lesson 2 problem solving practice multiplication and division equations answer key

2.2 Solve Equations using the Division and Multiplication Properties of Equality

Learning objectives.

By the end of this section, you will be able to:

• Solve equations using the Division and Multiplication Properties of Equality
• Solve equations that require simplification
• Translate to an equation and solve
• Translate and solve applications

Be Prepared 2.5

Before you get started, take this readiness quiz.

Simplify: −7 ( 1 −7 ) . −7 ( 1 −7 ) . If you missed this problem, review Example 1.68 .

Be Prepared 2.6

Evaluate 9 x + 2 9 x + 2 when x = −3 x = −3 . If you missed this problem, review Example 1.57 .

Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form x + a = b x + a = b or x − a = b x − a = b . We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Let’s look at our puzzle again with the envelopes and counters in Figure 2.5 .

In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since 6 ÷ 2 = 3 6 ÷ 2 = 3 ).

What equation models the situation shown in Figure 2.6 ? There are two envelopes, and each contains x x counters. Together, the two envelopes must contain a total of 6 counters.

We found that each envelope contains 3 counters. Does this check? We know 2 · 3 = 6 2 · 3 = 6 , so it works! Three counters in each of two envelopes does equal six!

This example leads to the Division Property of Equality .

The Division Property of Equality

For any numbers a , b , and c , and c ≠ 0 c ≠ 0 ,

When you divide both sides of an equation by any non-zero number, you still have equality.

Manipulative Mathematics

The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.

Example 2.13

Solve: 5 x = −27 . 5 x = −27 .

Try It 2.25

Solve: 3 y = −41 . 3 y = −41 .

Try It 2.26

Solve: 4 z = −55 . 4 z = −55 .

Consider the equation x 4 = 3 x 4 = 3 . We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

The Multiplication Property of Equality

For any numbers a , b , and c ,

If you multiply both sides of an equation by the same number, you still have equality.

Example 2.14

Solve: y −7 = −14 . y −7 = −14 .

Here y y is divided by −7 −7 . We must multiply by −7 −7 to isolate y y .

Try It 2.27

Solve: a −7 = −42 . a −7 = −42 .

Try It 2.28

Solve: b −6 = −24 . b −6 = −24 .

Example 2.15

Solve: − n = 9 . − n = 9 .

Try It 2.29

Solve: − k = 8 . − k = 8 .

Try It 2.30

Solve: − g = 3 . − g = 3 .

Example 2.16

Solve: 3 4 x = 12 . 3 4 x = 12 .

Since the product of a number and its reciprocal is 1, our strategy will be to isolate x x by multiplying by the reciprocal of 3 4 3 4 .

Try It 2.31

Solve: 2 5 n = 14 . 2 5 n = 14 .

Try It 2.32

Solve: 5 6 y = 15 . 5 6 y = 15 .

In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

Example 2.17

Solve: 8 15 = − 4 5 x . 8 15 = − 4 5 x .

Try It 2.33

Solve: 9 25 = − 4 5 z . 9 25 = − 4 5 z .

Try It 2.34

Solve: 5 6 = − 8 3 r . 5 6 = − 8 3 r .

Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Example 2.18

Solve: 14 − 23 = 12 y − 4 y − 5 y . 14 − 23 = 12 y − 4 y − 5 y .

Begin by simplifying each side of the equation.

Try It 2.35

Solve: 18 − 27 = 15 c − 9 c − 3 c . 18 − 27 = 15 c − 9 c − 3 c .

Try It 2.36

Solve: 18 − 22 = 12 x − x − 4 x . 18 − 22 = 12 x − x − 4 x .

Example 2.19

Solve: −4 ( a − 3 ) − 7 = 25 . −4 ( a − 3 ) − 7 = 25 .

Here we will simplify each side of the equation by using the distributive property first.

Try It 2.37

Solve: −4 ( q − 2 ) − 8 = 24 . −4 ( q − 2 ) − 8 = 24 .

Try It 2.38

Solve: −6 ( r − 2 ) − 12 = 30 . −6 ( r − 2 ) − 12 = 30 .

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Properties of Equality

When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.

Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Example 2.20

Translate and solve: The number 143 is the product of −11 −11 and y .

Begin by translating the sentence into an equation.

Try It 2.39

Translate and solve: The number 132 is the product of −12 and y .

Try It 2.40

Translate and solve: The number 117 is the product of −13 and z .

Example 2.21

Translate and solve: n n divided by 8 is −32 −32 .

Try It 2.41

Translate and solve: n n divided by 7 is equal to −21 −21 .

Try It 2.42

Translate and solve: n n divided by 8 is equal to −56 −56 .

Example 2.22

Translate and solve: The quotient of y y and −4 −4 is 68 68 .

Try It 2.43

Translate and solve: The quotient of q q and −8 −8 is 72.

Try It 2.44

Translate and solve: The quotient of p p and −9 −9 is 81.

Example 2.23

Translate and solve: Three-fourths of p p is 18.

Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

Try It 2.45

Translate and solve: Two-fifths of f f is 16.

Try It 2.46

Translate and solve: Three-fourths of f f is 21.

Example 2.24

Translate and solve: The sum of three-eighths and x x is one-half.

Try It 2.47

Translate and solve: The sum of five-eighths and x is one-fourth.

Try It 2.48

Translate and solve: The sum of three-fourths and x is five-sixths.

Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

Example 2.25

Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

Try It 2.49

Translate and solve:

Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?

Try It 2.50

At JB’s Bowling Alley, 6 people can play on one lane for $34.98. What is the cost for each person?

Example 2.26

Andreas bought a used car for $12,000. Because the car was 4-years old, its price was 3 4 3 4 of the original price, when the car was new. What was the original price of the car?

Try It 2.51

The annual property tax on the Mehta’s house is $1,800, calculated as 15 1,000 15 1,000 of the assessed value of the house. What is the assessed value of the Mehta’s house?

Try It 2.52

Stella planted 14 flats of flowers in 2 3 2 3 of her garden. How many flats of flowers would she need to fill the whole garden?

Section 2.2 Exercises

Practice makes perfect.

In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

8 x = 56 8 x = 56

7 p = 63 7 p = 63

−5 c = 55 −5 c = 55

−9 x = −27 −9 x = −27

−809 = 15 y −809 = 15 y

−731 = 19 y −731 = 19 y

−37 p = −541 −37 p = −541

−19 m = −586 −19 m = −586

0.25 z = 3.25 0.25 z = 3.25

0.75 a = 11.25 0.75 a = 11.25

−13 x = 0 −13 x = 0

24 x = 0 24 x = 0

x 4 = 35 x 4 = 35

z 2 = 54 z 2 = 54

−20 = q −5 −20 = q −5

c −3 = −12 c −3 = −12

y 9 = −16 y 9 = −16

q 6 = −38 q 6 = −38

m −12 = 45 m −12 = 45

−24 = p −20 −24 = p −20

− y = 6 − y = 6

− u = 15 − u = 15

− v = −72 − v = −72

− x = −39 − x = −39

2 3 y = 48 2 3 y = 48

3 5 r = 75 3 5 r = 75

− 5 8 w = 40 − 5 8 w = 40

24 = − 3 4 x 24 = − 3 4 x

− 2 5 = 1 10 a − 2 5 = 1 10 a

− 1 3 q = − 5 6 − 1 3 q = − 5 6

− 7 10 x = − 14 3 − 7 10 x = − 14 3

3 8 y = − 1 4 3 8 y = − 1 4

7 12 = − 3 4 p 7 12 = − 3 4 p

11 18 = − 5 6 q 11 18 = − 5 6 q

− 5 18 = − 10 9 u − 5 18 = − 10 9 u

− 7 20 = − 7 4 v − 7 20 = − 7 4 v

In the following exercises, solve each equation requiring simplification.

100 − 16 = 4 p − 10 p − p 100 − 16 = 4 p − 10 p − p

−18 − 7 = 5 t − 9 t − 6 t −18 − 7 = 5 t − 9 t − 6 t

7 8 n − 3 4 n = 9 + 2 7 8 n − 3 4 n = 9 + 2

5 12 q + 1 2 q = 25 − 3 5 12 q + 1 2 q = 25 − 3

0.25 d + 0.10 d = 6 − 0.75 0.25 d + 0.10 d = 6 − 0.75

0.05 p − 0.01 p = 2 + 0.24 0.05 p − 0.01 p = 2 + 0.24

−10 ( q − 4 ) − 57 = 93 −10 ( q − 4 ) − 57 = 93

−12 ( d − 5 ) − 29 = 43 −12 ( d − 5 ) − 29 = 43

−10 ( x + 4 ) − 19 = 85 −10 ( x + 4 ) − 19 = 85

−15 ( z + 9 ) − 11 = 75 −15 ( z + 9 ) − 11 = 75

Mixed Practice

In the following exercises, solve each equation.

9 10 x = 90 9 10 x = 90

5 12 y = 60 5 12 y = 60

y + 46 = 55 y + 46 = 55

x + 33 = 41 x + 33 = 41

w −2 = 99 w −2 = 99

s −3 = −60 s −3 = −60

27 = 6 a 27 = 6 a

− a = 7 − a = 7

− x = 2 − x = 2

z − 16 = −59 z − 16 = −59

m − 41 = −14 m − 41 = −14

0.04 r = 52.60 0.04 r = 52.60

63.90 = 0.03 p 63.90 = 0.03 p

−15 x = −120 −15 x = −120

84 = −12 z 84 = −12 z

19.36 = x − 0.2 x 19.36 = x − 0.2 x

c − 0.3 c = 35.70 c − 0.3 c = 35.70

− y = −9 − y = −9

− x = −8 − x = −8

In the following exercises, translate to an equation and then solve.

187 is the product of −17 −17 and m .

133 is the product of −19 −19 and n .

−184 −184 is the product of 23 and p .

−152 −152 is the product of 8 and q .

u divided by 7 is equal to −49 −49 .

r divided by 12 is equal to −48 −48 .

h divided by −13 −13 is equal to −65 −65 .

j divided by −20 −20 is equal to −80 −80 .

The quotient c c and −19 −19 is 38.

The quotient of b b and −6 −6 is 18.

The quotient of h h and 26 is −52 −52 .

The quotient k k and 22 is −66 −66 .

Five-sixths of y is 15.

Three-tenths of x is 15.

Four-thirds of w is 36.

Five-halves of v is 50.

The sum of nine-tenths and g is two-thirds.

The sum of two-fifths and f is one-half.

The difference of p and one-sixth is two-thirds.

The difference of q and one-eighth is three-fourths.

In the following exercises, translate into an equation and solve.

Kindergarten Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

Tickets Mollie paid $36.25 for 5 movie tickets. What was the price of each ticket?

Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

Sewing Nancy used 14 yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

Height Aiden is 27 inches tall. He is 3 8 3 8 as tall as his father. How tall is his father?

Real estate Bea earned $11,700 commission for selling a house, calculated as 6 100 6 100 of the selling price. What was the selling price of the house?

Everyday Math

Commission Every week Perry gets paid $150 plus 12% of his total sales amount over $1,250. Solve the equation 840 = 150 + 0.12 ( a − 1250 ) 840 = 150 + 0.12 ( a − 1250 ) for a , to find the total amount Perry must sell in order to be paid $840 one week.

Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation 0.49 s + 0.21 ​ ( s − 5 ) ​ ​ = 9.45 0.49 s + 0.21 ​ ( s − 5 ) ​ ​ = 9.45 for s , to find the number of 49-cent stamps Travis bought.

Writing Exercises

Frida started to solve the equation −3 x = 36 −3 x = 36 by adding 3 to both sides. Explain why Frida’s method will not solve the equation.

Emiliano thinks x = 40 x = 40 is the solution to the equation 1 2 x = 80 1 2 x = 80 . Explain why he is wrong.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction

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• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
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Solving Equations by Multiplication and Division

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Solve for x.

\(\textbf{1)}\) \( 4x=16 \) show answer the answer is \( x=4 \), \(\textbf{2)}\) \( 8x=48 \) show answer the answer is \( x=6 \), \(\textbf{3)}\) \( \frac{x}{4}=8 \) show answer the answer is \( x=32 \), \(\textbf{4)}\) \( \frac{x}{3}=-14 \) show answer the answer is \( x=-42 \), \(\textbf{5)}\) \( \frac{x}{2}=3 \) show answer the answer is \( x=6 \), \(\textbf{6)}\) \( 9x=3 \) show answer the answer is \( x=\frac{1}{3} \), see related pages\(\), \(\bullet\text{ equation calculator }\) \(\,\,\,\,\,\,\,\,\text{(symbolab.com)}\), \(\bullet\text{ solving equations by addition and subtraction}\) \(\,\,\,\,\,\,\,\,x+3=4…\), \(\bullet\text{ solving equations by multiplication and division}\) \(\,\,\,\,\,\,\,\,8x=48…\), \(\bullet\text{ solving multi-step equations}\) \(\,\,\,\,\,\,\,\,3x+2=14…\), \(\bullet\text{ solving equations with variables on both sides}\) \(\,\,\,\,\,\,\,\,3x+5=7x-3…\), \(\bullet\text{ solving equations with decimals}\) \(\,\,\,\,\,\,\,\,43.5+0.2x=51.1…\), \(\bullet\text{ solving equations with fractions}\) \(\,\,\,\,\,\,\,\,\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}…\), solving equations by multiplication and division is a common technique used in algebra to find the value of an unknown variable. the definition of solving equations by multiplication and division is the process of isolating the variable by using the inverse operations of multiplication and division. this means that if the variable is being multiplied by a number, we can divide both sides of the equation by that number to solve for the variable. similarly, if the variable is being divided by a number, we can multiply both sides of the equation by that number to solve for the variable. we learn about solving equations by multiplication and division in math class because it is a fundamental skill that is necessary for solving more complex equations and problems in higher level math courses. it also helps us understand the concept of solving equations in general, which is a key skill in many real-world situations. solving equations by multiplication and division is typically taught in middle or high school algebra classes. one common mistake when solving equations by multiplication and division is forgetting to distribute the inverse operation to all of the terms on the side of the equation that you are working on. for example, if you are solving for x in the equation "2x = 6", and you divide both sides by 2, you must remember to divide the 2 in front of the x as well as the 6 on the right side of the equation. a fun fact about solving equations by multiplication and division is that this technique has been used for centuries to solve mathematical problems. in fact, ancient civilizations like the greeks and romans used similar methods to solve equations in their own ways. it is not clear who specifically discovered the technique of solving equations by multiplication and division, as this method has been used and developed over time by many mathematicians. however, the concept of solving equations can be traced back to ancient civilizations like the egyptians and babylonians. some related topics to solving equations by multiplication and division include algebraic expressions, linear equations, and inverse operations. understanding these concepts can help you become more proficient at solving equations by multiplication and division. 5 real world examples of solving equations by multiplication and division a recipe calls for 2 cups of flour to make a certain number of cookies. you want to make a batch that is three times as large, so you need 32= >6 cups of flour. a car gets an average of 25 miles per gallon of gas. you have a 15-gallon tank and want to know how far you can drive on a full tank. you can drive 2515= >375 miles on a full tank. a store is offering a 20% discount on all clothing. if a shirt normally costs $50, how much will it cost with the discount the discounted price is 50*(1-.2)= >40 dollars. you have a rectangular garden that is 30 feet long and 20 feet wide. you want to divide it into four equal sections with a pathway running down the middle. the width of the pathway is x. the total width of the four sections is 30-2x, and each section must be 20/4=5 feet wide. setting the two expressions equal, we have 30-2x=5, so x= >12.5 feet. a bag contains x red marbles and y blue marbles. the total number of marbles is 15, and the ratio of red marbles to blue marbles is 3:2. we can set up the equation 3x/(3+2)=15/(3+2), which simplifies to 3x=15 and x= >5. we can then solve for y by substituting this value back into the equation y=15-5= >10. there are 5 red marbles and 10 blue marbles in the bag. 5 other math topics that use solving equations by multiplication and division solving systems of equations: a system of equations is a set of two or more equations that are solved simultaneously. one common method for solving systems of equations is to use the elimination method, which involves multiplying or dividing one or both of the equations by a constant and then adding or subtracting the resulting equations to eliminate one of the variables. rational expressions: a rational expression is an expression that is the ratio of two polynomials. solving equations involving rational expressions often requires the use of the principle of cross-multiplication, which involves multiplying both sides of the equation by the denominator of the rational expression in order to clear the fraction. quadratic equations: a quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. one common method for solving quadratic equations is the quadratic formula, which involves using the values of a, b, and c to calculate the two possible solutions for x. inequalities: an inequality is an expression that represents a relationship between two values that is not equal. solving inequalities often involves using the same techniques as those used to solve equations, such as multiplying or dividing both sides of the inequality by a constant or using the principle of cross-multiplication. proportions: a proportion is an equation that states that two ratios are equal. solving proportions often involves setting up a cross-multiplication equation and then solving for the unknown value., about andymath.com, andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

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One-Step Equation: Multiplication and Division Worksheets

These printable one-step equation worksheets involve the multiplication and division operation to solve them. Given below are separate exercises for equations which involve integers, fractions and decimals coefficients. The pdf worksheets are meticulously designed for 6th grade, 7th grade, and 8th grade students. You can access some of them for free.

Integers: Level 1

Solve the one-step equations by performing multiplication and division operation. 'Level 1' has simple equations for the beginners.

• Download the set

Integers: Level 2

In 'Level 2' the complexity of the problems increases. Solve each one-step equation by multiplying and dividing them. Each worksheet has 10 problems for practice.

Fractions: Level 1

These printable worksheets have proper and improper fractions in their coefficients, serving well in examining the skills of grade 6, grade 7, and grade 8 students.

Fractions: Level 2

In 'Level 2' fraction worksheets, mixed numbers become the coefficients of the given equations. Perform the multiplication and division operation to solve them.

Decimals: Multiplication and Division

Multiply and divide to solve each one-step equation. The terms used here are combinations of decimals and integers. Download these pdf worksheets to give ample practice to students.

One-Step Equations: Integers, Fractions and Decimals

A mix of integers, fractions and decimals are used to form equations in these worksheets. These worksheets are mixed review pages for children.

One Step Equation Word Problems Worksheets

Solve this extensive collection of one-step equation word problems that involves integers, fractions, and decimals. These worksheets are tailor-made for middle school students.

(15 Worksheets)

Related Worksheets

» Two-Step Equation

» Multi-Step Equation

» Equation Word Problems

» Translating Phrases

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This worksheets combine basic multiplication and division word problems. The division problems do not include remainders. These worksheets require the students to differentiate between the phrasing of a story problem that requires multiplication versus one that requires division to reach the answer.

Mixed Multiplication and Division Word Problems 1

Mixed Multiplication and Division Word Problems 2

Practicing the operations separately is a good start for each operation, but an important word problem skill is also figuring out which math operation is needed to solve a specific question. The worksheets in this section combine both multiplication word problems and division word problems on the same worksheet, so students not only need to solve the problem but they need to figure out how to do it.

By moving into these worksheets quickly, it avoids the crutch where students learn that they always need to add or always need to subtract the two values in a problem to get the answer. This forces students to really understand the intent of the problem, not just scan the text looking for numbers.

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Multiplication and division

Here you will learn about multiplication and division, including strategies on how to multiply and divide various types of rational numbers.

Students first learn about multiplication and division in the 3 rd grade and 4 th grade with their work with operations and algebraic thinking, as well as number and operations base ten and fractions.

What is multiplication and division?

Multiplication and division are two of the four basic operations. Multiplication is finding the product of two or more numbers, and division is finding the quotient of two numbers.

Multiplication is basically the repeated addition of equal groups.

For example, 4 equal groups of 3 :

In a multiplication equation, the answer to multiplying one number by another is called the product. The multiplicand is the quantity to be multiplied by the multiplier, which will give you a product.

The product will be 0 if either the multiplicand or multiplier is 0 .

Arrays are visual models that represent multiplication.

For example, this array shows 3 rows of 6 which is the same as 3 \times 6 .

3 \times 6=18

Step by step guide: Understanding Multiplication

Multiplication is commutative. The order in which the calculation is performed does not matter.

For example,

3\times{4}=4\times{3}=12

Multiplying multi-digit numbers

To multiply multi-digit numbers, you can use the algorithm or the area model.

The area model is a rectangular model where the product represents finding the area of the rectangle.

For example, multiply 42 \times 62 using an area model.

2400+120+80+4=2604

42 \times 62=2604

Step-by-step guide: Multiplying multi-digit numbers

Multiplicative comparisons

You can use multiplication to make comparisons between quantities. Multiplicative comparisons compare two quantities by showing that one quantity is how many times larger or smaller than another quantity.

Mike has 3 lollipops. Michelle has 4 times as many lollipops as Mike. How many lollipops does Michelle have?

Michelle has 4\times 3=12 lollipops.

Jillian has 24 inches of hair ribbon. Suzanne has half that amount. How long is Suzanne’s hair ribbon?

Suzanne’s ribbon is \cfrac{1}{2} \times 24=12 \text { inches }

Step by step guide: Multiplicative comparisons

[FREE] Multiplication And Division Worksheet (Grade 3 to 7)

Use this worksheet to check your grade 3 to 7 students’ understanding of multiplication and division. 15 questions with answers to identify areas of strength and support!

Multiplying rational numbers

You can multiply rational numbers. Rational numbers include multi-digit numbers, integers, fractions, and decimals. When multiplying positive and negative numbers, the following rules apply:

For example, (-3) \times(-5)=15

Step by step guide: Multiplying and dividing integers

Division shares or breaks a number into equal sized groups.

For example, the number 12 can be divided into 4 equal groups of 3 .

In a division equation, the answer you get when you divide one number by another is called the quotient.

The word quotient comes from Latin and means ‘how many times.’ When dividing, you are finding out ‘how many times’ a number goes into another number.

The quotient will only be 0 if the dividend is 0 but the divisor is not.

8 \div 0=\text {Does not exist }

Step by step guide: Understanding division

Unlike multiplication, division is not commutative. If the order of the numbers within the calculation changes, the result will change.

12 \div 4 ≠ 4 \div 12

To solve division problems with larger numbers, you can use long division.

For example, 452.1 \div 3

Step-by-step guide: Long division

Step by step guide: Dividing multi-digit numbers

Division can also be done with positive and negative integers, fractions, and decimals. When dividing positive and negative numbers, the following rules apply:

For example, (-20) \div (-5)=4

Step by step guide: Multiplying and dividing rational numbers

Common Core State Standards

How does this relate to 4 th – 7 th grade math?

• Grade 3: Operations and Algebraic Thinking ( 3.OA.A.1) Interpret products of whole numbers, for example, interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.
• Grade 3: Operations and Algebraic Thinking (3.OA.A.2) Interpret whole-number quotients of whole numbers, for example, interpret 56 \div 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
• Grade 3: Operations and Algebraic Thinking (3.OA.C.7) Fluently multiply and divide within 100 , using strategies such as the relationship between multiplication and division.
• Grade 4: Operations and Algebraic Thinking (4.OA.2) Multiply or divide to solve word problems involving multiplicative comparison, for example, by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
• Grade 4: Number and Operations – Fractions (4.NF.B.4) Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
• Grade 5: Number and Operations Base Ten (5.NBT.B.5) Fluently multiply multi-digit whole numbers using the standard algorithm.
• Grade 5: Number and Operations – Fractions (5.NF.B.7) Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
• Grade 6: Number System (6.NS.C.6) Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
• Grade 7: Number System (7.NS.A.2) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

How to do multiplication and division

There are several strategies to multiply and divide numbers. For more specific step-by-step guides, check out the individual pages linked in the “What is multiplication and division?” section above or read through the examples below.

In order to multiply using a visual model:

Draw the array.

• Count the objects in each row.
• Find the total .

In order to divide using a visual model:

Count the objects in each group.

• Write the answer .

In order to multiply and divide multi-digit numbers:

Perform the multiplication or division algorithm.

Write the answer.

In order to find multiplicative comparisons:

Draw a model.

Write an equation.

Solve the equation and label the answer.

Multiplication and division examples

Example 1: multiply using a model.

Use a visual model to multiply 5 \times 3.

5 x 3 is 5 rows of 3.

2 Count the objects in each row.

3 Find the total .

There are 3 chips in each row, 3+3+3+3+3=15

3 \times 5=15

Example 2: multiply using algorithm

Multiply 99 \times 7.

Using the algorithm,

99 \times 7=693

Example 3: multiply and divide rational numbers

Multiply 1.23 \times 3.2.

1.23 \times 3.2=3.936

Example 4: divide using a visual model

Divide: 9 \div 3

Example 5: dividing rational numbers

Divide: 15.4 \div 2

15.4 \div 2=7.7

Example 6: multiplicative comparison

Bobby has 3 baseball cards. Joey has five times as many cards as Bobby. How many cards does Joey have?

3 \times 5= \, ?

Joey has 15 baseball cards.

Teaching tips for multiplication and division

• Use manipulatives to reinforce the conceptual understanding of multiplication and division.
• Include real world scenarios so that students can connect the mathematical concepts to the world around them.
• Reinforce to students that the concept of multiplication and division is the same regardless if the numbers are whole numbers or rational numbers.
• Using the area model for multiplication and division can be a fun way for students to understand multiplication and division while also reinforcing math facts.
• To practice multiplication facts, consider using digital and non-digital games instead of flashcards. Game playing is a fun way for students to remember the times tables.

Easy mistakes to make

• Confusing the rules for multiplying and dividing positive and negative numbers For example, multiplying (-4)\times (-8) and getting a product of -32 instead of 32 .
• Misinterpreting the meaning of key words in word problems resulting in using the incorrect operation For example, thinking that the word “of” means to divide instead of multiply.

Practice multiplication and division questions

1) Which multiplication expression represents this array?

Count the number of objects in each row.

There are 5 objects in each row which is 5+5.

5+5 is the same as 2 \times 5.

So, 2 \times 5 is the correct expression.

2) Multiply 104 \times 3.

Use the algorithm for multiplying multi-digit numbers, regrouping when necessary.

104 \times 3=312

3) Multiply 53 \times 32.

You can use the area model to multiply 53 \times 32.

Add the products together: 1500+100+90+6=1696

53 \times 32=1696

4) Use the array to find the quotient of 16 \div 4 .

Divide the array into 4 equal groups and then count how many objects are in each group.

16 \div 4 = 4

5) Divide 128 \div 4.

Divide the numbers using the algorithm for long division.

128 \div 4= 32

6) Chris has 3 pencils. Pam has four times as many pencils as Chris. How many pencils does Pam have?

Draw a picture.

4 \times 3=12

Pam has 12 pencils.

Multiplication and division FAQs

Yes, the rules for multiplying and dividing positive and negative numbers hold true regardless if the numbers are whole numbers or rational numbers. Step-by-step guide : Negative numbers

Knowing your multiplication facts and division facts helps when solving problems.

Repeated subtraction is a way for students to begin to develop an understanding of division.

There is not one best strategy to use when multiplying multi-digit numbers. Some strategies, can be quicker than others, but not better.

Knowing your multiplication tables helps to answer questions faster than when you do not know them.

The commutative property of multiplication is: 5 \times 3=3 \times 5 , the order of the numbers does not matter.

The next lessons are

• Types of numbers
• Rounding numbers
• Factors and multiples

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