 86%,

5

12

_________, _________, _________, _________, _________

(e) 1.96, 68.5%, 0.4, 1

32,

11

10, 20%

_________, _________, _________, _________, _________, _________

(f) 0.276, 57.6%, 8

16, 1.1,

31

32, 30.2%

_________, _________, _________, _________, _________, _________

(g) 5

18

, 209%, 0.89, 1

4, 0.096, 64.5%

_________, _________, _________, _________, _________, _________

0 2 1 3

NSSAL 28 Draft

©2012 C. D. Pilmer

(h) 0.956, 91%, 1

100, 50.3%, 0.08,

6

14

_________, _________, _________, _________, _________, _________

(i) 1

13

, 28%, 0.9, 3

5, 194%, 1.02,

7

7

_________, _________, _________, _________, _________, _________, _________

(j) 214%, 0.34, 1

5,

72

8, 94.5%,

71

12, 1.092

_________, _________, _________, _________, _________, _________, _________

6. Open-ended Questions (i.e. more than one acceptable answer)

Your Answer

(a) Provide a percent that is between 34% and 35%.

(b) Provide a percent that is between 2

5 and

3

5.

(c) Provide a percent that is between 0.78 and 0.81.

(d) Provide a decimal number that is between 7% and 9.3%.

(e) Provide a decimal number that is between 8.2 and 8.3.

(f) Provide a decimal number that is between 1

110

and 2

110

.

(g) Provide a mixed number that is between 145% and 156%.

(h) Provide a mixed number that is between 3

24

and 3.

(i) Provide a mixed number that is between 3.01 and 3.25

(Have your instructor check your answers to question 6.)

NSSAL 29 Draft

©2012 C. D. Pilmer

Adding and Subtracting Decimal Numbers

In the Whole Number Operations Bridging Unit, we learned how to add multi-digit whole

numbers. To accomplish this, we start by stacking the numbers vertically such that

corresponding place values line up (e.g. units with units, tens with tens) and add from right to

left. If the sum in any corresponding place value is 10 or greater, we regroup (i.e. carry the

excess to the next larger place value).

e.g. 158 + 265

Answer:

Add the Units Add the Tens Add the Hundreds

1

1 5 8

2 6 5

3

1 1

1 5 8

2 6 5

2 3

1 1

1 5 8

2 6 5

4 2 3

8 units plus 5 units is 13

units. Regroup the 13 to 1

ten and 3 units.

1 ten plus 5 tens plus 6 tens

is 12 tens. Regroup the 12

to 1 hundred and 2 tens.

1 hundred plus 1 hundred

plus 2 hundreds is 4

hundreds

Adding Decimals

We follow the same procedure when adding decimal numbers. We start by stacking the numbers

vertically such that the corresponding place values line up (i.e. tenths with tenths, hundredths

with hundredths, etc.). Again we add from right to left. If the sum in any corresponding place

value is 10 or greater, we regroup (i.e. carry the excess to the next larger place value).

e.g. 0.67 + 2.84

Answer:

Add the Hundredths Add the Tenths Add the Units

1

0 . 6 7

2 . 8 5

2

1 1

0 . 6 7

2 . 8 5

. 5 2

1

0 . 6 7

2 . 8 5

3 . 5 2

7 hundredths plus 5

hundredths is 12

hundredths. Regroup the

12 to 1 tenth and 2

hundredths.

1 tenth plus 6 tenths plus 8

tenths is 15 tenths.

Regroup the 15 to 1 unit

and 5 tenths. Transfer

down the decimal point.

1 unit plus 2 units is 3

units. The final answer is

3.52.

NSSAL 30 Draft

©2012 C. D. Pilmer

e.g. 0.471 + 4.89 + 0.055

Answer:

Add the Thousandths Add the Hundredths

0 . 4 7 1

4 . 8 9

0 . 0 5 5

6

2

0 . 4 7 1

4 . 8 9

0 . 0 5 5

1 6

1 thousandth plus 5 thousandths is 6

thousandths.

7 hundredths plus 9 hundredths plus 5

hundredths is 21 hundredths. Regroup

the 21 to 2 tenths and 1 hundredth.

Add the Tenths Add the Units

1 2

0 . 4 7 1

4 . 8 9

0 . 0 5 5

. 4 1 6

1 2

0 . 4 7 1

4 . 8 9

0 . 0 5 5

5 . 4 1 6

2 tenths plus 4 tenths plus 8 tenths is

14 tenths. Regroup the 14 to 1 unit and

4 tenths. Transfer down the decimal

point.

1 unit plus 4 units is 5 units. The final

answer is 5.416.

e.g. 2.95 + 14.86 + 0.7

Answer:

Add the Hundredths Add the Tenths Add the Units Add the Tens

1

2 . 9 5

1 4 . 8 6

0 . 7

1

2 1

2 . 9 5

1 4 . 8 6

0 . 7

. 5 1

2 1

2 . 9 5

1 4 . 8 6

0 . 7

8 . 5 1

2 1

2 . 9 5

1 4 . 8 6

0 . 7

1 8 . 5 1

NSSAL 31 Draft

©2012 C. D. Pilmer

In the Whole Number Operations Bridging Unit, we learned how to subtract multi-digit whole

numbers. To accomplish this, we start by stacking the numbers vertically such that

corresponding place values line up (e.g. units with units, tens with tens) and subtract from right

to left. If the digit being subtracted is larger than the digit from which it is being subtracted,

regroup (i.e. borrow) one from the digit in the next larger place value.

e.g. 392 - 145

Answer:

Subtract the Units

Subtract the Tens Subtract the Hundreds

8 12

3 9 2

1 4 5

7

8 12

3 9 2

1 4 5

4 7

8 12

3 9 2

1 4 5

2 4 7

We cannot take 5 units

from 2 units. Therefore we

regroup (i.e. borrow) 1

from the tens.

8 tens minus 4 tens is 4

tens.

3 hundreds minus 1

hundred is 2 hundreds.

Subtracting Decimals

We follow the same procedure when subtracting decimal numbers. We start by stacking the

numbers vertically such that the corresponding place values line up (i.e. tenths with tenths,

hundredths with hundredths, etc.). Again we work from right to left. If the digit being

subtracted is larger than the digit from which it is being subtracted, regroup (i.e. borrow) one

from the digit in the next larger place value.

e.g. 5.96 - 3.45

Answer:

Subtract the Hundredths Subtract the Tenths Subtract the Units

5 . 9 6

3 . 4 5

1

5 . 9 6

3 . 4 5

. 5 1

5 . 9 6

3 . 4 5

2 . 5 1

6 hundredths minus 5

hundredths is 1 hundredth.

9 tenths minus 4 tenths is 5

tenths. Transfer down the

decimal point.

5 units minus 3 units is 2


2.51.

NSSAL 32 Draft

©2012 C. D. Pilmer

e.g. 7.63 - 2.18

Answer:

Subtract the Hundredths Subtract the Tenths Subtract the Units

5 13

7 . 6

3

2 . 1 8

5

5 13

7 . 6

3

2 . 1 8

. 4 5

5 13

7 . 6

3

2 . 1 8

5 . 4 5

We cannot take 8

hundredths from 3

hundredths. Therefore we

regroup (i.e. borrow) 1

from the tenths. 13

hundredths minus 8

hundredths is 5 hundredths.

5 tenths minus 1 tenth is 4

tenths. Transfer down the

decimal point.

7 units minus 2 units is 5


5.45.

e.g. 40.59 - 12.7

Answer:

This question is a little more challenging because in the second step (i.e. subtracting the

tenths), we cannot initially regroup (i.e. borrow) from the units because there are zero units.

That means we have to regroup from the tens to the units, and then the units to the tenths.

Subtract the Hundredths Subtract the Tenths

4 0 . 5 9

1 2 . 7

9

9

3 10

15

4 0 . 5 9

1 2 . 7

. 8 9

Subtract the Units Subtract the Tens

9

3 10

15

4 0 . 5 9

1 2 . 7

7 . 8 9

9

3 10

15

4 0 . 5 9

1 2 . 7

2 7 . 8 9

NSSAL 33 Draft

©2012 C. D. Pilmer

e.g. 6.35 - 0.728

Answer:

Initially change 6.35 to 6.350; they are equivalent decimals.

Subtract the Thousandths Subtract the Hundredths

4 10

6 . 3 5 0

0 . 7 2 8

2

4 10

6 . 3 5 0

0 . 7 2 8

2 2

Subtract the Tenths Subtract the Units

5 13 4 10

6 . 3 5 0

0 . 7 2 8

. 6 2 2

5 13 4 10

6 . 3 5 0

0 . 7 2 8

5 . 6 2 2

Questions


1. Complete the indicated operation. Show all your work.

(a) 12.72 + 34.16 (b) 62.53 + 7.31

(c) 38.6 + 50.27 (d) 6.423 + 0.39

NSSAL 34 Draft

©2012 C. D. Pilmer

(e) 6.39 + 35.572 (f) 142.8 + 87.53

(g) 0.265 + 6.81 + 38.7 (h) 7.46 + 0.085 + 0.93

2. Complete the indicated operation. Show all your work.

(a) 64.87 - 21.52 (b) 6.95 - 2.91

(c) 3.547 - 1.819 (d) 13.52 - 7.8

(e) 9.28 - 0.415 (f) 7.042 - 0.36

NSSAL 35 Draft

©2012 C. D. Pilmer

(g) 28.049 - 6.27 (h) 7.406 - 0.85

3. A container filled with water weighs 4.56 kilograms. Once the water is removed, the

container weighs 0.89 kilograms. What was the weight of the water that was removed?

4. Jack gained 1.36 kilograms in the first week and 2.06 kilograms in the

second week. How much weight did he gain over that two week period?

5. The odometer on Akira's car initially read 23 467.4 kilometers. After driving 825.7

kilometres, what would be the new odometer reading?

6. Montez's time on the 100 metre dash was 10.54 seconds. Hinto's time was 10.92 seconds.

How many seconds earlier did Montez arrive at the finishing line as compared to Hinto?

NSSAL 36 Draft

©2012 C. D. Pilmer

Multiplying Decimal Numbers

In the Level III Whole Number Operations Unit, we learned three techniques for multiplying

whole numbers: traditional algorithm, multiplying using the expanded form, and the lattice

method. You chose the method you preferred; the same will apply here in this section.

e.g. 4967

Traditional

Algorithm

Using

Expanded

Form

Lattice Method

6 7

4 9

6 0 3

2 6 8 0

3 2 8 3

940

760

0042

082

045

36

3823

Multiplying Decimals

1. Multiply the decimals as though they were whole numbers (i.e. initially ignore the decimal

points)

2. The decimal point in the product is placed so that the number of decimal places in the

product is equal to the sum of the number of decimal places in the factors.

Note: When multiplying decimals, you do not need to line up the decimal points, unlike question

involving addition and subtraction of decimal numbers.

Example 1

Complete the indicated operation.

(a) 67 4.9 (b) 6.7 4.9 (c) 0.67 4.9 (d) 0.67 0.49

Answers:

These questions were chosen because all of their solutions rely on knowing that

67 49 3283 , which was calculated above.

(a) 67 - zero decimal places

4.9 - one decimal place

The final answer should have one decimal place (0 + 1 = 1)

Therefore: 67 4.9 328.3

(b) 6.7 - one decimal place


The final answer should have two decimal places (1 + 1 = 2)

Therefore: 6.7 4.9 32.83

2

3 2

3 4

6 5

8 4

2

3 8

carry

1

carry 1

6 7

9

4

NSSAL 37 Draft

©2012 C. D. Pilmer

(c) 0.67 - two decimal places


The final answer should have three decimal places (2 + 1 = 3)

Therefore: 0.67 4.9 3.283

(d) 0.67 - two decimal places

0.49 - two decimal places

The final answer should have four decimal places (2 + 2 = 4)

Therefore: 0.67 0.49 0.3283

Example 2

Complete the following operation. Show all your work.

49.7 0.53

Answer:

Change the question to 497 53 ; we will deal with the decimals points in a later step. Again,

you can choose one of the three multiplication techniques that you prefer.

Traditional

Algorithm

Using Expanded Form Lattice Method

4 9 7

5 3

1 4 9 1

2 4 8 5 0

2 6 3 4 1

400 90 7

50 3

2 1

2 7 0

1 2 0 0

3 5 0

4 5 0 0

2 0 0 0 0

2 6 3 4 1

We will now consider the decimal points.


0.53 - two decimal places

The final answer should have three decimal places (1 + 2 = 3)

Therefore: 49.7 0.53 26.341

To determine whether the answer is reasonable, round the decimal numbers to numbers that

are more manageable. We could round 49.7 to 50, and round 0.53 to 0.5. Since

50 0.5 25 , then we can assume that the answer of 26.341 is reasonable.

4 9

3

5

4 2

0

7

5

3

5

2

1 2 7

3

2 1

1 4

2

6

1 1

NSSAL 38 Draft

©2012 C. D. Pilmer

Questions


1. Complete the indicated operation. Show all your work. Please note that we have addition,

subtraction and multiplication questions in here.

(a) 3.7 6.5 (b) 7.43 2.6

(c) 0.45 0.52 (d) 3.4 + 18.92

(e) 1.73 4 (f) 0.49 9.1

NSSAL 39 Draft

©2012 C. D. Pilmer

(g) 83.91 - 7.28 (h) 63 0.29

(i) 0.453 0.78 (j) 90.36 - 12.5

2. For each question, we have provided three possible solutions. Use your estimation skills to

determine which of three the correct answer is. We do not want you to work these out on

paper or use a calculator. Instead, we want you to round the decimals to numbers that are

more manageable, and estimate the final answer in your head. For example, 9.13 4.9 could

be changed to 9 5 , which has a product of 45. You would then look for the answer that is

close to 45.

(a) 7.08 3.2 10.28 22.656 2.2125

(b) 82.53 6.89 75.64 89.42 568.6317

(c) 153.6 23.7 129.9 194.3 177.3

(d) 4.3 89.5 384.85 512.75 319.65

(e) 452.5 351.7 924.2 734.2 804.2

(f) 51.3 49.68 2548.584 3264.374 1956.924

NSSAL 40 Draft

©2012 C. D. Pilmer

3. Six questions are supplied below. You must use your estimation skills to determine which

arrow on the number line below best represents the solution to each of the six questions.

Question Arrow Question Arrow

0.98 2.1 0.326 2.21

5.23 4.37 2.34 1.98

1.1 1.97 0.49 2.88

4. Use your estimation skills to match up each question with each answer.

Questions Answers

(a) 2.93 + 3.208 32.185

(b) 16.08 - 5.239 311.74

(c) 7.85 4.1 21.78

(d) 47.9 + 32.7 6.138

(e) 29.58 - 7.8 126.7

(f) 39.8 6.1 80.6

(g) 98.3 + 28.4 1110.9

(h) 409.8 - 98.06 10.841

(i) 52.9 21 242.78

5. John's car holds 48.7 litres of gas. If his vehicle can travel 15

kilometres on a litre of gas, how far can it travel on a full tank?

6. Kadeer had $14.15 in his iTunes account. If he purchases a song

for $1.29 from iTunes, how much will be left in his account?

0 2 1 3

a b c d e f

NSSAL 41 Draft

©2012 C. D. Pilmer

7. If Meera makes $12.65 per hour, how much will she make, before

deductions, in a 38 hour work week?

8. The shrub is 38.7 cm tall. They expect that it will grow an

additional 3.5 cm over the year. What is the expected height of the

tree in a year's time?

9. If each plastic pellet weighs 0.58 grams, how much does 45 pellets

weigh?

NSSAL 42 Draft

©2012 C. D. Pilmer

Dividing Decimal Numbers

In this section, we will show you two ways to solve these types of division question. One

technique uses the traditional algorithm; the other uses the partial quotient method. You chose

the method you prefer.

Dividing a Decimal Number by a Whole Number

With a first few questions we are only going to be looking at questions where we are dividing a

decimal number by a whole number (e.g. 165.2 7 , 4.23 9 ).

Example 1

Complete the operation 165.2 7 .

Answer:

In these explanations, and the ones that follow, we will be using the

terms divisor, quotient, and dividend. These terms have been

described in the diagram on the right. For this particular question,

the dividend is 165.2, the divisor is 7, and the quotient is the final

answer.

Traditional Algorithm Partial Quotient Method

Do the long division as you would

with whole numbers, then place the

decimal point in the quotient directly

above the decimal point in the

dividend.

23.6

7 165.2

-14

25

- 21

42

- 42

0

Therefore: 165.2 7 23.6

Initially ignore the decimal point and pretend that

you are dividing two whole numbers.

236

7 1652

1400

252

210

42

42

0

Now move the decimal point in the quotient, the

same number of places and in the same direction

as the decimal point in the dividend. In this case,

the dividend should be 165.2, where the decimal

point is one place to the left. Therefore our

quotient should be 23.6; notice that the decimal

point is also one place to the left.

Therefore: 165.2 7 23.6

This answer looks reasonable because we know that 140 7 20 , therefore we would expect

that 165.2 7 would be a little more than 20.

quotient

dividenddivisor

200

30

6

NSSAL 43 Draft

©2012 C. D. Pilmer

Example 2

Complete the operation 4.23 9 .

Answer:


Do the long division as you would

with whole numbers, then place the

decimal point in the quotient directly

above the decimal point in the

dividend.

0.47

9 4.23

- 36

63

- 63

0

Therefore: 4.23 9 0.47

Initially ignore the decimal point and pretend that

you are dividing two whole numbers.

47

9 423

360

63

63

0

In this case, the dividend should be 4.23, where

the decimal point is two places to the left.

Therefore our quotient should be 0.47; notice that

the decimal point is also two places to the left.

Therefore: 4.23 9 0.47

This answer looks reasonable because we know that 4.5 9 or 4.5

9 is equal to

1

2 or 0.5.

Therefore we would expect that 4.23 9 is slightly less than 0.5.

Example 3

Match each division question with the appropriate answer. We are not asking you to work these

out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation

skills.

Questions Answers

(a) 2090.2 7 39.48

(b) 51.84 6 0.5325

(c) 315.84 8 298.6

(d) 2.13 4 51.93

(e) 467.37 9 8.64

Answers:

We know that 2100 7 300 , therefore 2090.2 7 is likely equal to 298.6.



We know that 2 4 or 2

4 is equal to 0.5, therefore 2.13 4 likely equals 0.5325.


40

7

NSSAL 44 Draft

©2012 C. D. Pilmer

Briefly Revisiting Fractions

We know that fractions are one way of expressing the operation of division.

e.g. 3

3 44 e.g.

99 5

5 e.g.

2727 100

100

We also know that equivalent fractions can be created by multiplying or dividing the numerator

and denominator of a fraction by the same number.

e.g. 12 12 3 4

15 15 3 5

e.g.

35 35 5 7

20 20 5 4

e.g.

20 20 10 2

30 30 10 3

e.g. 7 7 2 14

8 8 2 16

e.g.

2 2 3 6

5 5 3 15

e.g.

9 9 10 90

4 4 10 40

We will use both of these pieces of knowledge to help us understand the first step in dividing a

decimal number by another decimal number.

Consider the question 9 0.6 .

The question 9 0.6 can be expressed as 9

0.6.

We could create an equivalent fraction by multiplying the numerator and denominator

by 10.

9 9 10 90

0.6 0.6 10 6

The 90

6 can be expressed as 90 6 .

We have shown that the answer (i.e. quotient) to 9 0.6 (or 0.6 9 ) is equal to answer

to 90 6 (or 6 90 ).

Consider the question 8.1 0.03 .

The question 8.1 0.03 can be expressed as 8.1

0.03.


by 100.

8.1 8.1 100 810

0.03 0.03 100 3

The 810

3 can be expressed as 810 3 .

We have shown that the answer (i.e. quotient) to 8.1 0.03 (or 0.03 8.1 ) is equal to

answer to 810 3 (or 3 810 ).

NSSAL 45 Draft

©2012 C. D. Pilmer

Consider the question 7.675 0.5 .

The question 7.675 0.5 can be expressed as 7.675

0.5.


by 10.

7.675 7.675 10 76.75

0.5 0.5 10 5

The 76.75

5 can be expressed as 76.75 5 .

We have shown that the answer (i.e. quotient) to 7.675 0.5 (or 0.5 7.675 ) is equal

to answer to 76.75 5 (or 5 76.75 ).

We can take this and apply it to a variety of division questions.

e.g. 3.01 3.01 10 30.1

0.7 3.01 3.01 0.7 7 30.10.7 0.7 10 7

e.g. 5 5 10 50

0.8 5 5 0.8 8 500.8 0.8 10 8

e.g. 9.28 9.28 100 928

0.02 9.28 9.28 0.02 2 9280.02 0.02 100 2

e.g. 0.0387 0.0387 100 3.87

0.09 0.0387 0.0387 0.09 9 3.870.09 0.09 100 9

e.g. 0.0224 0.0224 1000 22.4

0.004 0.0224 0.0224 0.004 4 22.40.004 0.004 1000 4

Let's look at all the division questions with equivalent quotients that we have discussed in the

last two pages.

0.6 9 6 90 0.03 8.1 3 810 0.5 7.675 5 76.75 0.7 3.01 7 30.1

0.8 5 8 50 0.02 9.28 2 928 0.09 0.0387 9 3.87 0.004 0.0224 4 22.4

Notice that every case we started with a question where we were dividing by a decimal number,

but in the end we had changed the question to one where we were dividing by a whole number.

We should be able to solve new question as we have already learned how to divide a decimal by

a whole number.

So how could we describe the process of changing a division question from one where we are

dividing by a decimal number to one where we are dividing by a whole number? We obviously

do not want to go through the lengthy process of converting the division question to fraction

question, creating an equivalent fraction, and then converting from a fraction question to a

NSSAL 46 Draft

©2012 C. D. Pilmer

division question. Instead, we use the following shortcut. Start by

moving the decimal point to the right in the divisor until the divisor is a

whole number. Then move the decimal point to the right in the dividend

the same number of places as was moved for the divisor. If you move

both one place to the right, it is equivalent to multiplying the numerator

and denominator of a fraction by 10. If you move both two places to the right, it is equivalent to

multiplying the numerator and denominator of a fraction by 100.

Dividing a Whole Number or Decimal Number by a Decimal Number

Step 1: Move the decimal point to the right in the divisor until the divisor is a whole number.

Step 2: Move the decimal point to the right in the dividend the same number of places as was

done in Step 1.

Step 3: Divide through using the procedure that you prefer for dividing a decimal number by a

whole number (i.e. what we did in Examples 1 and 2)

Example 4

Complete the operation 21.87 0.9 .

Answer:

Regardless of whether you prefer the traditional algorithm or the partial quotient method, you

must start by changing the divisor (0.9) to a whole number. This is accomplished by moving

the decimal point one place to the right in the divisor. We must then move the decimal point

one place to the right in the dividend. This means that the question changes from 21.87 0.9

to 218.7 9


24.3

9 218.7

-18

38

- 36

27

- 27

0

If 218.7 9 24.3 ,

then 21.87 0.9 24.3

243

9 2187

1800

387

360

27

27

0



as the decimal point in the dividend.

If 218.7 9 24.3 ,

then 21.87 0.9 24.3

quotient

dividenddivisor

200

40

3

NSSAL 47 Draft

©2012 C. D. Pilmer

Example 5

Complete the operation 0.2922 0.06 .

Answer:

Start by changing the divisor (0.06) to a whole number. This is accomplished by moving the

decimal point two places to the right in the divisor. We must then move the decimal point

two places to the right in the dividend. This means that the question changes from

0.2922 0.06 to 29.22 6 .


4.87

6 29.22

- 24

52

- 48

42

- 42

0

If 29.22 6 4.87 ,

then 0.2922 0.06 4.87

487

6 2922

2400

522

480

42

42

0



as the decimal point in the dividend.

If 29.22 6 4.87 ,

then 0.2922 0.06 4.87

Example 6

Match each division question with the appropriate answer. In many cases, you may wish to

move the decimal points in both divisor and dividend to make the question more manageable.

We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are

asking you to use your estimation skills.

Questions Answers

(a) 58.08 0.8 209.3

(b) 0.3474 0.06 0.48

(c) 83.72 0.4 5.79

(d) 880.2 9 72.6

(e) 0.0336 0.07 97.8

Answers:

Change 58.08 0.8 to 580.8 8 . We know that 560 8 70 , therefore 580.8 8 (or

58.08 0.8 ) is likely equal to 72.6.


0.3474 0.06 ) is likely equal to 5.79.

400

80

7

NSSAL 48 Draft

©2012 C. D. Pilmer


83.72 0.4 ) is likely equal to 209.3


Change 0.0336 0.07 to 3.36 7 . We know that 3.5 7 or 3.5

7 equals 0.5. Therefore

it is likely that 3.36 7 (or 0.0336 0.07 ) is equal to 0.48.

Questions


1. Complete each of the operations. Show all your work.

(a) 32.04 6 (b) 240.3 9

(c) 2.415 5 (d) 0.651 7

NSSAL 49 Draft

©2012 C. D. Pilmer

2. Match each division question with the appropriate answer. We are not asking you to work

these out using paper-and-pencil or a calculator; rather, we are asking you to use your

estimation skills.

Questions Answers

(a) 389.6 8 0.506

(b) 49.02 6 48.7

(c) 2.024 4 4.69

(d) 257.8 9 8.17

(e) 32.83 7 28.62

3. In each case, four division questions have been provided. From the last three division

questions, circle the one which has the same quotient (i.e. generates the same answer) as the

first division question. You do not want you to work any of these out using paper-and-pencil

or a calculator.

(a) 0.6 45.36 6 4.536 6 453.6 6 4536

(b) 8 7.36 0.8 73.6 0.8 0.736 0.8 736

(c) 0.5 385.6 5 3856 5 38.56 5 3.856

(d) 8 27.345 0.8 273.45 0.8 2734.5 0.8 2.7345

(e) 0.5 49 5 4.9 5 490 5 0.49

(f) 0.06 8.2 6 820 6 82 6 0.082

(g) 3 182.6 0.03 18260 0.03 18.26 0.03 1.826

(h) 0.07 58.2 7 582 7 5820 7 0.582

4. Complete the following operations. Show all your work. Please note that we have also

included a few addition, subtraction, and multiplication questions.

(a) 4.48 0.8 (b) 2.58 0.03

NSSAL 50 Draft

©2012 C. D. Pilmer

(c) 7.59 12.8 (d) 66.01 0.7

(e) 4.7 6.8 (f) 3.4 0.05

(g) 0.2616 0.3 (h) 183.2 59.16

NSSAL 51 Draft

©2012 C. D. Pilmer

5. Match each question with the appropriate answer. Note that we have included addition,

subtraction, multiplication, and division questions. With some of the division questions, you

may wish to move the decimal points in both divisor and dividend to make the question more

manageable. We are not asking you to work these out using paper-and-pencil or a calculator;

rather, we are asking you to use your estimation skills.

Questions Answers

(a) 639.1 7 39.6

(b) 289.4 315.7 122.04

(c) 23.76 0.6 0.48

(d) 20.5 61.8 91.3

(e) 0.24 0.5 254.8

(f) 453.6 198.8 6.2

(g) 0.496 0.08 817

(h) 0.9 135.6 1266.9

(i) 32.68 0.4 605.1

6. Six questions are supplied below. You must use your estimation skills to determine which

arrow on the number line below best represents the solution to each of the six questions.


6.65 7 3.05 2.97

1.49 1.04 0.31 5.2

0.145 0.05 1.278 0.6

7. The prize money of $169.80 has to be shared equally by 6 people.

How much money does each person get?

0 2 1 3

a b c d e f

NSSAL 52 Draft

©2012 C. D. Pilmer

8. Hamid uses 0.2 kg of ground beef when making a single hamburger

patty. How many patties can he make using 3.64 kg of ground beef?

9. Ryan makes $13.50 per hour. How much will he make, before deductions, if he works 6.5

hours?

10. Tylena paid $5.79 for 0.6 kg of meat. How much would one kilogram

of the meat cost?

11. Jessie cycled 85.3 km on day one, 93.6 km on day two, and 78.8 km on

day three. How far did she cycle in that three day period?

NSSAL 53 Draft

©2012 C. D. Pilmer

Estimation Questions Involving Percentages

We use percentages every day when we work out the price of an item after taxes, determine the

sale price of an item, and calculate the tip for your waiter or waitress. In many cases, we use our

estimation skills when addressing these real-life situations.

In Nova Scotia, when you purchase most items, you have to pay a 15% harmonized sales

tax on those goods. For example, a bedroom suite advertised at $1395 will be subject to

sales tax. It is important that you be able to estimate the tax on that purchase and the total

cost of the purchase.

A discount is a reduction in a price. When a discount on an item is offered, the rate of

discount is often advertised as a percent of the regular price. For example a sofa,

regularly priced at $799, may be advertised as 25% off during a particular sale. It is

important to be able to estimate the cost of the sofa after the discount so that you are not

overcharged for that item.

When you go out to a restaurant for a meal, you are expected to tip the waiter or waitress

for good service. Typically people tip between 15% (good service) and 20% (exceptional

service). It is important that you be able to mentally calculate these tips so that the waiter

or waiter receives appropriate amount for their level of service.

Below we solve a variety of estimation questions involving percentages. As with any estimation

question, there are a variety of ways of obtaining a reasonable estimate. In our solutions, we

have only provided one reasonable estimate. We have tried to use the most common approach in

each case, but we recognize that there are other perfectly acceptable techniques.

Example 1

Your bill at a local restaurant is $68.95. The waitress offered exceptional service and you decide

to give a tip of approximately 20%. How much money should she receive?

Answer:

Round $68.95 to $70.

We know that 10% of $70 is $7.

Therefore 20% of $70 is $14.

The tip for the waitress should be approximately $14.

Example 2

Jorell needs to purchase a new mattress for his bed. It costs $795 before taxes. He has to pay

15% tax. Approximately how much will he pay in taxes, and what will be the approximate cost

of this purchase?

Answer:

Round $795 to $800

We know that 10% of $800 is $80, and that 20% of $800 is $160.

Therefore 15% of $800 would have to be half way between $80 and $160. That means

the tax on the mattress would be approximately $120.

That means that the total cost who be slightly less than $920 ($800 + $120).

NSSAL 54 Draft

©2012 C. D. Pilmer

Example 3

Lei is going to purchase a sofa that is regularly priced at $1195.

Today the sofa is marked down by 30%. What is the approximate

sales price of this item?

Answer:

Round $1195 to $1200.

If the price is reduced by 30%, then 70% of the price is retained.

If 10% of $1200 is $120, then 70% of $1200 is found by multiplying $120 by 7.

Since 120 7 840 , then we can conclude that the sale price of the sofa is approximately

$840.

Example 4

The new jeans, Nasrin is interested in, regularly cost $79. Today they are marked down by 25%.

How much will she pay approximately for these jeans including the 15% sales tax?

Answer:

Round $79 to $80.

25% off is the same as one-quarter off.

One-quarter 1

i.e. 4

of $80 is $20.

If the price is reduced by $20, then the approximate sale price of the jeans is $60.

Now we need to determine the 15% sales tax. We know that 10% of $60 is $6, and that

20% of $60 is $12. Therefore 15% of $60 will be halfway between $6 and $12. The

sales tax will be approximately $15 on this item.

The total cost of the jeans will be approximately $75 ($60 + $15).

Questions

Do not use a calculator on any of these questions.

1. Solve each of the following. No work needs to be shown (i.e. Do it in your head.).

(a) What is 10% of 40? _______ (b) What is 10% of 120? _______

(c) What is 10% of 500? _______ (d) What is 10% of 1400? _______

(e) What is 20% of 40? _______ (f) What is 20% of 120? _______

(g) What is 20% of 500? _______ (h) What is 20% of 1400? _______

(i) What is 15% of 40? _______ (j) What is 15% of 120? _______

(k) What is 15% of 500? _______ (l) What is 15% of 1400? _______

(m) What is 30% of 40? _______ (n) What is 30% of 120? _______

(o) What is 25% of 400? _______ (p) What is 25% of 1200? _______

NSSAL 55 Draft

©2012 C. D. Pilmer

2. Manish's bill at a local restaurant is $49.45. The waiter offered

exceptional service so Manish decides to give a tip of approximately

20%. How much money should the waiter receive?

3. Krys is purchasing a fall jacket for her son. It costs $39.95. Approximately how much will

she have to pay after taxes (15%) for this item?

4. Ryan is purchasing a DVD boxed set of Season 13 of The Simpsons. It normally sells for

$29.95 but today it is marked down by 30%. What is the approximate sale price of this item?

5. Alice received satisfactory service at the restaurant and therefore felt it was reasonable to

leave a 15% tip on her $81.35 bill. Approximately how much should she leave?

6. All spring stock was marked down by 40% in a local clothing store.

Approximately how much would one pay, after taxes (15%), for a

spring dress regularly costing $59.95?

7. The $160 electronic device was marked down by 25% because a newer model of the same

device was now on the market. Approximately how much will it cost for this discounted

device after the paying sales tax (15%)?

NSSAL 56 Draft

©2012 C. D. Pilmer

8. The demand for a particular running shoe was much higher than the

manufacturer expected. They were originally going to sell the shoes

for $89.95 a pair. The manufacturer decides to increase the price by

20%. If they do this, what would be approximate new cost of the

shoes before taxes?

9. The regular price of a season's pass for skiing is $295. If you purchase the pass early, you

can save 30%. What is the approximate total cost, after taxes (15%), if you purchase this

early-bird season's pass?

10. Eight questions are supplied below. You must use your estimation skills to determine which

arrow on the number line below best represents the solution to each of the eight questions.


25% of 11.90 0.52 0.496

0.784 0.4 10% of 21.50

1.43 1.316 20% of 2.99

30% of 4.90 2.1 1.513


Percent Game found in the appendix of this resource. Record in the chart below whom you

played and who won.

Opponent Winner

Round #1

Round #2

0 2 1 3

a b c d e f g h

NSSAL 57 Draft

©2012 C. D. Pilmer

Calculator Questions

This is one of the few times in this course where we will allow you to use a

calculator to solve problems. Our rationale is that you should know all the

fundamentals concerning decimals and percentages at this point in time, and that

we now want to expose you to multi-step problems with "messier" numbers that

are better handled with a calculator, as opposed to using paper-and-pencil techniques.

Example 1

John was born on July 8, 1955. In an attempt to get his twelve grandchildren to remember his

birthday, John gives each child $78.55 cash at Christmas. How much money should he take out

of his account to cover his grandchildren's gifts?

Answer:

Simply multiply 12 by 78.55 on the calculator.

John needs to take out $942.60 to cover the gifts.

Example 2

Suzzette has a 53.5 litre container of water. She wants to know how many 2.45 litre containers

she can completely fill using the larger container of water.

Answer:

Using a calculator: 53.5 2.45 21.8 (rounded to one decimal place)

Normally we would round 21.8 up to 22, but this question is asking us how many

"containers she can completely fill." For this reason, we will round down and say that

she can completely fill 21 containers.

Example 3

In week one, Carissa's expenses were $496.65, and her earnings were $757.50. The money not

spent went into her savings. In week two she hopes to save twice as much money as week one.

If she is able to do this, how much money will go into her savings in week two?

Answer:

Her Savings on Week One: 757.50 - 496.65 = $260.85

Her Desired Savings on Week Two: 2 260.85 = $521.70

Example 4

Barb is purchasing at shirt priced at $18.95, a pair of jeans at $46.95, and a

knapsack at $39.95. What is the total cost after tax (15%)?

Answer:

Find the total before tax.

18.95 + 46.95 + 39.95 = 105.85

Determine the tax.

15% of 105.85 = 0.15 105.85 15.88 (rounded to the second decimal point)

Find the total after tax.

105.85 + 15.88 = $121.73

NSSAL 58 Draft

©2012 C. D. Pilmer

Example 5

At one particular store, customers can receive a 30% discount if their

purchases before the discount total $150 or more. Shelly plans on

purchasing a $89.45 set of bath towels and facecloths, a $49.95 set of

blinds for her bedroom window, and a $17.45 toaster. What is her

total cost after tax (15%)?

Answer:

Find the total before taxes.

89.95 + 49.95 + 17.45 = 157.35

Since their total purchase exceeds $150, they are able to receive the 30% discount.

If the price is reduced by 30%, then 70% of the price is retained. Take 70% of $157.35 to

find the new total (before taxes).

70% of 157.35 = 0.70 157.35 110.15 (rounded off)

Determine the tax.

15% of 110.15 = 0.15 110.15 16.52 (rounded off)

Find the total after tax.

110.15 + 16.52 = $126.67

Example 6

Meera works 47 hours this week. She gets $15.60 per hour for the first 40

hours. She gets "time-and-a-half" for any hours after the 40 hours; this is

considered overtime. How much will she earn, before deductions, for this

work week?

Answer:

Earnings for the First 40 Hours of Work: 40 15.60 = $624

Hourly Earnings at Time-and-a-Half: 1.5 15.60 = $23.40 per hour

Earnings for the 7 Hours of Overtime: 7 23.40 = $163.80

Total Earnings: 624 + 163.80 = $787.80

Example 7

Nashi makes $13.20 per hour plus a 2.5% commission on all her sales. If she works 36 hours

and makes $6490 worth of sales, what will be her earnings before deductions?

Answer:

Earnings from Hourly Wage: 36 13.20 $475.20

Commission Earnings: 2.5% of $6490 = 0.025 6490 $162.25

Total Earnings: 475.20 + 162.25 = $637.45

NSSAL 59 Draft

©2012 C. D. Pilmer

Questions

Calculators are permitted with these questions. Show how you solved the each question.

1. If a long-distance phone provider offers a rate of $0.12 per minute, how long can you talk for

$2.76?

2. Suppose it costs $27.50 per day plus $0.11 per kilometer for a rental car. What is the total

bill if you have the car for three days and travel 657 kilometres?

3. Masato purchases a loft of bread ($2.65), a can of beans ($2.29) and hot dogs ($3.89). If he

pays with a $10 bill, how much change will he receive? Please note that sales tax is not

applied to food.

4. The garden center marked all plants down by 60% for their end-of-season sale. How much

would you have to pay after taxes (15%) for a plant that normally cost $129.99?

5. Rana purchases a sweat top ($18.99), jeans ($34.99), and running shoes ($37.99) for each of

her twin boys. What will be the total cost after paying the sales tax (15%)?

6. Last week Meera was making $13.40 per hour and working 36 works. This week her hours

increased to 42 hours and her hourly wage increased to $14.70 per hour. How much will she

make, before deductions, over this two week period?

NSSAL 60 Draft

©2012 C. D. Pilmer

7. Kendrick makes $12.50 per hour plus a 3% commission on all his sales. If he worked 38

hours and sold $5840 worth of merchandise this week, what would be his earnings before

deductions?

8. Harris, a pipefitter working in oil project in northern Alberta, has to work a 12 hour shift on a

statutory holiday. For doing so, his employer will pay him time-and-a-half for the first 8

hours and double-time for the remaining 4 hours. If his normal hourly rate is $32.60, how

much will he make, before deductions, for this 12 hour shift?

9. The Boxing Day sale at a local clothing boutique advertised 40% off all purchases. Kimi

wanted to purchase a blouse, regularly priced at $29.95, and a sweater, regularly priced at

$37.59. If she purchases both during the sale, what is the total cost including sales tax

(15%)?

10. Kevin travelled to friend's cottage using his car. The car's odometer initially read 33 407.2

kilometres. Upon arriving at the cottage, the odometer read 33 598.9 kilometres. If he used

13.5 litres of fuel during the trip, how many kilometres per litre did his car achieve on this

trip?

NSSAL 62 Draft

©2012 C. D. Pilmer

Connect Four Fraction Decimal Equivalency Game

Number of Players: Two

Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically

or diagonally.

Instructions:

1. Roll a die to see which player will go first.

2. The first player looks at the board and decides which square he/she wishes to capture. The

square with a specified decimal is captured by creating the equivalent fraction using the

numerator and denominator strips at the bottom of the page. One paper clip is placed on

each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the

denominator, then they can capture one square labeled 0.75 (3

4 is equivalent to 0.75). They

either mark the square with an X or place a colored counter on the square. There may be

other squares with that same difference but only one square can be captured at a time.

3. Now the second player is ready to capture a square but he/she can only move one of the

paperclips. They then mark the square with the equivalent decimal using an O or a different

colored marker. If a player cannot move a single paperclip to capture a square, a paperclip

must still be moved in order to ensure that the game can continue.

4. Play alternates until one player connects four squares. Remember that only one player clip is

moved at a time. If none of the players is able to connect four, then the winner is the

individual who has captured the most squares.

Game Board:

0.4 1 0.2 0.4 1 0.5

0.25 0.1 0.75 0.8 0.6 0.2

0.3 0.6 0.4 0.25 0.3 0.4

0.75 0.2 0.3 0.5 0.2 1

0.2 0.8 0.25 0.1 0.8 0.4

0.1 0.5 0.6 1 0.3 0.75

Numerator (Top) Strip: Denominator (Bottom) Strip:

1 2 3 4 4 5 10

NSSAL 63 Draft

©2012 C. D. Pilmer

Connect Four Fraction Percent Equivalency Game



or diagonally.

Instructions:


2. The first player looks at the board and decides which square he/she wishes to capture. The

square with a specified percent is captured by creating the equivalent fraction using the

numerator and denominator strips at the bottom of the page. One paper clip is placed on

each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the

denominator, then they can capture one square labeled 75% (3

4 is equivalent to 75%). They

either mark the square with an X or place a colored counter on the square. There may be

other squares with that same difference but only one square can be captured at a time.


paperclips. They then mark the square with the equivalent decimal using an O or a different

colored marker. If a player cannot move a single paperclip to capture a square, a paperclip

must still be moved in order to ensure that the game can continue.

4. Play alternates until one player connects four squares. Remember that only one player clip is



Game Board:

40% 10% 20% 100% 40% 50%

25% 100% 25% 80% 60% 20%

30% 60% 40% 50% 30% 40%

75% 20% 30% 25% 80% 100%

20% 80% 75% 10% 20% 40%

10% 50% 100% 60% 30% 75%

Numerator (Top) Strip: Denominator (Bottom) Strip:

1 2 3 4 4 5 10

NSSAL 64 Draft

©2012 C. D. Pilmer

Connect Four Percentage Game



or diagonally.

Instructions:


2. The first player looks at the board and decides which square he/she wishes to capture. They

place two paperclips on two strips below; one on the "Percentage" strip and one on the "Of"

strip. Take the percentage of that number and capture the appropriate square (e.g. 20% of 40

allows one to capture an "8" square). They either mark the square with an X or place a

colored counter on the square. There may be other squares with that same value but only one

square can be captured at a time.


paperclips. They then mark the square with that value using an O or a different colored

marker. If a player cannot move a single paperclip to capture a square, a paperclip must still

be moved in order to ensure that the game can continue.

4. Play alternates until one player connects four squares. Remember that only one paperclip is



Game Board:

10 16 10 12 8 20

30 8 3 24 15 10

2 5 18 4 25 30

25 20 10 6 16 8

6 4 12 2 3 5

18 24 15 20 12 4

Percentage: Of:

10% 15% 20% 25% 20 40 80 100 120

NSSAL 65 Draft

©2012 C. D. Pilmer

Answers

Introduction to Decimals (pages 1 to 5)

1. (a) 0.5 (b) 0.3 (c) 0.13

(d) 0.41 (e) 0.7 (f) 0.29

2.

5

10 0.5

46

1000

81

10

0.07

7

100 0.046 1.8

37

1000 0.037

9

100 0.09 1.26

72

10

93

100

261

100

2.7 3.09

56

1000

6043

1000 3.604 0.056

3. (a) 0.256 (b) 0.06 (c) 3.9

(d) 1.097 (e) 2.007 (f) 13.58

4. (a) 45

100 (b)

4

10 (c)

5084

1000

(d) 8

1100

(e) 3

21000

(f) 59

61000

5. (a) 35.6

(b) 7.09

(c) 0.58

(d) 1000.15

(e) 206.309

(f) 70.1

(g) 5.037

(h) 400.029

NSSAL 66 Draft

©2012 C. D. Pilmer

6. Fraction

Decimal Numerals Words

(a) 32.8

832

10 thirty-two and eight tenths

(b) 0.472

472

1000 four hundred seventy-two thousandths

(c) 13.067

6713

1000 thirteen and sixty-seven thousandths

(d) 7.59

597

100 seven and fifty-nine hundredths

(e) 327.09

9327

100 three hundred twenty-seven and nine hundredths

7. (a) 40 2 0.8 and 1

4 10 2 1 810

(b) 9 0.3 0.01 and 1 1

9 1 3 110 100

(c) 300 2 0.4 0.02 0.009 and 1 1 1

3 100 2 1 4 2 910 100 1000

(d) 10 8 0.03 0.004 or 1 1

1 10 8 1 3 4100 1000

(e) 4000 200 9 0.07 or 1

4 1000 2 100 9 1 7100

Comparing Decimals (pages 6 to 12)

1. (a) 1

2 (b) 0 (c) 1

(d) 0 (e) 1

2 (f) 1

(g) 1

2 (h) 1 (i) 0

(j) 1 (k) 0 (l) 1

2

(m) 1 (n) 1

2 (o) 0

NSSAL 67 Draft

©2012 C. D. Pilmer

2. (a) 0.89 (b) 7

8

(c) 4

8 (d) 0.907

(e) 3

3 (f) 0.879

(g) 13

12 (h) 0.58

3. (a) 0.7 (b) 0.52

(c) 0.198 (d) 1.24

(e) 2.4 (f) 0.09

(g) 3.1 (h) 0.57

(i) 5.618 (j) 2.09

(k) 7.08 (l) 12.988

(m) 0.409 (n) 31.3005

(o) 3.01 (p) 7.809

(q) 15.35 (r) 0.75

(s) 15

16 (t) 0.81

(u) 0.51 (v) 5

26

(w) 4.7 (x) 11

312

4. The numbers should be placed along the number line in the following order.

0.1, 0.6, 0.8, 1.3, 1.4, 1.9, 2.2, 2.5, 2.7, 3.1


0.097, 0.54, 1.039, 1.46, 1.75, 1.95, 2.4, 2.62, 2.89, 3.05


0.07, 3

8,

7

10 , 1.2, 1.57,

12

16, 2.44 2.78,

43

100, 3.3

7. (a) 0.2, 0.4, 0.9, 1.3, 1.6

(b) 0.08, 0.55, 0.59, 1.14, 1.23

(c) 0.09, 0.52, 0.8, 0.83, 1.01, 1.1

(d) 0.19, 0.2, 0.26, 0.3, 0.98, 1

(e) 0.006, 0.08, 0.2, 0.209, 0.24, 0.72

(f) 0.05, 0.092, 0.4, 0.619, 0.64, 0.7, 0.78

(g) 7

100, 0.3,

1

2, 0.542, 0.85, 0.862, 0.9

NSSAL 68 Draft

©2012 C. D. Pilmer

(h) 0.16, 0.201, 0.6, 0.649, 8

8,

431

1000, 1.3

(i) 1

16, 0.4, 0.48, 0.509, 1.002, 1.1,

51

8

Equivalent Fractions and Decimals (pages 13 to 16)

1. (a) 3

4 (b)

13

2 (c)

41

5

(d) 1

23

(e) 1

75

(f) 2

53

2. (a) 9.25 (b) 0.4 (c) 6.3

(d) 7.75 (e) 9.5 (f) 8.6

3. (a) 0.65 (b) 0.875

(c) 1

13

(d) 2.6

(e) 2

43

(f) 4

65

(g) 2.304 (h) 1

75

4. The numbers should appear on the number line in the following order.

1

25,

1

3,

2

5, 0.9, 1.44,

31

4, 2.069, 2.539,

72

8,

13

5, 3.37

5. The numbers should appear on the number line in the following order.

0.098, 9

16,

95

100,

21

5,

21

3, 1.87, 2.2, 2.43, 2.71, 3.12,

13

4

6. (a) 0.09, 5

8,

93

100,

11

4, 1.6,

111

12, 2.4,

32

5

(b) 41

100, 0.587,

11

3, 1.58,

31

4,

12

16, 2.7,

92

10

(c) 9

16,

4

5, 0.892,

11

10,

191

20, 2.011,

12

4, 2.6

7. In each case we have supplied three possible answers; there are many more than the ones we

have listed.

(a) 3.41, 3.459, 3.499

NSSAL 69 Draft

©2012 C. D. Pilmer

(b) 3

24

, 3

25

, 2

23

(c) 1.3, 1.25, 1.239

(d) 0.7, 0.74, 0.859

(e) 1

35

, 2

310

, 1

34

(f) 2.9, 2.83, 2.999, 3.005

Introduction to Percent (pages 17 to 24)

1. (a) 17%, 0.17, 17

100 (b) 43%, 0.43,

43

100 (c) 29%, 0.29,

29

100

2. There are a range of acceptable answers.

(a) 40% to 47%

(b) 11% to 19%

(c) 85% to 94%

(d) 63% to 73%

(e) 4% to 7%

3. (a) 0.79 (b) 0.16

(c) 0.09 (d) 1.45

(e) 0.294 (f) 0.07

(g) 2.08 (h) 0.817

(i) 0.045 (j) 0.008

4. (a) 19% (b) 48%

(c) 173% (d) 69.2%

(e) 6% (f) 209%

(g) 7.3% (h) 154.8%

(i) 0.2% (j) 170%

5. (a) 39

100 (b)

91

100

(c) 4

25 (d)

291

100

(e) 7

220

(f) 51

1000

(g) 23

500 (h)

241

500

(i) 1

250 (j)

1033

500

NSSAL 70 Draft

©2012 C. D. Pilmer

6. (a) 75% (b) 140%

(c) 350% (d) 233.3%

(e) 80% (f) 325%

(g) 166.6% (h) 220%

7. Percent Fraction Decimal Percent Fraction Decimal

(a) 83% 83

100 0.83 (b) 67%

67

100 0.67

(c) 460% 3

45

4.6 (d) 139% 39

1100

0.39

(e) 5% 1

20 0.05 (f) 175%

31

4 1.75

(g) 71.9% 719

1000 0.719 (h) 216.3%

1632

1000 2.163

8. (a) 40%

(b) 1%

(c) 45%

(d) 96%

(e) 15%

9. Remember we are not grouping equivalent decimal, fractions, and percentages in the same

boxes; rather, we are completing an estimation activity where we match the numbers to the

most appropriate diagram.

(a) 51%, 0.52, 1

2 (b) 26%, 0.24,

1

4 (c) 69%, 0.72,

7

10

(d) 140%, 1.43, 3

17

(e) 81%, 0.8, 7

9 (f) 260%, 2.57,

52

8

(g) 218%, 2.16, 1

26

(h) 89%, 0.91, 7

8 (i) 194%, 1.93,

191

20

NSSAL 71 Draft

©2012 C. D. Pilmer

10. 93

100

17

1000 1.7% 1.387 138.7%

93%

53

100

0.8% 0.008 0.53

32

4

13.1% 280% 275%

131

1000

42

5

24

3 4.6

3

47

100 347%

3

10 30%

11. There are 8 people in the Sampson family. Of those, 6 are female. That means that

percentage of females in this family is 75%, which can also be represented by the fraction 3

4.

The percentage of males in this family is 25%, which can also be represented by the fraction

1

4.

Comparing Fractions, Decimals, and Percentages (pages 25 to 28)

1. Closest

to:

Closest

to:

Closest

to:

(a) 98% 1 (b) 11% 0 (c) 45% 1

2

(d) 0.02 0 (e) 0.899 1 (f) 0.6 1

2

(g) 8

9 1 (h)

11

20

1

2 (i)

1

16 0

(j) 0.3% 0 (k) 1.05 1 (l) 102% 1

(m) 13

24

1

2 (n) 56.2%

1

2 (o)

16

15 1

NSSAL 72 Draft

©2012 C. D. Pilmer

2. (a) 83% (b) 14.7%

(c) 136% (d) 3.1%

(e) 0.48 (f) 83%

(g) 2.3% (h) 0.65

(i) 105% (j) 1.45

(k) 7

8 (l)

9

16

(m) 135% (n) 1

10

(o) 7

16 (p) 93.5%

(q) 2

3 (r) 215%

(s) 4

5 (t) 0.48

(u) 81.2% (v) 8.3%

3. The numbers should occur in this order along the number line (from left to right).

1

20, 0.422, 54.7%, 0.93, 1.099, 125%, 180%, 200%,

12

4,

52

8,

13

16

4. The numbers should occur in this order along the number line (from left to right).

9.7%, 3

6, 0.713, 96%, 1.389, 155%,

191

20, 215%, 2.85,

32

4, 2.85, 3.2

5. (a) 0.7%, 5.8%, 32%, 91.2%, 124%

(b) 0.1, 14%, 64%, 0.745, 0.82

(c) 8.2%, 57.2%, 0.61, 123%, 1.45

(d) 3.8%, 5

12, 0.792, 86%,

9

10

(e) 1

32, 20%, 0.4, 68.5%,

11

10, 1.96

(f) 0.276, 30.2%, 8

16, 57.6%,

31

32, 1.1

(g) 0.096, 1

4, 64.5%, 0.89,

51

8, 209%

(h) 1

100, 0.08,

6

14, 50.3%, 91%, 0.956,

(i) 28%, 3

5, 0.9,

7

7, 1.02,

11

3, 194%

(j) 1

5, 0.34, 94.5%, 1.092,

71

12, 214%,

72

8

NSSAL 73 Draft

©2012 C. D. Pilmer

Adding and Subtracting Decimal Numbers (pages 29 to 35)

1. (a) 46.88 (b) 69.84

(c) 88.87 (d) 6.813

(e) 41.962 (f) 230.33

(g) 45.775 (h) 8.475

2. (a) 43.35 (b) 4.04

(c) 1.728 (d) 5.72

(e) 8.865 (f) 6.682

(g) 21.779 (h) 6.556

3. 3.67 kg

4. 3.42 kg

5. 24 293.1 km

6. 0.38 seconds

Multiplying Decimal Numbers (pages 36 to 41)

1. (a) 24.05 (b) 19.318

(c) 0.234 (d) 22.32

(e) 6.92 (f) 4.459

(g) 76.63 (h) 18.27

(i) 0.35334 (j) 77.86

2. (a) 22.656

(b) 75.64

(c) 177.3

(d) 384.85

(e) 804.2

(f) 2548.584

3. Question Arrow Question Arrow

0.98 2.1 d 0.326 2.21 e

5.23 4.37 b 2.34 1.98 a

1.1 1.97 f 0.49 2.88 c

NSSAL 74 Draft

©2012 C. D. Pilmer

4. Question Answers

(a) 2.93 + 3.208 (c) 32.185

(b) 16.08 - 5.239 (h) 311.74

(c) 7.85 4.1 (e) 21.78

(d) 47.9 + 32.7 (a) 6.138

(e) 29.58 - 7.8 (g) 126.7

(f) 39.8 6.1 (d) 80.6

(g) 98.3 + 28.4 (i) 1110.9

(h) 409.8 - 98.06 (b) 10.841

(i) 52.9 21 (f) 242.78

5. 730.5 km

6. $12.86

7. $480.70

8. 42.2 cm

9. 26.1 grams

Dividing Decimal Numbers (pages 42 to 52)

1. (a) 32.04 (b) 26.7

(c) 2.415 (d) 0.093

2. Questions Answers

(a) 389.6 8 (c) 0.506

(b) 49.02 6 (a) 48.7

(c) 2.024 4 (e) 4.69

(d) 257.8 9 (b) 8.17

(e) 32.83 7 (d) 28.62

3. (a) 6 453.6

(b) 0.8 0.736

(c) 5 3856

(d) 0.8 2.7345

(e) 5 490

NSSAL 75 Draft

©2012 C. D. Pilmer

(f) 6 820

(g) 0.03 1.826

(h) 7 5820

4. (a) 5.6 (b) 86

(c) 20.39 (d) 94.3

(e) 31.96 (f) 68

(g) 0.872 (h) 124.04

5. (a) 639.1 7 (c) 39.6

(b) 289.4 315.7 (h) 122.04

(c) 23.76 0.6 (e) 0.48

(d) 20.5 61.8 (a) 91.3

(e) 0.24 0.5 (f) 254.8

(f) 453.6 198.8 (g) 6.2

(g) 0.496 0.08 (i) 817

(h) 0.9 135.6 (d) 1266.9

(i) 32.68 0.4 (b) 605.1


6.65 7 b 3.05 2.97 a

1.49 1.04 e 0.31 5.2 c

0.145 0.05 f 1.278 0.6 d

7. $28.30

8. 18.2, but we round down such that the final answer is 18 hamburgers.

9. $87.75

10. $9.65

11. 257.7 km

Estimation Questions Involving Percentages (pages 53 to 56)

1. (a) 4 (b) 12

(c) 50 (d) 140

(e) 8 (f) 24

(g) 100 (h) 280

(i) 6 (j) 18

(k) 75 (l) 210

(m) 12 (n) 36

NSSAL 76 Draft

©2012 C. D. Pilmer

(o) 100 (p) 300

Answers are likely to vary slightly from learner to learner on questions 2 through 9. This is to be

expected with estimation questions. As long as your answer is close to our answer, assume that

you estimation technique was perfectly valid.

2. Approximately $10




6. Approximately $42 ($36 + $6)

7. Approximately $138 ($120 + $18)

8. Approximately $108 ($90 + $18)

9. Approximately $230 ($200 + $30)


25% of 11.90 g 0.52 0.496 c

0.784 0.4 e 10% of 21.50 f

1.43 1.316 a 20% of 2.99 b

30% of 4.90 d 2.1 1.513 h

Calculator Questions (pages 57 to 60)

1. 23 minutes

2. $154.77

3. $1.17

4. $59.80

5. Hint: Rember that we are purchasing each of these items for her two boys.

Answer: $211.54

6. $1099.80

7. $650.20

decimal numbers and percent unit - forbes math€¦ · decimal numbers and percent unit (pilot...

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