# Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors

Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors: Before knowing the numerical expressions, Grade 6 students must be familiar with composite numbers, factor pairs, and prime numbers. Refer to the concept of vocabulary terms, subtracting and adding the fractions, mixed fractions in the below sections.

Know the various terms like what is exponent, perfect square, power, composite numbers, prime numbers with the help of Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors. You can also get all the answers to the questions which are available in the Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors pdf format. Scroll to the below sections to check example problems, answer PDFs, etc.

## Big Ideas Math Book 6th Grade Answer Key Chapter 1 Numerical Expressions and Factors

BIM 6th Grade Chapter 1 Numerical Expression and Factors Answer key helps you in easy and quick learning. Download Big Ideas Math Book 6th Grade Answer Key Chapter 1 Numerical Expressions and Factors pdf to kickstart your preparation. Get the solutions for all the questions in this article. There are various topics included in numerical expressions like Least Common Multiple, Prime Factorisation, Powers and Exponents, Order of Operations, Greatest Common Factor, Exponents and Powers. Click on the below links and prepare for the exam as per the topics.

Lesson 1: Powers and Exponents

Lesson 2: Order of Operations

Lesson 3: Prime Factorization

Lesson 4: Greatest Common Factor

Lesson 5: Least Common Multiple

Chapter: 1 – Numerical Expressions and Factors

### Numerical Expressions and Factors Steam Video/Performance Task

Filling Piñatas

Common factors can be used to make identical groups of objects. Can you think of any situations in which you would want to separate objects into equal groups? Are there any common factors that may be more useful than others? Can you think of any other ways to use common factors?
watch the STEAM Video “Filling Piñatas.” Then answer the following questions. The table below shows the numbers of party favors that Alex and Enid use to make piñatas.

Question 1.
When ﬁnding the number of identical piñatas that can be made, why is it helpful for Alex and Enid to list the factors of each number given in the table?

Answer: By using the list of the factors of all the numbers Alex and Enid can make identical groups of the objects.

Question 2.
You want to create 6 identical piñatas. How can you change the numbers of party favors in the table to make this happen? Can you do this without changing the total number of party favors?

Answer: You can change the number of party favors to create 6 identical pinatas.
There are 100 Mints. So divide it into two identical groups.
Change the number of mints to 50. And add 50 to new identical pinatas.

Setting the Table

After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be asked to plan a fundraising event with the items below.
72 chairs
48 balloons
24 flowers
36 candles

You will ﬁnd the greatest number of identical tables that can be prepared, and what will be in each centerpiece. When making arrangements for a party, should a party planner always use the greatest number of identical tables possible? Explain why or why not.

72 chairs = 2 × 36
= 2 × 2 × 18
= 2 × 2 × 2 × 9
= 2 × 2 × 2 × 3 × 3
2, 2, 2, 3, 3
Therefore, 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 are the factors of 72.
48 balloons = 2 × 24
= 2 × 2 × 12
= 2 × 2 × 2 × 6
= 2 × 2 × 2 × 2 × 3
The positive Integer factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
The factors of number 24 are 1, 2, 3, 4, 6, 8, 12, 24.
The factors of number 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36
The greatest number of identical tables possible are 1, 2, 3, 4, 6, 12.

### Numerical Expressions and Factors Getting Ready for Chapter 1

Chapter Exploration

Work with a partner. In Exercises 1–3, use the table.

1. Cross out the multiples of 2 that are greater than 2. Do the same for 3, 5, and 7.
2. The numbers that are not crossed out are called prime numbers. The numbers that are crossed out are called composite numbers. In your own words, describe the characteristics of prime numbers and composite numbers.
3. MODELING REAL LIFE Work with a partner. Cicadas are insects that live underground and emerge from the ground after x or x + 4 years. Is it possible that both x and x +4 are prime? Give some examples.

The numbers that are not crossed are 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71, 73, 79, 83, 89, 97.
These are not multiples of any numbers. So, the above numbers are the prime numbers.

Vocabulary

The following vocabulary terms are deﬁned in this chapter. Think about what each term might mean and record your thoughts.

i. First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.
ii. “Factors” are numbers we can multiply together to get another number. When we find the factors of two or more numbers, and then find some factors are the same, then they are the “common factors”.
iii. A common multiple is a whole number that is a shared multiple of each set of numbers. The multiples that are common to two or more numbers are called the common multiples of those numbers. The smallest positive number is a multiple of two or more numbers.

### Lesson 1.1 Powers and Exponents

Exploration 1
Writing Expressions Using Exponents
Work with a partner. Copy and complete the table.

i. In your own words, describe what the two numbers in the expression 35 mean.

Answer: 35 means the number 3 repeats 5 times.
3 × 3 × 3 × 3 × 3 = 243

EXPLORATION 2
Using a Calculator to Find a Pattern

Work with a partner. Copy the diagram. Use a calculator to ﬁnd each value. Write one digit of the value in each box. Describe the pattern in the digits of the values.

1.1 Lesson

A power is a product of repeated factors. The base of a power is the repeated factor. The exponent of a power indicates the number of times the base is used as a factor.

Try It
Write the product as a power.
Question 1.
2 × 2 × 2

Two cubed or three to the two. Here 2 is repeated three times.

Question 2.

Six to the six. Here 6 is repeated six times.

Question 3.
15 × 15 × 15 × 15

15 to the power 4. Here 15 is repeated four times.

Question 4.

20 to the power 7. Here 20 is repeated seven times.

Try It
Find the value of the power.

Question 5.
63

Answer: 6 × 6 × 6 = 216
The value of the power 63 is 216.

Question 6.
92

Answer: 9 × 9 = 81
The value of the power 92 is 81

Question 7.
34

Answer: 3 × 3 × 3 × 3
The value of the power 34 is 81.

Question 8.
182

The value of the power 182 is 324.

Try It
Determine whether the number is a perfect square.

Question 9.
25

Yes, 25 is the perfect square.
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 10.
2

Answer: 2 is not a perfect square. 2 cannot be expressed as the square of a number from the same number system.

Question 11.
99

Answer: 99 is not a perfect square. 99 cannot be expressed as the square of a number from the same number system.

Question 12.
36

36 is a perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.
FINDING VALUES OF POWERS
Find the value of the power.

Question 13.
82

Answer: 8 × 8 = 64
The value of the power 82 is 64.

Question 14.
35

Answer: 3 × 3 × 3 × 3 × 3 = 243
The value of the power 35 is 243.

Question 15.
113

Answer: 11 × 11 × 11 = 1331
The value of the power 113 is 1331.

Question 16.
VOCABULARY
How are exponents and powers different?

An expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

Question 17.
VOCABULARY
Is 10 a perfect square? Is 100 a perfect square? Explain.

Answer: 10 is not a perfect square.
A perfect square is a number that is generated by multiplying two equal integers by each other.
100 is a perfect square. Because 10 × 10 = 100.

Question 18.
WHICH ONE DOESN’T BELONG?
Which one does not belong with the other three? Explain your reasoning.

24 = 2 × 2 × 2 × 2 = 16
32 = 3 × 3 = 9
3 + 3 + 3 + 3 = 3 × 4
5.5.5 = 125
The 3rd option does not belong to the other three expressions.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 19.
A square solar panel has an area of 16 square feet. Write the area as a power. Then ﬁnd the side lengths of the panel.

Explanation:
Given that,
A square solar panel has an area of 16 square feet.
A = s × s
16 = s²
4² = s²
s = 4
Thus the side length of the panel is 4 feet.

Question 20.
The four-square court shown is a square made up of four identical smaller squares. What is the area of the court?

Given,
The four-square court shown is a square made up of four identical smaller squares.
The side of each square is 6 feet.
6 + 6 = 12 feet
The area of the court is 12 ft × 12 ft
A = 144 square feet
Thus the area of the court is 144 square feet.

Question 21.
DIG DEEPER!
Each face of a number cube is a square with a side length of 16 millimeters. What is the total area of all of the faces of the number cube?

Given that,
Each face of a number cube is a square with a side length of 16 millimeters.
Area of the cube = 6 a²
A = 6 × 16 × 16
A = 1536 sq. mm

### Powers and ExponentsPractice 1.1

Review & Refresh

Multiply.

Question 1.
150 × 2

Explanation:
Multiply the two numbers 150 and 2.
First multiply 2 with ones place 2 × 0 = 0
Next multiply with tens place 2 × 50 = 100
Next multiply with hundreds place 2 × 100 = 200
200 + 100 = 300

Question 2.
175 × 8

Explanation:
Multiply the two numbers 175 and 8.
First, multiply 2 with ones place 8 × 5 = 40
Next multiply with tens place 8 × 70= 560
Next multiply with hundreds place 8 × 100 = 800
800 + 560 + 40 = 1400

Question 3.
123 × 3

Explanation:
Multiply the two numbers 123 and 3.
First multiply 2 with ones place 3 × 3 = 9
Next multiply with tens place 3 × 20 = 60
Next multiply with hundreds place 3 ×100 = 300
300 + 60 + 9 = 369

Question 4.
151 × 9

Explanation:
Multiply the two numbers 151 and 9.
First multiply 2 with ones place 9 × 1 = 9
Next multiply with tens place 9 × 50 = 450
Next multiply with hundreds place 9 × 100 = 900
900 + 450 + 9 = 1359

Write the sentence as a numerical expression.

Question 5.
Add 5 and 8, then multiply by 4.

Answer: The numerical expression for the above sentence is 5 + 8 × 4

Question 6.
Subtract 7 from 11, then divide by 2.

Answer: The numerical expression for the above sentence is 11 – 7 ÷ 2
Round the number to the indicated place value.

Question 7.
4.03785 to the tenths

Answer: The number 4.03785 nearest to the tenths is 4.0

Question 8.
12.89503 to the hundredths

Answer: The number 12.89503 nearest to the hundredths is 12.90

Complete the sentence.

Question 9.

Explanation:
(1/10) × 30 = 30/10 = 3
The product of 1/10 and 30 is 3.

Question 10.

Explanation:
(4/5) × 25 = 4 × 5 = 20
The product of 4/5 and 25 is 20.

Concepts, Skills, & Problem Solving

WRITING EXPRESSIONS USING EXPONENTS
Copy and complete the table. (See Exploration 1, p. 3.)

WRITING EXPRESSIONS AS POWERS
Write the product as a power.

Question 15.
9 × 9

Answer: The exponential form of the given expression is 9²

Question 16.
13 × 13

Answer: The exponential form of the given expression is 13²

Question 17.
15 × 15 × 15

Answer: The exponential form of the given expression is 15³

Question 18.
2.2.2.2.2

Answer: The exponential form of the given expression is 25

Question 19.
14 × 14 × 14

Answer: The exponential form of the given expression is 14³

Question 20.
8.8.8.8

Answer: The exponential form of the given expression is 84

Question 21.
11 × 11 × 11 × 11 × 11

Answer: The exponential form of the given expression is 115

Question 22.
7.7.7.7.7.7

Answer: The exponential form of the given expression is 76

Question 23.
16.16.16.16

Answer: The exponential form of the given expression is 164

Question 24.
43 × 43 × 43 × 43 × 43

Answer: The exponential form of the given expression is 435

Question 25.
167 × 167 × 167

Answer: The exponential form of the given expression is 167³

Question 26.
245.245.245.245

Answer: The exponential form of the given expression is 2454

FINDING VALUES OF POWERS
Find the value of the power.

Question 27.
52

Answer: The value of the powers 52 is 5 × 5 = 25

Question 28.
43

Answer: The value of the powers 43 is 4 × 4 × 4 = 64

Question 29.
62

Answer: The value of the powers 62 is 6 × 6 = 36

Question 30.
17

Answer: The value of the powers 17 is 1 × 1 × 1 × 1 × 1 × 1 × 1 = 1

Question 31.
03

Answer: The value of the powers 03 is 0 × 0 × 0 = 0

Question 32.
84

Answer: The value of the powers 84 is 8 × 8 × 8 × 8 = 4096

Question 33.
24

Answer: The value of the powers 24 is 2 × 2 × 2 × 2 = 64

Question 34.
122

Answer: The value of the powers 122 is 12 × 12 = 144

Question 35.
73

Answer: The value of the powers 73 is 7 × 7 × 7 = 343

Question 36.
54

Answer: The value of the powers 54 is 5 × 5 × 5 × 5 = 625

Question 37.
25

Answer: The value of the powers 25 is 2 × 2 × 2 × 2 × 2 = 32

Question 38.
142

Answer: The value of the powers 142 is 14 × 14 = 196

USING TOOLS
Use a calculator to ﬁnd the value of the power.

Question 39.
76

Answer: 7 × 7 × 7 × 7 × 7 × 7 = 117649

Question 40.
48

Answer: 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 256

Question 41.
124

Answer: 12 × 12 × 12 × 12 = 20736

Question 42.
175

Answer: 17 × 17 × 17 × 17 × 17 = 1419857

Question 43.
YOU BE THE TEACHER

83 is nothing but 8 repeats 3 times.
83 = 8 × 8 × 8 = 512

IDENTIFYING PERFECT SQUARES
Determine whether the number is a perfect square.

Question 44.
8

Answer: 8 is not the perfect square. 8 cannot be expressed as the square of a number from the same number system.

Question 45.
4

Answer: 4 is a perfect square.
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 46.
81

A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 47.
44

Answer: 44 is not the perfect square. 44 cannot be expressed as the square of a number from the same number system

Question 48.
49

Answer: 49 is a perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 49.
125

Answer: 125 is not the perfect square. 125 cannot be expressed as the square of a number from the same number system

Question 50.
150

Answer: 150 is not the perfect square. 150 cannot be expressed as the square of a number from the same number system

Question 51.
144

Answer: 144 is the perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 52.
MODELING REAL LIFE
On each square centimeter of a person’s skin, there are about 392 bacteria. How many bacteria does this expression represent?

Given,
On each square centimeter of a person’s skin, there are about 392 bacteria.
392  = 39 × 39 = 1521 centimeters
Thus the bacteria represents 1521 centimeters.

Question 53.
REPEATED REASONING
The smallest ﬁgurine in a gift shop is 2 inches tall. The height of each ﬁgurine is twice the height of the previous ﬁgurine. What is the height of the tallest ﬁgurine?

Given that,
The smallest ﬁgurine in a gift shop is 2 inches tall. The height of each ﬁgurine is twice the height of the previous ﬁgurine.
The second ﬁgurine is twice that of the first ﬁgurine = 2 × 2 = 4 inches
The third ﬁgurine is twice that of the second ﬁgurine = 4 × 4 = 16 inches
The fourth ﬁgurine is twice that of the third ﬁgurine = 16 × 16 = 256 inches
Thus the height of the tallest ﬁgurine is 256 inches.

Question 54.
MODELING REAL LIFE
A square painting measures 2 meters on each side. What is the area of the painting in square centimeters?

Given that,
A square painting measures 2 meters on each side.
Area of the square = s × s
A = 2 m × 2 m = 4 sq. meters
Thus the area of the painting in square centimeters is 4.

Question 55.
NUMBER SENSE
Write three powers that have values greater than 120 and less than 130.

11² = 11(11) = 121; this is between 120 and 130.
5³ = 5(5)(5) = 25(5) = 125; this is between 120 and 130.
2⁷ = 2(2)(2)(2)(2)(2)(2) = 4(2)(2)(2)(2)(2) = 8(2)(2)(2)(2) = 16(2)(2)(2) = 32(2)(2) = 64(2) = 128; this is between 120 and 130.

Question 56.
DIG DEEPER!
A landscaper has 125 tiles to build a square patio. The patio must have an area of at least 80 square feet.

a. What are the possible arrangements for the patio?

Given that a square patio of at least 80 square feet is to be built from 125 tiles of length 12 inches or 1 foot.
Since there are 125 tiles and the patio has a shape of a square of at least 80 square feet, then the possible dimensions of the patio are
9 ft × 9 ft = 81 ft
10 ft × 10 ft = 100 ft, and
11 ft × 11 ft = 121 ft.

b. How many tiles are not used in each arrangement?

For a patio of dimensions, 9ft by 9ft, the number of tiles that will not be used is given by 125 – 81 = 44
For a patio of dimensions, 10ft by 10ft, the number of tiles that will not be used is given by 125 – 100 = 25
For a patio of dimensions, 11ft by 11ft, the number of tiles that will not be used is given by 125 – 121 = 4

Question 57.
PATTERNS
Copy and complete the table. Describe what happens to the value of the power as the exponent decreases. Use this pattern to ﬁnd the value of 40.

40 = 1
Thus the value of 40 is 1.

Question 58.
REPEATED REASONING
How many blocks do you need to add to Square 6 to get Square 7? to Square 9 to get Square 10? to Square 19 to get Square 20? Explain.

You need to add 14 blocks to get square 7. The square 7 contains 7 × 7 = 49 blocks
You need to add 32 blocks to get square 9. The square 9 contains 9 × 9 = 81 blocks
You need to add 19 blocks to get square 10. The square 10 contains 10 × 10 = 100 blocks
You need to add 261 blocks to get square 19. The square 19 contains 19 × 19 = 361 blocks
You need to add 39 blocks to get square 20. The square 20 contains 20 × 20 = 400 blocks

### Lesson 1.2 Order of Operations

Order of Operations

EXPLORATION 1
Comparing Different Orders

Work with a partner. Find the value of each expression by using different orders of operations. Are your answers the same?

The answers for all the expressions are not the same. The values of each expression will change if you change the order of operations.
a. 3 + 2 × 2
5 × 2 = 10
3 + 2 × 2
3 + 4 = 7
b. Subtract then multiply
18 – 3 × 3
15 × 3 = 45
Multiply, then subtract
18 – 3 × 3
18 – 9 = 9
c. Multiply, then subtract
8 × 8 – 2
64 – 2 = 62
Subtract, then Multiply
8 × 8 – 2
8 × 6 = 48
6 × 6 + 2
36 + 2 = 38
6 × 6 + 2
6 × 8 = 48

EXPLORATION 2
Determining Order of Operations
Work with a partner.
a. Scientiﬁc calculators use a standard order of operations when evaluating expressions. Why is a standard order of operations needed?

Answer: The order of operations is a rule that tells you the right order in which to solve different parts of a math problem. The order of operations is important because it guarantees that people can all read and solve a problem in the same way.

b. Use a scientiﬁc calculator to evaluate each expression in Exploration 1. Enter each expression exactly as written. For each expression, which order of operations is correct?

a. 3 + 2 × 2  – Multiply, then add
b. 18 – 3 × 3 – Multiply, then subtract
c. 8 × 8 – 2 – Multiply, then subtract
d. 6 × 6 + 2 – Multiply, then add

c. What order of operations should be used to evaluate 3 + 22, 18 − 32, 82 − 2, and 62 + 2?

Solve the expressions by using the calculator.
a. 3 + 2 × 2
3 + 4 = 7
b. 18 – 3 × 3
18 – 9 = 9
c. 8 × 8 – 2
64 – 2 = 62
d. 6 × 6 + 2
36 + 2 = 38
d. Do 18 ÷ 3.3 and 18 ÷ 32 have the same value? Justify your answer.

Explanation:
18 ÷ 3.3
(18 ÷ 3) × 3
6 × 3 = 18
18 ÷ 32  = 2
By using the calculator you can find the difference.
e. How does evaluating powers ﬁt into the order of operations?

When an expression has parentheses and powers, evaluate it in the following order: contents of parentheses, powers from left to right, multiplication and division from left to right, and addition and subtraction from left to right.

1.2 Lesson

A numerical expression is an expression that contains numbers and operations. To evaluate, or ﬁnd the value of, a numerical expression, use a set of rules called the order of operations.
Key Idea
order of operations

1. Perform operations in grouping symbols.
2. Evaluate numbers with exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Try It
a. Evaluate the expression.

Question 1.
7.5 + 3

Explanation:
You have to evaluate the expression from left to right.
7(5 + 3) = 7 × 8
= 56

Question 2.
(28 – 20) ÷ 4

Explanation:
You have to evaluate the expression from left to right.
28 – 20 = 8
8 ÷ 4 = 2

Question 3.
[6 + (15 – 10)] × 5

Explanation:
You have to evaluate the expression from left to right.
[6 + (15 – 10)] × 5
[6 + 5] × 5
11 × 5 = 55

Try It
Evaluate the expression.

Question 4.
6 + 24 – 1

Explanation:
You have to evaluate the expression from left to right.
6 + 24 – 1
6 + (16 – 1)
6 + 15 = 21
6 + 24 – 1 = 21

Question 5.
4.32 + 18 – 9

Explanation:
You have to evaluate the expression from left to right.
4.32 + (18 – 9)
4.32 + 9
4 × 9 + 9
36 + 9 = 45

Question 6.
16 + (52 – 7) ÷ 3

Explanation:
You have to evaluate the expression from left to right.
16 + (52 – 7) ÷ 3
16 + (25 – 7) ÷ 3
16 + (18) ÷ 3
16 + (18 ÷ 3)
16 + 6 = 22

Thee symbols × and . are used to indicate multiplication. You can also use parentheses to indicate multiplication. For example, 3(2 +7) is the same as 3 × (2 + 7).

Try It
Evaluate the expression.

Question 7.
50 + 6(12 ÷ 4) – 82

Explanation:
You have to evaluate the expression from left to right.
50 + 6(12 ÷ 4) – 82
50 + 6(3) – 82
50 + 18 – 82
50 + 18 – 64
68 – 64
4

Question 8.

Explanation:
You have to evaluate the expression from left to right.
5² – 1/5 (10 – 5)
5² – 1/5 (5)
5² – 1
25 – 1
24

Question 9.

Explanation:
You have to evaluate the expression from left to right.
8(2+5) = 8 × 7
(8 × 7)/7 = 8

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 10.
7 + 2.4

Explanation:
You have to evaluate the expression from left to right.
7 + 2 × 4 = 7 + 8 = 15

Question 11.
8 ÷ 4 × 2

Explanation:
You have to evaluate the expression from left to right.
8 ÷ 4 = 2
2 × 2 = 4

Question 12.
3(5 + 1) ÷ 32

Explanation:
You have to evaluate the expression from left to right.
3(5 + 1) ÷ 32
3 × 6 ÷ 32
18 ÷ 9 =2

Question 13.
WRITING
Why does 12 − 8 ÷ 2 = 2?

12 − 8 ÷ 2
4 ÷ 2 = 2

Question 14.
REASONING
Describe the steps in evaluating the expression 8 ÷ (6 − 4) + 32.

8 ÷ (6 − 4) + 32
8 ÷ 2 + 32
4 + 9 = 13

Question 15.
WHICH ONE DOESN’T BELONG?
Which expression does not belong with the other three? Explain your reasoning.

Answer: (52 – 8) × 2 does not belong to the other three. Because the order of operations and expressions are different for the fourth option.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 16.
A square plot of land has side lengths of 40 meters. An archaeologist divides the land into 64 equal parts. What is the area of each part?

Given that,
A square plot of land has side lengths of 40 meters.
An archaeologist divides the land into 64 equal parts.
Side of the square field = 40m
Area of the square field = s × s
A = 40m × 40m
A = 1600 sq.m
Area of each part of the square = 1600/64 = 25 sq.m

Question 17.
A glass block window is made of two different-sized glass squares. The window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches. Find the total area of the small glass squares.

Given,
A glass block window is made of two different-sized glass squares.
The window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches.
40 × 10 = 400

Question 18.
DIG DEEPER!
A square vegetable garden has side lengths of 12 feet. You plant ﬂowers in the center portion as shown. You divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers. What is the area of the onion section?

A square vegetable garden has side lengths of 12 feet.
You plant flowers in the center portion of the garden, a square that has side lengths of 4 feet.
You divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers.
Given that,
→ side of the square vegetable garden is = 12 feet.
So,
→ Area of square vegetable garden = (side)² = (12)² = 144 feet².
Now, given that, inside this area, there is a square of side 4 feet reserved for flowers.
So,
→ The area of the flower section = (side)² = (4)² = 16 feet².
Therefore,
→ The rest of the garden that is intended for vegetables is = The total garden area – The flower section area = 144 – 16 = 128 feet².
Now, this remaining area is to be divided into four equal sections.
So,
→ The area of the onion section = (1/4) of remaining area = (1/4) × 128 = 32 feet².

### Order of Operations Practice 1.2

Review & Refresh

Write the product as a power.

Question 1.
11 × 11 × 11 × 11

Answer: The exponent for the product 11 × 11 × 11 × 11 is 114

Question 2.
13 × 13 × 13 × 13 × 13

Answer: The exponent for the product 13 × 13 × 13 × 13 × 13 is 135
Find the missing dimension of the rectangular prism.

Question 3.

Given that,
l = 6 in.
b = 4 in.
h = ?
v = 192 cu. in
Volume of the rectangular prism = lbh
192 = 6 × 4 × h
h = 192/24
h = 8
Thus the height of the rectangular prism is 8 inches.

Question 4.

Given that,
h = 9m
b = 3m
v = 135 cu. m
l = ?
The volume of the rectangular prism = lbh
135 = l × 3 × 9
135 = l × 27
l = 5m

Tell whether the number is prime or composite.

Question 5.
9

A natural number greater than 1 that is not prime is called a composite number.

Question 6.
11

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Question 7.
23

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Concepts, Skills, & Problem Solving

COMPARING DIFFERENT ORDERS
Find the value of the expression by using different orders of operations. Are your answers the same? (See Exploration 1, p. 9.)

Question 8.

4 + 6 × 6
10 × 6 = 60
4 + 6 × 6
4 + 36 = 40

Question 9.

5 × 5 – 3
5 × 2 = 10
5 × 5 – 3
25 – 3 = 22

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 10.
5 + 18 ÷ 6

Explanation:
First, divide then divide.
5 + 3 = 8

Question 11.
(11 – 3) ÷ 2 + 1

Explanation:
First, subtract and divide.
(11 – 3) ÷ 2 + 1
8 ÷ 2 + 1
4 + 1 = 5

Question 12.
45 ÷ 9 × 2

Explanation:
The first divide then multiply.
45 ÷ 9 × 2
5 × 2 = 10

Question 13.
62 – 3.4

Explanation:
Multiply then subtract
62 – 3.4
36 – 12
24

Question 14.
42 ÷ (15 – 23)

Explanation:
Subtract then divide.
42 ÷ (15 – 8)
42 ÷ 7
6

Question 15.
42.2 + 8.7

Explanation:
42.2 + 8.7
16 × 2 + 56
32 + 56 = 88

Question 16.
(52 – 2) × 15 + 4

Explanation:
(52 – 2) × 15 + 4
(25 – 2) × 1 + 4
23 + 4 = 27

Question 17.
4 + 2 × 32 – 9

Explanation:
4 + 2 × 32 – 9
4 + 18 – 9
4 + 9 = 13

Question 18.
8 ÷ 2 × 3 + 42 ÷ 4

Explanation:
8 ÷ 2 × 3 + 42 ÷ 4
(4 × 3) + (16 ÷ 4)
12 + 4
16

Question 19.
32 + 12 ÷ (6 – 3) × 8

Explanation:
32 + 12 ÷ (6 – 3) × 8
9 + (12 ÷ (6 – 3)) × 8
9 + (12 ÷ 3) × 8
9 + 4 × 8
9 + 32
41

Question 20.
(10 + 4) ÷ (26 – 19)

Explanation:
(10 + 4) ÷ (26 – 19)
14 ÷ 7
2

Question 21.
(52 – 4).2 – 18

Explanation:
((52 – 4).2) – 18
((25 – 4) × 2) – 18
(21 × 2) – 18
42 – 18
24

Question 22.
2 × [(16 – 8) × 2]

Explanation:
2 × [(16 – 8) × 2]
2 × [8 × 2]
2 × 16
32

Question 23.
12 + 8 × 33 – 24

Explanation:
12 + 8 × 33 – 24
12 + (8 × 27) – 24
12 + 216 – 24
12 + 192 = 204

Question 24.
62 ÷ [(2 + 4) × 23]

Explanation:
62 ÷ [(2 + 4) × 23]
36 ÷ [(2 + 4) × 23]
36 ÷ 6 × 8
6 × 8
48

YOU BE THE TEACHER

Question 25.

9 + 3 × 3²
9 + (27)
36

Question 26.

19 – 6 + 12
13 + 12
25

Question 27.
PROBLEM SOLVING
You need to read 20 poems in 5 days for an English project. Each poem is 2 pages long. Evaluate the expression 20 × 2 ÷ 5 to ﬁnd how many pages you need to read each day.

Given,
You need to read 20 poems in 5 days for an English project. Each poem is 2 pages long.
20 × 2 ÷ 5
40 ÷ 5 = 8
Thus you need to read 8 pages each day.

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 28.
12 – 2(7 – 4)

12 -(2 × (7 – 4))
12 – (2 × 3)
12 – 6 = 6

Question 29.
4(3 + 5) – 3(6 -2)

4(3 + 5) – 3(6 -2)
4 × 8 – 3 × 4
32 – 12
20

Question 30.

6 + 1/4 (12 -8)
6 + 1/4(4)
6 + 1
7

Question 31.
92 – 8(6 + 2)

81 – (8(6 + 2))
81 – (8 × 8)
81 – 64
17

Question 32.
4(3 – 1)3 + 7(6) – 52

4(3 – 1)3 + 7(6) – 52
4(2)3 + 7(6) – 52
4 × 8 + 42 – 25
32 + 42 – 25 = 49

Question 33.

8[(1 1/6 + 5/6) ÷ 4]
[8[7/6 + 5/6] ÷ 4]
8[12/6] ÷ 4
8[2 ÷ 4]
8(1/2)
4

Question 34.

49 – 2((11-3)/8)
49 – 2 (8/8)
49 – 2
47

Question 35.
8(7.3 + 3.7 – 8) ÷ 2

8(7.3 + 3.7 – 8) ÷ 2
(8(7.3 + 3.7 – 8)) ÷ 2
8 (11 – 8) ÷ 2
8 × 3 ÷ 2
24 ÷ 2
12

Question 36.
24(5.2 – 3.2) ÷ 4

24(5.2 – 3.2) ÷ 4
16 (5.2 – 3.2) ÷ 4
16 (2) ÷ 4
32 ÷ 4
8

Question 37.

36(3+5)/4
36 × 8/4
36 × 2
72

Question 38.

(144 – 24 + 1)/121
121/121
1

Question 39.

26 ÷ 2 + 5 = 18
18/6 = 3

Question 40.
PROBLEM SOLVING
Before a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave. Evaluate the expression 8 + 5(4) − 3(2) to ﬁnd how many people are in the theater.

Given,
Before a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave.
8 + (5 × 4) – (3 × 2)
8 + 20 – 6
28 – 6
22

Question 41.
MODELING REAL LIFE
The front door of a house is painted white and blue. Each window is a square with a side length of 7 inches. What is the area of the door that is painted blue?

Given,
The front door of a house is painted white and blue. Each window is a square with a side length of 7 inches.
Area of the square = s × s
A = 7 in × 7 in
A = 49 sq. inches
Therefore the area of the door that is painted blue is 49 sq. inches.

Question 42.
PROBLEM SOLVING
You buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school. After using a $10 gift card, how much do you owe? Explain how you solved the problem. Answer: Given, You buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school. Cost of 1 notebook =$2
6 notebooks = 6 × $2 =$12
Cost of 1 folder = $1 10 folders = 10 ×$1 = $10 Cost of 1 pack of pencils =$3
Cost of 1 lunch box = $8 So the total cost is$11 + $10 +$3 + $8 =$31
You used $10 gift card.$31 – $10 =$21
2nd prize: $25 to 25th caller 3rd prize: free concert tickets to 100th caller So, in order to get all three prizes the caller must be 15th, 25th, and 100th caller at the same time. But to find when the radio station will give first all three prizes we calculate L.C.M. of ( 15, 25, 100 ) that is 300 Hence, the station first gives away all three prizes to the 300th caller. Question 43. LOGIC You and a friend are running on treadmills. You run 0.5 mile every 3 minutes, and your friend runs 2 miles every 14 minutes. You both start and stop running at the same time and run a whole number of miles. What are the least possible numbers of miles you and your friend can run? Answer: If you run 0.5 miles every 3 minutes then you run 1 mile every 6 minutes. If your friend runs 2 miles every 14 minutes then your friend runs 1 mile every 7 minutes. You will both then both run a whole number of minutes for a time that is a multiple of 6 and 7. The least common multiple of 6 and 7 is 42 so the least possible time you and your friend could run for and both run a whole number of miles is then 42 minutes. Since you run 1 mile every 6 minutes, in 42 minutes you will run 42/6=7 miles. Since your friend runs 1 mile every 7 minutes, in 42 minutes your friend will run 42/7=6 miles. Question 44. VENN DIAGRAM Refer to the Venn diagram. a. Copy and complete the Venn diagram. b. What is the LCM of 16, 24, and 40? c. What is the LCM of 16 and 40? 24 and 40? 16 and 24? Explain how you found your answers. Answer: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. LCM of 16, 40: Multiples of 16: 16, 32, 48, 64, 80, 96, 112 Multiples of 40: 40, 80, 120, 160 Therefore, LCM(16, 40) = 80 LCM of 24, 40: Multiples of 24: 24, 48, 72, 96, 120, 144, 168 Multiples of 40: 40, 80, 120, 160, 200 Therefore, LCM(24, 40) = 120 LCM of 16, 24: Multiples of 16: 16, 32, 48, 64, 80 Multiples of 24: 24, 48, 72, 96 Therefore, LCM(16, 24) = 48 CRITICAL THINKING Tell whether the statement is always, sometimes, or never true. Explain your reasoning. Question 45. The LCM of two different prime numbers is their product. Answer: Always true Example: 3 × 5 = 15 Question 46. The LCM of a set of numbers is equal to one of the numbers in the set. Answer: The LCM of a set of numbers is equal to one of the numbers in the set. Always Sometimes Never true. Question 909193: The LCM of a set of numbers is equal to one of the numbers in the set. This is sometimes true. Question 47. The GCF of two different numbers is the LCM of the numbers. Answer: Another way to find the LCM of two numbers is to divide their product by their greatest common factor ( GCF ). Example 2: Find the least common multiple of 18 and 20. The common factors are 2 and 3 . ### Numerical Expressions and Factors Connecting Concepts Getting Ready for Chapter Connecting Concepts Using the Problem-Solving Plan Question 1. A sports team gives away shirts at the stadium. There are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium. What is the greatest number of groups that can be formed using every shirt? Understand the Problem You know the number of large shirts and two relationships among the numbers of small, medium, and large shirts. You are asked to ﬁnd the greatest number of identical groups that can be formed using every shirt. Make a plan Break the problem into parts. First use multiplication to ﬁnd the number of each size shirt. Then ﬁnd the GCF of these numbers. Solve and Check Use the plan to solve the problem. Then check your solution. Answer: Given, A sports team gives away shirts at the stadium. There are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium. 60 × 1.6 = 96 60 × 1.5 = 90 The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 Then the greatest common factor is 6. Question 2. An escape artist ﬁlls the tank shown with water. Find the number of cubic feet of water needed to ﬁll the tank. Then ﬁnd the number of cubic yards of water that are needed to ﬁll the tank. Justify your answer. Answer: Given, An escape artist ﬁlls the tank shown with water. side = 6 ft We know that The volume of the cube = s³ V = 6ft × 6ft × 6ft V = 216 cubic ft. Performance Task Setting the Table At the beginning of this chapter, you watched a STEAM video called “Filling Piñatas.” You are now ready to complete the performance task for this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task. Answer: Factors of 50 – 1, 2, 5, 10, 25, and 50 Factors of 12 – 1, 2, 3, 4, 6, 12 Factors of 16 – 1, 2, 4, 8, 16 The factors of number 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 100 are 1,2,4,5,10,20,25,50 and 100. ### Numerical Expressions and Factors Chapter Review Review Vocabulary Write the deﬁnition and give an example of each vocabulary term. Graphic Organizers You can use an Information Frame to organize and remember concepts. Here is an example of an Information Frame for the vocabulary term power. Choose and complete a graphic organizer to help you study the concept. 1. perfect square 2. numerical expression 3. order of operations 4. prime factorization 5. greatest common factor (GCF) 6. least common multiple (LCM) Answer: 1. A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system. Examples of Numbers that are Perfect Squares. 25 is a perfect square. 2. A numerical expression is a mathematical sentence involving only numbers and one or more operation symbols. Examples of operation symbols are the ones for addition, subtraction, multiplication, and division. 3. Order of operations refers to which operations should be performed in what order, but it’s just convention. 4. “Prime Factorization” is finding which prime numbers multiply together to make the original number. 5. Greatest Common Factor. The highest number that divides exactly into two or more numbers. 6. Least Common Multiple. The smallest positive number that is a multiple of two or more numbers. Chapter Self-Assessment As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal. 1.1 Powers and Exponents (pp. 3-8) Write the product as a power. Question 1. 3 × 3 × 3 × 3 × 3 × 3 Answer: The product of 3 × 3 × 3 × 3 × 3 × 3 is 36 Question 2. 5 × 5 × 5 Answer: The product of 5 × 5 × 5 is 53 Question 3. 17 . 17 . 17 . 17 . 17 Answer: The product of 17 . 17 . 17 . 17 . 17 is 175 Question 4. 33 Answer: 3 × 3 × 3 Question 5. 26 Answer: 2 × 2 × 2 × 2 × 2 × 2 Question 6. 44 Answer: 4 × 4 × 4 × 4 Question 7. Write a power that has a value greater than 23 and less than 33. Answer: The power that has a value greater than 23 and less than 33 is 4² Question 8. Without evaluating, determine whether 25 or 42 is greater. Explain. Answer: 25 > 42 Explanation: The exponent with the highest number will be greater. Question 9. The bases on a softball ﬁeld are square. What is the area of each base? Answer: Given, The bases on a softball ﬁeld are square. s = 15 inches We know that, Area of the square = s × s A = 15 × 15 A = 225 sq. in Thus the area of each base is 225 sq. in. 1.2 Order of Operations (pp. 9–14) Evaluate the expression. Question 10. 3 × 6 – 12 ÷ 6 Answer: 16 Explanation: You have to evaluate from left to right. (3 × 6) – (12 ÷ 6) 18 – 2 = 16 Question 11. 30 ÷ (14 – 22) × 5 Answer: 15 Explanation: You have to evaluate from left to right. 30 ÷ (14 – 4) × 5 30 ÷ 10 × 5 3 × 5 = 15 Question 12. Answer: 15 Explanation: You have to evaluate from left to right. 2.3 + 3.7 = 6 5(6)/2 = 30/2 = 15 Question 13. Answer: 37 Explanation: You have to evaluate from left to right. 7² + 5 = 49 + 5 = 54 1/2 × 54 = 27 4³ – 27 = 64 – 27 = 37 Question 14. 20 (32 – 4) ÷ 50 Answer: 2 Explanation: You have to evaluate from left to right. (32 – 4) = 9 – 4 = 5 20 × 5 ÷ 50 100 ÷ 50 = 2 Question 15. 5 + 3(42 – 2) ÷ 6 Answer: 12 Explanation: You have to evaluate from left to right. (42 – 2) = 16 – 2 = 14 5 + 3(14) ÷ 6 5 + 42 ÷ 6 5 + 7 = 12 Question 16. Use grouping symbols and at least one exponent to write a numerical expression that has a value of 80. Answer: 6 + (9² – 7) = 80 1.3 Prime Factorization (pp. 15–20) List the factor pairs of the number. Question 17. 28 Answer: The factor pairs of the number 28 are 1, 2, 4, 7, 14, 28 Explanation: 28 = 1 × 28 2 × 14 4 × 7 7 × 4 14 × 2 28 × 1 Question 18. 44 Answer: The factor pairs of the number 44 are 1, 2, 4, 11, 44. Explanation: 44 = 1 × 44 2 × 22 4 × 11 11 × 4 44 × 1 Question 19. 96 Answer: The factor pairs of the number 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 Explanation: 1 and 96 are a factor pair of 96 since 1 x 96= 96 2 and 48 are a factor pair of 96 since 2 x 48= 96 3 and 32 are a factor pair of 96 since 3 x 32= 96 4 and 24 are a factor pair of 96 since 4 x 24= 96 6 and 16 are a factor pair of 96 since 6 x 16= 96 8 and 12 are a factor pair of 96 since 8 x 12= 96 12 and 8 are a factor pair of 96 since 12 x 8= 96 16 and 6 are a factor pair of 96 since 16 x 6= 96 24 and 4 are a factor pair of 96 since 24 x 4= 96 32 and 3 are a factor pair of 96 since 32 x 3= 96 48 and 2 are a factor pair of 96 since 48 x 2= 96 96 and 1 are a factor pair of 96 since 96 x 1= 96 Question 20. There are 36 graduated cylinders to put away on a shelf after science class. The shelf can ﬁt a maximum of 20 cylinders across and 4 cylinders deep. The teacher wants each row to have the same number of cylinders. List the possible arrangements of the graduated cylinders on the shelf. Answer: Given, There are 36 graduated cylinders to put away on a shelf after science class. The shelf can ﬁt a maximum of 20 cylinders across and 4 cylinders deep. The teacher wants each row to have the same number of cylinders. There are three possible arrangements of the graduated cylinders on the shelf. 1. 4 rows of 9 graduated cylinders 2. 3 rows of 12 graduated cylinders. 3. 2 rows of 18 graduated cylinders. Write the prime factorization of the number. Question 21. 42 Answer: The prime factorization of the number 42 is 2 × 3 × 7 Explanation: 42 = 2 × 21 = 2 × 3 × 7 Question 22. 50 Answer: The prime factorization of the number 2 × 5 × 5 Explanation: 50 = 2 × 25 = 2 × 5 × 5 Question 23. 66 Answer: The prime factorization of the number 2 × 3 × 11 Explanation: 66 = 2 × 33 = 2 × 3 × 11 1.4 Greatest Common Factor (pp. 21–26) Find the GCF of the numbers using lists of factors. Question 24. 27, 45 Answer: GCF is 9 Explanation: The factors of 27 are: 1, 3, 9, 27 The factors of 45 are: 1, 3, 5, 9, 15, 45 Then the greatest common factor is 9. Question 25. 30, 48 Answer: GCF is 6 Explanation: The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Then the greatest common factor is 6. Question 26. 28, 48 Answer: GCF is 4 Explanation: The factors of 28 are: 1, 2, 4, 7, 14, 28 The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Then the greatest common factor is 4. Find the GCF of the numbers using prime factorizations. Question 27. 24, 80 Answer: The prime factorization is the product of the circled primes. So the prime factorization of 24 is 24 = 2 · 2 · 2 · 3 = 2³ . 3 The prime factorization is the product of the circled primes. So the prime factorization of 80 is 80 = 2 x 2 x 2 x 2 x 5 = 2² . 2² . 5 Question 28. 52, 68 Answer: The prime factorization is the product of the circled primes. So the prime factorization of 52 is 2 x 2 x 13 = 2² . 13 The prime factorization is the product of the circled primes. So the prime factorization of 68 is 2 × 2 × 17 = 2². 17 Question 29. 32, 56 Answer: The prime factorization is the product of the circled primes. So the prime factorization of 32 is 2 x 2 x 2 x 2 x 2 = 25 The prime factorization is the product of the circled primes. So the prime factorization of 56 is 2 x 2 x 2 x 7 = 2³ . 7 Question 30. Write a pair of numbers that have a GCF of 20. Answer: The prime factors of 20 are 2 x 2 x 5. The GCF of 20 is 5. Question 31. What is the greatest number of friends you can invite to an arcade using the coupon such that the tokens and slices of pizza are equally split between you and your friends with none left over? How many slices of pizza and tokens will each person receive? Answer: (n/4)-1 Explanation: Total slices = n Total number of people =4 Each people may be eat = n/4 slices Here Harris eats 1 slice fewer Then Harris eats (n/4)-1 slices 1.5 Least Common Multiple (pp. 27–32) Find the LCM of the numbers using lists of multiples. Question 32. 4, 14 Answer: 28 Explanation: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36 Multiples of 14: 14, 28, 42, 56 Therefore, LCM(4, 14) = 28 Question 33. 6, 20 Answer: 60 Explanation: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. Multiples of 20: 20, 40, 60, 80, 100 The LCM of 6, 20 is 60 Question 34. 12, 28 Answer: 84 Explanation: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96 Multiples of 28: 28, 56, 84, 112, 140, 168, 196 LCM of 12, 28 is 84 Find the LCM of the numbers using prime factorizations. Question 35. 6, 45 Answer: Prime Factorisation of 6: 2 × 3 Prime Factorisation of 45: 3 × 3 × 5 LCM is 2 × 3 × 3 × 3 × 5 = 60 Question 36. 10, 12 Answer: 60 Prime factorization of 10: 2 × 5 Prime factorization of 12: 2 × 2 × 3 LCM is 5 × 2 × 2 × 3 = 60 Question 37. 18, 27 Answer: Prime factorization of 18: 2 × 3 × 3 Prime factorization of 27: 3 × 3 × 3 LCM is 2 × 3 × 3 × 3 = 54 Question 38. Find the LCM of 8, 12, and 18. Answer: 72 Prime Factorisation of 8: 2 × 2 × 2 Prime Factorisation of 12: 2 × 2 × 3 Prime factorization of 18: 2 × 3 × 3 LCM = 72 Question 39. Write a pair of numbers that have an LCM of 84. Answer: 84 and 12 Explanation: The LCM of 84 and 12 is 84. Prime factorization of 12: 2 × 2 × 3 Prime factorization of 84: 2 × 2 × 3 × 7 The Least Common Multiple is 2 × 2 × 3 × 7 = 84 Question 40. Write three numbers that have an LCM of 45. Answer: 3, 15, 45 Explanation: The prime factorization of 15: 3 × 5 The prime factorization of 45: 3 × 3 × 5 The LCM of 3, 15, 45 is 45. Question 41. You water your roses every sixth day and your hydrangeas every ﬁfth day. Today you water both plants. In how many days will you water both plants on the same day again? Answer: 30 Explanation: Given, You water your roses every sixth day and your hydrangeas every ﬁfth day. Today you water both plants. 6 × 5 = 30 You water both plants for 30 days on the same day again. Question 42. Hamburgers are sold in packages of 20, while buns are sold in packages of 12. What are the least numbers of packages you should buy in order to have the same number of hamburgers and buns? Answer: Given, Hamburgers are sold in packages of 20, while buns are sold in packages of 12. At least 5 packages of buns and 3 packages of hamburgers. 20×3=60 12×5=60 So that is how you get the answer by seeing if they have any integers in common. Question 43. A science museum is giving away a magnetic liquid kit to every 50th guest and a plasma ball to every 35th guest until someone receives both prizes. a. Which numbered guest will receive both a magnetic liquid kit and a plasma ball? Answer: A magnetic liquid kit prize every 50 guests and a plasma ball every 35 guests. 1.Guest 50th 2.Guest 100th 3.Guest 150th 4.Guest 200th 5.Guest 250th 6.Guest 300th 7.Guest 350th 8.Guest 400th and so on, in case no coincidence would happen. b. How many people will receive a plasma ball? Answer: 1.Guest 35th 2.Guest 70th 3.Guest 105th 4.Guest 140th 5.Guest 175th 6.Guest 210th 7.Guest 245th 8.Guest 280th 9.Guest 315th 10.Guest 350th. As you can see, Guest 350th will be the first one to receive both prizes, and including him or her, a total of ten guests will receive the plasma ball until that moment. There wasn’t any coincidence before Guest 350th. ### Numerical Expressions and Factors Practice Test Question 1. Find the value of 23. Answer: 23 can be written as 2 × 2 × 2 = 8 Thus the value of 23 is 8. Question 2. Evaluate Answer: 5 + 4(12 – 2) = 5 + 4(10) = 5 + 40 (5 + 40)/3² = 45/9 = 5 Thus the value of is 5. Question 3. Write 264.264.264 as a power Answer: 264.264.264 can be written as 264³ Question 4. List the factor pairs of 66. Answer: The factor pairs of 66 are (1,66) (2, 33) (6, 11) 66 = 1 × 66 66 = 2 × 33 66 = 6 × 11 Question 5. Write the prime factorization of 56. Answer: 56 = 2 × 28 = 2 × 2 × 14 = 2 × 2 × 2 × 7 Thus the prime factorization of 56 is 2 × 2 × 2 × 7 Find the GCF of the numbers. Question 6. 24, 54 Answer: GCF is 6 The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54 Then the greatest common factor is 6. Question 7. 16, 32, 72 Answer: GCF is 8 The factors of 16 are: 1, 2, 4, 8, 16 The factors of 32 are: 1, 2, 4, 8, 16, 32 The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Then the greatest common factor is 8. Question 8. 52, 65 Answer: GCF is 13 The factors of 52 are: 1, 2, 4, 13, 26, 52 The factors of 65 are: 1, 5, 13, 65 Then the greatest common factor is 13. Find the LCM of the numbers. Question 9. 9, 24 Answer: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 Multiples of 24: 24, 48, 72, 96, 120 Therefore, LCM(9, 24) = 72 Question 10. 26, 39 Answer: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 26: 26, 52, 78, 104, 130 Multiples of 39: 39, 78, 117, 156 Therefore, LCM(26, 39) = 78 Question 11. 6, 12, 14 Answer: Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96 Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108 Multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112 Therefore, LCM(6, 12, 14) = 84 Question 12. You have 16 yellow beads, 20 red beads, and 24 orange beads to make identical bracelets. What is the greatest number of bracelets that you can make using all of the beads? Answer: To find how many identical bracelets you can make, you need to find a common denominator. In this case, all three numbers; 16, 20, and 24, can be divided by four. So you now know you can have four bracelets. Then you take your numbers of each color beads and divide them by four so you know how many of each color will be on the bracelets. In the end, you have four bracelets, each with 4 yellow beads, 5 red beads and 6 orange beads Question 13. A bag contains equal numbers of green marbles and blue marbles. You can divide all of the green marbles into groups of 12 and all the blue marbles into groups of 16. What is the least number of each color of marble that can be in the bag? Answer: Given, A bag contains equal numbers of green marbles and blue marbles. You can divide all of the green marbles into groups of 12 and all the blue marbles into groups of 16. To solve this problem, we need to find for the LCM of each number. That is: 12: 12, 24, 36, 48, 60 16: 16, 32, 48, 64, 80 So we can see that the LCM is 48. Therefore the least number of each color of marble must be 48. Question 14. The ages of the members of a family are 65, 58, 27, 25, 5, and 2 years old. What is the total admission price for the family to visit the zoo? Answer: The ages of the members of a family are 65, 58, 27, 25, 5, and 2 years old. We can find the total admission price for the family to visit the zoo by following the above table.$10 + $12 +$27 + $12 +$8 + $8 =$77

Question 15.
A competition awards prizes for fourth, third, second, and ﬁrst place. The fourth place winner receives $5. Each place above that receives a prize that is ﬁve times the amount of the previous prize. How much prize money is awarded? Answer: A competition awards prizes for fourth, third, second, and ﬁrst place. The fourth place winner receives$5. Each place above that receives a prize that is ﬁve times the amount of the previous prize.
Each place above that receives a prize that is five times the amount of the previous prize
So we can say that;
Pn = 5 ×  P(n+1)
Where n ⇒ Number Place
Pn = Price received by Number place
Substituting the values of n as 3,2,1 to find the price of third second first place winner.
P3 = 5 ×  P(3+1) = 5 × P4 = 5 × 5 = 25
P2 = 5 ×  P(2+1) = 5 × P3 = 5 × 25 = 125
P1 = 5 ×  P(1+1) = 5 × P2 = 5 × 125 = 625
Now We will find the Total Prize money awarded.
Total Prize money awarded = 625 + 125 + 25 + 5 = 780
Hence A total of $780 price money was awarded. Question 16. You buy tealight candles and mints as party favors for a baby shower. The tealight candles come in packs of 12 for$3.50. The mints come in packs of 50 for $6.25. What is the least amount of money you can spend to buy the same number of candles and mints? Answer: The least amount of money it can be spent is$125.

Explanation:
First, we write the prime factorization of each number:
12= 2·2·3
15= 2·5·5
Then, we search for each different factor which appears the greater number of times. The factor 2 appears in both factorizations so the least common multiple is:
LCM= 2·2·3·5·5=300
Hence, the total quantity of packs of each thing is:
Candles: 300÷12=25
Mints: 300÷50=6
The least amount of money it can be spent is:
T=25×$3.50 + 6×$6.25= $87.5 +$37.5= \$125

### Numerical Expressions and Factors Cumulative Practice

Question 1.
Find the value of 8 × 135?

Answer: Multiply the two numbers 8 and 135
8 × 135 = 1080

Question 2.
Which number is equivalent to the expression blow?
3.23 – 8 ÷ 4

Explanation:
Given the expression 3.23 – 8 ÷ 4
3 × 8 -(8÷4)
24 – (2)
24 – 2 = 22

Question 3.
The top of an end table is a square with a side length of 16 inches. What is the area of the tabletop?

Explanation:
Given that
The top of an end table is a square with a side length of 16 inches.
Area of the square = s × s
A = 16 × 16
A = 256 in²
Thus the correct answer is option I.

Question 4.
You are ﬁlling baskets using 18 green eggs, 36 red eggs, and 54 blue eggs. What is the greatest number of baskets that you can ﬁll so that the baskets are identical and there are no eggs left over?
A. 3
B. 6
C. 9
D. 18

Explanation:
Given,
You are ﬁlling baskets using 18 green eggs, 36 red eggs, and 54 blue eggs.
18/n = 36/n = 54/n
Factors of 18 are 2,3,6,9,18
Factors of 36 are 2,3,4,6,9,12,18.
Factors of 54 are 2,3,6,9,18,27,54.
The common multiples of 18,36 and 54 are 2,3,6,9,18.
Thus the greatest among them is 18.
Thus the correct answer is option D.

Question 5.
What is the value of 23.32.5?

23.32.5
2³ = 8
3² = 9
8 × 9 × 5 = 360

Question 6.
You hang the two strands of decorative lights shown below.

Both strands just changed color. After how many seconds will the strands change color at the same time again?
F. 3 seconds
G. 30 seconds
H. 90 seconds
I. 270 seconds

Explanation:

Strand I: Changes between red and blue every 15 seconds
Strand II: Changes between green and gold every 18 seconds
18 – 15 = 3 seconds
Thus the correct answer is option F.

Question 7.
Point P is plotted in the coordinate plane below.

What are the coordinates of Point P ?
A. (5, 3)
B. (4, 3)
C. (3, 5)
D. (3, 4)

By seeing the above graph we can find the coordinates of point p.
The X-axis is on 3 and Y-axis is on 5.
Thus the correct answer is option c.

Question 8.
What is the prime factorization of 1100?
F. 2 × 5 × 11
G. 22 × 52 × 11
H. 4 × 52 × 11
I. 22 × 5 × 55

Prime factorization of 1100: 2 × 550
2 × 2 × 275
2 × 2 × 5 × 55
2 × 2 × 5 × 5 × 11
Thus the correct answer is option b.

Question 9.
What is the least common multiple of 3, 8, and 10?
A. 24
B. 30
C. 80
D. 120

Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126
Multiples of 8:
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136
Multiples of 10:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 140.
The common multiple among the three is 120.
Thus the correct answer is option D.

Question 10.
What is the area of the shaded region of the ﬁgure below?

The above figure is square.
s = 4 yd
Area of the square = s × s
A = 4 yd × 4 yd
A = 16 sq. yd
The area of the outer box.
s = 9 yd
Area of the square = s × s
A = 9 yd × 9 yd
A = 81 sq. yd
The area of the shaded region is 81 – 16 = 65 sq. yd
Thus the correct answer is option G.

Question 11.
Which expression represents a prime factorization?
A. 4 × 4 × 7
B. 22 × 21 × 23
C. 34 × 5 × 7
D. 5 × 5 × 9 × 11

Prime factorization:
22 × 21 × 23
2, 21, 23 is a prime number.
Thus the correct answer is option B.

Question 12.
Find the greatest common factor for each pair of numbers.

What can you conclude about the greatest common factor of 10, 15, and 21? Explain your reasoning.