problem solving using integer

Integer Word Problems Worksheets

An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses. Integers are closed under the operations of addition and multiplication . Integer word problems worksheets provide a variety of word problems associated with the use and properties of integers. 

Benefits of Integers Word Problems Worksheets

We use integers in our day-to-day life like measuring temperature, sea level, and speed limit. Translating verbal descriptions into expressions is an essential initial step in solving word problems. Deposits are normally represented by a positive sign and withdrawals are denoted by a negative sign. Negative numbers are used in weather forecasting to show the temperature of a region. Solving these integers word problems will help us relate the concept with practical applications.

Download Integers Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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• Word Problems On Integers

Integers: Word Problems On Integers

An arithmetic operation is an elementary branch of mathematics. Arithmetical operations include addition, subtraction, multiplication and division. Arithmetic operations are applicable to different types of numbers including integers.

Integers are a special group of numbers that do not have a fractional or a decimal part. It includes positive numbers, negative numbers and zero.  Arithmetic operations on integers are similar to that of whole numbers. Since integers can be positive or negative numbers i.e. as these numbers are preceded either by a positive (+) or a negative sign (-), it makes them a little confusing concept. Therefore, they are different from whole numbers . Let us now see how various arithmetical operations can be performed on integers with the help of a few word problems. Solve the following word problems using various rules of operations of integers.

Word problems on integers Examples:

Example 1: Shyak has overdrawn his checking account by Rs.38.  The bank debited him Rs.20 for an overdraft fee.  Later, he deposited Rs.150.  What is his current balance?

Solution:  Given,

Total amount deposited= Rs. 150

Amount overdrew by Shyak= Rs. 38

Amount charged by bank= Rs. 20

⇒ Debit amount= -20

Total amount debited = (-38) + (-20) = -58

Current balance= Total deposit +Total Debit

Hence, the current balance is Rs. 92.

Example 2: Anna is a microbiology student. She was doing research on optimum temperature for the survival of different strains of bacteria. Studies showed that bacteria X need optimum temperature of -31˚C while bacteria Y need optimum temperature of -56˚C. What is the temperature difference?

Solution: Given,

Optimum temperature for bacteria X = -31˚C

Optimum temperature for bacteria Y= -56˚C

Temperature difference= Optimum temperature for bacteria X – Optimum temperature for bacteria Y

⇒ (-31) – (-56)

Hence, temperature difference is 25˚C.

Example 3: A submarine submerges at the rate of 5 m/min. If it descends from 20 m above the sea level, how long will it take to reach 250 m below sea level?

Initial position = 20 m    (above sea level)

Final position = 250 m    (below sea level)

Total depth it submerged = (250+20) = 270 m

Thus, the submarine travelled 270 m below sea level.

Time taken to submerge 1 meter = 1/5 minutes

Time taken to submerge 270 m = 270 (1/5) = 54 min

Hence, the submarine will reach 250 m below sea level in 54 minutes.

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Integer Word Problems

Welcome to the fascinating world of integer word problems! Don’t let the fancy name scare you off; these problems might be easier and more fun than you think. Simplifying them is handy in daily life, and they’ll reappear in various forms throughout your academic journey. Let’s dive into the fundamental components.

What are Integer Word Problems?

In essence, integer word problems are mathematical problems involving number-related questions in the form of a story or practical situation. Specifically, these problems use integers — whole numbers that can be positive, negative, or zero. For instance, you might be asked how many more books Mike read than Sarah if Mike reads 15 and Sarah reads 7. Since you’re subtracting 7 from 15, you’re dealing with an integer word problem.

Importance of Solving Integer Word Problems

Mastering integer word problems plays a significant role in building your mathematical expertise. They help improve your problem-solving skills and enhance your ability to think logically and critically. Moreover, these problems are a cornerstone of real-world situations. Whether you are calculating the distance between two cities, determining profit and loss in business, or even figuring out temperature changes, integers and their problems come into play.

How to Solve Integer Word Problems

Are you ready to tackle integer word problems? Here are a few steps:

• Understand the Problem:  Breaking the problem into smaller parts makes it less daunting. Take your time to understand what the problem is about.
• Identify the Key Information:  Highlight or underline important facts or figures in the problem. Look for clues indicating whether you’re dealing with addition, subtraction, or a combination.
• Formulate a Plan:  Write down your actions to arrive at the solution.
• Execute Your Plan:  Apply the actions you’ve mapped out to solve the problem.
• Verify Your Answer:  Always double-check your outcome. Does it make sense in terms of the problem?

In dealing with integer word problems, practice is critical. The more problems you tackle, the more proficient you become. Happy problem-solving!

Basic Concepts of Integers

As a student or math enthusiast, knowing and mastering the basic concepts of integers will help you understand and tackle integer word problems better. In this section, we’ll delve into the definitions of integers, further distinguishing between positive and negative integers.

Defining Integers

Integers  are a number category that includes all the whole numbers, their opposites (negative counterparts), and zero. They are distinct from fractions, decimals, and percents. An integer can be a zero, a positive, or a negative whole number. The set of integers is denoted mathematically as {…, -3, -2, -1, 0, 1, 2, 3}. These numbers form the backbone of many mathematical operations and concepts, especially in algebra.

Positive and Negative Integers

Positive and negative integers make understanding and calculating many real-world situations better and more efficient.

Positive integers , often natural numbers , are numbers greater than zero. They are frequently used to denote weight, distance, or money values. However, not all situations can be expressed with positive numbers; sometimes, we must resort to negative ones.

Negative integers are the opposites of natural numbers, excluding zero, and fall below zero on the number line. They are typically used when something is decreased, removed, or lost. An excellent example of using negative integers is in banking, where they represent debt. Or in meteorology, where they represent temperatures below zero.

Understanding the concept of positive and negative integers is paramount because they are central to successfully dealing with integer word problems. In the next segment, we will dive deeper into strategies for solving these problems, so tighten your seatbelts as we explore a fun section of the mathematical world.

Addition and Subtraction Word Problems

When it comes to integers, understanding how to add and subtract these numbers is crucial, taking center stage in everyday mathematical operations. While learning, students begin grappling with word problems – mathematical problems presented in the form of a narrative or story – which include real-world scenarios. These serve as a bridge for children and adults to apply theoretical knowledge practically.

Adding and Subtracting Integers

In terms of  adding integers , there are a few rules to remember. If the integers have the same sign, add their absolute values and keep the standard sign. On the flip side, when the integers have different signs, subtract the smaller absolute value from the larger one and give the solution the sign of the number with the more considerable absolute value.

Subtracting integers , however, involves an additional step. More specifically, any subtraction can be reinterpreted as an addition. To subtract an integer, add its opposite. For example, to subtract -3 from 5 (5 – -3), we add 3 to 5 (5 + 3), with the sum coming to 8.

Real-life Examples of Addition and Subtraction Word Problems

Let’s explore a few word problems that imitate daily life scenarios. Suppose a child has £5 and they want to buy a toy that costs £10. How many more pounds do they need? The problem here is 10 – 5, which equals 5. Thus, the child needs five more pounds.

In another situation, imagine the temperature was 5 degrees Celsius in the morning and dropped 3 degrees by the afternoon. What’s the temperature now? Here, we have 5 – 3 = 2. The answer is 2 degrees Celsius.

These examples illustrate how adding and subtracting integers can help us solve practical problems and better understand the world. We encourage you to find your examples and practice to enhance your understanding and mastery of this fundamental mathematical skill.

Multiplication and Division Word Problems

As the journey of discovery with integers continues, multiplication and division of these numbers become an integral part of our everyday mathematical activities. Understanding how to tackle word problems – mathematical problems in narrative form – becomes critical. Specifically, multiplication and division integer word problems provide the groundwork for applying knowledge practically in real-world situations.

Multiplying and Dividing Integers

Multiplying integers might initially seem complex , but it becomes straightforward once you grasp the core concept. When multiplying two integers, the result will be positive if the signs are the same (positive or negative). However, if the signs are different (positive and negative), the result will be a negative integer.

Dividing integers  follows a similar concept. If the integers have the same sign, the quotient is positive, and if they have different signs, it is negative.

Application of Multiplication and Division Word Problems

Now, let’s see how these concepts apply in real-world scenarios. Suppose a person has $20 and wants to buy as many chocolates as possible, with each chocolate bar costing $4. In this case, they’d need to divide 20 by 4. The question boils down to 20 ÷ 4, which equals 5. So, they can buy five chocolate bars.

Considering multiplication, imagine a scenario where a store sells packages of bottled drinking water. Each package contains six bottles, and the store has twenty packages. To calculate the total number of bottles, you would multiply 6 (bottles per package) by 20 (number of packages), getting 6 x 20 = 120. So, the store has 120 bottled water.

These real-world examples show how multiplication and division word problems offer practical ways to understand and apply mathematical knowledge. Engaging with these problems enhances understanding of fundamental math concepts and promotes problem-solving skills crucial for daily life.

Multi-Step Word Problems

In a journey through mathematics, we commonly encounter complex multi-step word problems. These problems often involve multiple operations using integers , such as addition, subtraction, multiplication, and division. Solving these tasks enhances problem-solving skills, logical thinking, and mathematical proficiency. This part will delve into complex integer word problems and introduce strategies for solving multi-step problems.

Complex Integer Word Problems

Complex integer word problems  involve more than one mathematical operation, often requiring a systematic approach to reach the solution. For instance, imagine a scenario where a garden filled with 120 roses and petunias is being prepared for a garden show. There are twice as many roses as there are petunias. The question is, “How many petunias are there?”

Here, the problem will be solved in two steps. First, understanding that the number of roses is twice that of petunias. That means, if we denote the number of petunias as ‘p,’ then the number of roses is ‘2p’. The total quantity of flowers (120) is the sum of roses and petunias, leading to the equation 2p + p = 120. Solving this equation provides the number of petunias. Since multi-step word problems rely heavily on integers, understanding their operation rules is essential.

Strategies for Solving Multi-Step Word Problems

Solving multi-step word problems  can seem daunting, but a systematic approach simplifies the task. Below are vital strategies:

• Understand the Problem:  Read the problem carefully, ensure you understand what it’s asking, and identify the operations needed.
• Develop a Plan:  Break down the problem into smaller, manageable steps. Form equations if needed.
• Solve:  Carry out each operation. Ensure your calculations are correct at each step.
• Check Your Answer:  Review your solution, ensuring you answered the initial question correctly. Doing this validates that your solution aligns with the problem’s conditions.

Remember, practice significantly improves problem-solving skills and the ability to tackle complex multi-step word problems involving integers. Happy problem-solving!

Common Mistakes and Tips for Success

In particular, integer word problems can sometimes throw you off course. Like every journey, it is customary to make mistakes along the way. However, understanding and learning from these common errors can help you avoid detours and get you on the fast track to mastery.

Common Errors in Solving Integer Word Problems

Misinterpretation  is one of the most common mistakes when handling integer word problems. Often, students need to understand the operations required or interpret the relationship between the integers presented in the problem.

Inaccurate Calculations  – Integers include both positive and negative numbers, and it is easy to miscalculate when it comes to subtraction, addition, or other operations involving such numbers. For example, subtracting a negative integer leads to an addition instead.

Helpful Tips and Tricks for Solving Integer Word Problems

Once you’re aware of common pitfalls, arm yourself with the right strategies to navigate your way through complex integer word problems adeptly.

Thorough Understanding:  Read the integer word problem carefully and understand what is being asked. It can be helpful to jot down essential information or even draw diagrams to visualize the problem.

Plan:  Make a plan. Break the problem down into smaller, solvable parts and create equations representing each step of the problem.

Check Your Work:  After solving, double-check your calculations to ensure accuracy. Compare your answer with the original question to see if it makes sense.

Practice:  Just like anything, practice makes perfect. The more problems you solve, the more comfortable you become with integers and their operations.

Always remember making mistakes is part of the learning process. By staying aware and utilizing strategies, you’ll soon find yourself an expert at solving integer word problems. Happy Practicing!

Practice Exercises

Knowing the common errors and tips for solving integer word problems, it is time to put that knowledge into practice. With the right amount of practice, anyone can enhance their skills in solving such problems. With that in mind, let’s tackle some practice exercises to understand integer word problems further.

Practice Problems for Integer Word Problems

Here are some various types of integer word problems. Remember to read carefully, understand what’s asked, and plan your solution before jumping into the problem.

• Maria has $15 in her pocket. She spends $7 on a movie and $6 on snacks. Write an integer to represent Maria’s money situation and calculate how much she has left.
• At the start of the week, the temperature is 5 degrees. The temperature then drops by 7 degrees the next day. What is the temperature now?
• A company lost $2000 this year, 3 times the amount they lost last year. How much did the company lose last year?

Step-by-Step Solutions for Practice Exercises

Let’s walk through the solutions together to help you understand how these problems are solved.

• Maria has $15. She spent $7 and $6. This expenditure is a loss, so we represent it with negative integers. So, the situation becomes: 15 + (-7) + (-6) = 2. Maria has $2 left.
• The temperature is 5 degrees initially. Then, it drops by 7 degrees (a decrease is a negative operation). So, the situation is 5 + (-7) = -2 degrees. The temperature is now -2 degrees.
• Let’s denote the amount of money the company lost last year as x. We know that 3x = $2000. So, x = $2000 / 3 = $666.67. The company lost around $666.67 last year.

Do more exercises and get comfortable with solving integer word problems. It may take some time, but you will get there with consistent practice. Remember, avoiding rushing and breaking the problem into smaller parts can be very helpful. Practicing will make you better at solving integer word problems effectively and efficiently. Happy learning!

Emerging victorious in integer word problems opens up an exciting facet of mathematical knowledge. After all, these problems translate mathematical concepts into real-world scenarios, thereby cultivating critical thinking skills. Let’s explore the benefits of mastering integer word problems and round off with a few parting thoughts.

Benefits of Mastering Integer Word Problems

Boosts Problem-solving Skills:  Integer word problems are an ideal way to sharpen problem-solving skills. They compel one to think logically and systematically about how to apply mathematical operations accurately.

Enhances Numerical Literacy: With a firm grasp of integers, people can better comprehend numerical information daily. For instance, understanding debt and assets or gain and loss in finance becomes clearer.

Encourages Diversity of Thought:  Integer word problems offer multiple ways to find a solution, fostering creativity. It encourages diverse approaches to problem-solving.

Promotes Practical Application:  Integers have ubiquitous applications in diverse fields, including science, engineering, and information technology. Being comfortable with integer word problems equips one with skills applicable to these areas.

Final Thoughts on Integer Word Problems

Integer word problems seem daunting initially, but their mastery is a matter of regular practice and strategy. Break down the problem, identify what operation is warranted, and then move towards a solution progressively. Remember to cross-check the answer, as it ensures correctness.

Remember, it’s perfectly fine to make mistakes while learning. They are merely stepping stones to success. So, stay patient, persist in your efforts, and remember the tips shared. You will soon gain a commendable prowess over integer word problems. The confidence and skills you gain here will be beneficial throughout your mathematical journey.

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3.6: Solve Equations Using Integers; The Division Property of Equality

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Learning Objectives

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

• Determine whether an integer is a solution of an equation
• Solve equations with integers using the Addition and Subtraction Properties of Equality

Model the Division Property of Equality

• Solve equations using the Division Property of Equality
• Translate to an equation and solve

Be Prepared 3.10

Before you get started, take this readiness quiz.

Evaluate x + 4 when x = −4 . Evaluate x + 4 when x = −4 . If you missed this problem, review Example 3.22.

Be Prepared 3.11

Solve: y − 6 = 10 . Solve: y − 6 = 10 . If you missed this problem, review Example 2.33.

Be Prepared 3.12

Translate into an algebraic expression 5 5 less than x . x . If you missed this problem, review Table 1.3.

Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

How to determine whether a number is a solution to an equation.

• Step 1. Substitute the number for the variable in the equation.
• Step 2. Simplify the expressions on both sides of the equation.
• If it is true, the number is a solution.
• If it is not true, the number is not a solution.

Example 3.60

Determine whether each of the following is a solution of 2 x − 5 = −13 : 2 x − 5 = −13 :

• ⓐ x = 4 x = 4
• ⓑ x = −4 x = −4
• ⓒ x = −9 . x = −9 .

Since x = 4 x = 4 does not result in a true equation, 4 4 is not a solution to the equation.

Since x = −4 x = −4 results in a true equation, −4 −4 is a solution to the equation.

Since x = −9 x = −9 does not result in a true equation, −9 −9 is not a solution to the equation.

Try It 3.119

Determine whether each of the following is a solution of 2 x − 8 = −14 : 2 x − 8 = −14 :

• ⓐ x = −11 x = −11
• ⓑ x = 11 x = 11
• ⓒ x = −3 x = −3

Try It 3.120

Determine whether each of the following is a solution of 2 y + 3 = −11 : 2 y + 3 = −11 :

• ⓐ y = 4 y = 4
• ⓑ y = −4 y = −4
• ⓒ y = −7 y = −7

Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

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

Properties of Equalities

Example 3.61.

Solve: y + 9 = 5 . y + 9 = 5 .

Check the result by substituting −4 −4 into the original equation.

Since y = −4 y = −4 makes y + 9 = 5 y + 9 = 5 a true statement, we found the solution to this equation.

Try It 3.121

y + 11 = 7 y + 11 = 7

Try It 3.122

y + 15 = −4 y + 15 = −4

Example 3.62

Solve: a − 6 = −8 a − 6 = −8

The solution to a − 6 = −8 a − 6 = −8 is −2 . −2 .

Since a = −2 a = −2 makes a − 6 = −8 a − 6 = −8 a true statement, we found the solution to this equation.

Try It 3.123

a − 2 = −8 a − 2 = −8

Try It 3.124

n − 4 = −8 n − 4 = −8

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. Now we’ll see how to solve equations that involve division.

We will model an equation with envelopes and counters in Figure 3.21.

Here, 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?

To determine the number, separate the counters on the right side into 2 2 groups of the same size. So 6 6 counters divided into 2 2 groups means there must be 3 3 counters in each group (since 6 ÷ 2 = 3 ) . 6 ÷ 2 = 3 ) .

What equation models the situation shown in Figure 3.22? There are two envelopes, and each contains x x counters. Together, the two envelopes must contain a total of 6 6 counters. So the equation that models the situation is 2 x = 6 . 2 x = 6 .

We can divide both sides of the equation by 2 2 as we did with the envelopes and counters.

We found that each envelope contains 3 counters. 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.

Figure 3.23 shows another example.

Now we have 3 Figure 3.24.

The equation that models the situation is 3 x = 12 . 3 x = 12 . We can divide both sides of the equation by 3 . 3 .

Does this check? It does because 3 · 4 = 12 . 3 · 4 = 12 .

Manipulative Mathematics

Example 3.63.

Write an equation modeled by the envelopes and counters, and then solve it.

There are 4 envelopes, 4 envelopes, or 4 4 unknown values, on the left that match the 8 counters 8 counters on the right. Let’s call the unknown quantity in the envelopes x . x .

There are 2 counters 2 counters in each envelope.

Try It 3.125

Write the equation modeled by the envelopes and counters. Then solve it.

Try It 3.126

Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality . When you divide both sides of an equation by any nonzero number, you still have equality.

Division Property of Equality

For any numbers a , b , c , and c ≠ 0 , If a = b then a c = b c . For any numbers a , b , c , and c ≠ 0 , If a = b then a c = b c .

Example 3.64

Solve: 7 x = −49 . Solve: 7 x = −49 .

To isolate x , x , we need to undo multiplication.

Check the solution.

Therefore, −7 −7 is the solution to the equation.

Try It 3.127

8 a = 56 8 a = 56

Try It 3.128

11 n = 121 11 n = 121

Example 3.65

Solve: −3 y = 63 . −3 y = 63 .

To isolate y , y , we need to undo the multiplication.

Since this is a true statement, y = −21 y = −21 is the solution to the equation.

Try It 3.129

−8 p = 96 −8 p = 96

Try It 3.130

−12 m = 108 −12 m = 108

Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

Example 3.66

Translate and solve: five more than x x is equal to −3 . −3 .

Check the answer by substituting it into the original equation.

x + 5 = −3 −8 + 5 = ? −3 −3 = −3 ✓ x + 5 = −3 −8 + 5 = ? −3 −3 = −3 ✓

Try It 3.131

Translate and solve:

Seven more than x x is equal to −2 −2 .

Try It 3.132

Eleven more than y is equal to 2. Eleven more than y is equal to 2.

Example 3.67

Translate and solve: the difference of n n and 6 6 is −10 . −10 .

n − 6 = −10 −4 − 6 = ? −10 −10 = −10 ✓ n − 6 = −10 −4 − 6 = ? −10 −10 = −10 ✓

Try It 3.133

The difference of p p and 2 2 is −4 −4 .

Try It 3.134

The difference of q q and 7 7 is −3 −3 .

Example 3.68

Translate and solve: the number 108 108 is the product of −9 −9 and y . y .

108 = −9 y 108 = ? −9 ( −12 ) 108 = 108 ✓ 108 = −9 y 108 = ? −9 ( −12 ) 108 = 108 ✓

Try It 3.135

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

Try It 3.136

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

ACCESS ADDITIONAL ONLINE RESOURCES

• One-Step Equations With Adding Or Subtracting
• One-Step Equations With Multiplying Or Dividing

Section 3.5 Exercises

Practice makes perfect.

In the following exercises, determine whether each number is a solution of the given equation.

4 x − 2 = 6 4 x − 2 = 6

• ⓐ x = −2 x = −2
• ⓑ x = −1 x = −1
• ⓒ x = 2 x = 2

4 y − 10 = −14 4 y − 10 = −14

• ⓐ y = −6 y = −6
• ⓑ y = −1 y = −1
• ⓒ y = 1 y = 1

9 a + 27 = −63 9 a + 27 = −63

• ⓐ a = 6 a = 6
• ⓑ a = −6 a = −6
• ⓒ a = −10 a = −10

7 c + 42 = −56 7 c + 42 = −56

• ⓐ c = 2 c = 2
• ⓑ c = −2 c = −2
• ⓒ c = −14 c = −14

Solve Equations Using the Addition and Subtraction Properties of Equality

In the following exercises, solve for the unknown.

n + 12 = 5 n + 12 = 5

m + 16 = 2 m + 16 = 2

p + 9 = −8 p + 9 = −8

q + 5 = −6 q + 5 = −6

u − 3 = −7 u − 3 = −7

v − 7 = −8 v − 7 = −8

h − 10 = −4 h − 10 = −4

k − 9 = −5 k − 9 = −5

x + ( −2 ) = −18 x + ( −2 ) = −18

y + ( −3 ) = −10 y + ( −3 ) = −10

r − ( −5 ) = −9 r − ( −5 ) = −9

s − ( −2 ) = −11 s − ( −2 ) = −11

In the following exercises, write the equation modeled by the envelopes and counters and then solve it.

In the following exercises, solve each equation using the division property of equality and check the solution.

5 x = 45 5 x = 45

4 p = 64 4 p = 64

−7 c = 56 −7 c = 56

−9 x = 54 −9 x = 54

−14 p = −42 −14 p = −42

−8 m = −40 −8 m = −40

−120 = 10 q −120 = 10 q

−75 = 15 y −75 = 15 y

24 x = 480 24 x = 480

18 n = 540 18 n = 540

−3 z = 0 −3 z = 0

4 u = 0 4 u = 0

In the following exercises, translate and solve.

Four more than n n is equal to 1.

Nine more than m m is equal to 5.

The sum of eight and p p is −3 −3 .

The sum of two and q q is −7 −7 .

The difference of a a and three is −14 −14 .

The difference of b b and 5 5 is −2 −2 .

The number −42 is the product of −7 and x x .

The number −54 is the product of −9 and y y .

The product of -15 and f f is 75.

The product of −18 and g g is 36.

−6 plus c c is equal to 4.

−2 plus d d is equal to 1.

Nine less than m m is −4.

Thirteen less than n n is −10 −10 .

Mixed Practice

In the following exercises, solve.

• ⓐ x + 2 = 10 x + 2 = 10
• ⓑ 2 x = 10 2 x = 10
• ⓐ y + 6 = 12 y + 6 = 12
• ⓑ 6 y = 12 6 y = 12
• ⓐ −3 p = 27 −3 p = 27
• ⓑ p − 3 = 27 p − 3 = 27
• ⓐ −2 q = 34 −2 q = 34
• ⓑ q − 2 = 34 q − 2 = 34

a − 4 = 16 a − 4 = 16

b − 1 = 11 b − 1 = 11

−8 m = −56 −8 m = −56

−6 n = −48 −6 n = −48

−39 = u + 13 −39 = u + 13

−100 = v + 25 −100 = v + 25

11 r = −99 11 r = −99

15 s = −300 15 s = −300

100 = 20 d 100 = 20 d

250 = 25 n 250 = 25 n

−49 = x − 7 −49 = x − 7

64 = y − 4 64 = y − 4

Everyday Math

Cookie packaging A package of 51 cookies 51 cookies has 3 3 equal rows of cookies. Find the number of cookies in each row, c , c , by solving the equation 3 c = 51 . 3 c = 51 .

Kindergarten class Connie’s kindergarten class has 24 children. 24 children. She wants them to get into 4 4 equal groups. Find the number of children in each group, g , g , by solving the equation 4 g = 24 . 4 g = 24 .

Writing Exercises

Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation 3 x = 15 ? 3 x = 15 ? Explain why or why not.

Suppose you are using envelopes and counters to model solving the equations x + 4 = 12 x + 4 = 12 and 4 x = 12 . 4 x = 12 . Explain how you would solve each equation.

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

Raoul started to solve the equation 4 y = 40 4 y = 40 by subtracting 4 4 from both sides. Explain why Raoul’s method will not solve the equation.

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

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Integer Word Problems

In these lessons, we will look at Integer Word Problems that have more than two unknowns.

In another set of lessons, we have some examples of Integer Word Problems that involve two unknowns .

Related Pages Consecutive Integer Word Problems Consecutive Integers 1 Consecutive Integers 2 More Algebra Word Problems

Integer Problems With More Than Two Unknowns

Integer Problems with three unknowns are not necessarily more difficult than integer word problems with two unknowns . You just have to be careful when relating the different unknowns.

Example: Jane and her friends were selling cookies. They sold 4 more boxes the second week than they did the first. On the third week, they doubled the sale of their second week. Altogether, they sold a total of 352 boxes. How many boxes did they sell in the third week?

Solution: Step 1: Sentence: They sold 4 more boxes the second week than they did the first. On the third week, they doubled the sale of their second week.

Assign variables :

Example: The sum of three numbers is 12. The first is five times the second and the sum of the first and third is 9. Find the numbers.

Advanced Consecutive Integer Problems Example: (1) Find three consecutive positive integers such that the sum of the two smaller integers exceed the largest integer by 5.

(2) The sum of a number and three times its additive inverse is 16. Find the number.

Example: The largest of five consecutive even integers is 2 less than twice the smallest. Which of the following is the largest integer?

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How to Solve Integers and Their Properties

Last Updated: April 6, 2024

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 30,679 times.

An integer is a set of natural numbers, their negatives, and zero. However, some integers are natural numbers, including 1, 2, 3, and so on. Their negative values are, -1, -2, -3, and so on. So integers are the set of numbers including (…-3, -2, -1, 0, 1, 2, 3,…). An integer is never a fraction, decimal, or percentage, it can only be a whole number. To solve integers and use their properties, learn to use addition and subtraction properties and use multiplication properties.

Using Addition and Subtraction Properties

• a + b = c (where both a and b are positive numbers the sum c is also positive)
• For example: 2 + 2 = 4

• -a + -b = -c (where both a and b are negative, you get the absolute value of the numbers then you proceed to add, and use the negative sign for the sum)
• For example: -2+ (-2)=-4

• a + (-b) = c (when your terms are of different signs, determine the larger number's value, then get the absolute value of both terms and subtract the lesser value from the larger value. Use the sign of the larger number for the answer.)
• For example: 5 + (-1) = 4

• -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value)
• For example: -5 + 2 = -3

• An example of the additive identity is: a + 0 = a
• Mathematically, the additive identity looks like: 2 + 0 = 2 or 6 + 0 = 6

• The additive inverse is when a number is added to the negative equivalent of itself.
• For example: a + (-b) = 0, where b is equal to a
• Mathematically, the additive inverse looks like: 5 + -5 = 0

• For example: (5+3) +1 = 9 has the same sum as 5+ (3+1) = 9

Using Multiplication Properties

• When a and b are positive numbers and not equal to zero: +a * + b = +c
• When a and b are both negative numbers and not equal to zero: -a*-b = +c

• However, understand that any number multiplied by zero, equals zero.

• For example: a(b+c) = ab + ac
• Mathematically, this looks like: 5(2+3) = 5(2) + 5(3)
• Note that there is no inverse property for multiplication because the inverse of a whole number is a fraction, and fractions are not an element of integer.

Joseph Meyer

The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.

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Unit 4: Integers

Negative numbers.

• Negative symbol as opposite (Opens a modal)
• Intro to negative numbers (Opens a modal)
• Number opposites challenge Get 3 of 4 questions to level up!
• Interpreting negative numbers (temperature and elevation) Get 3 of 4 questions to level up!

Comparing integers

• Ordering negative numbers (Opens a modal)
• Negative numbers, variables, number line (Opens a modal)
• Ordering rational numbers Get 3 of 4 questions to level up!
• Compare rational numbers using a number line Get 3 of 4 questions to level up!

Adding and subtracting

• Adding numbers with different signs (Opens a modal)
• Adding & subtracting negative numbers (Opens a modal)
• Adding & subtracting negative numbers Get 5 of 7 questions to level up!
• Addition & subtraction: find the missing value Get 3 of 4 questions to level up!

Module 10: Linear Equations

Using a problem-solving strategy to solve number problems, learning outcomes.

• Solve number problems
• Solve consecutive integer problems

Solving Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference , of , and and .

The difference of a number and six is thirteen. Find the number.

https://ohm.lumenlearning.com/multiembedq.php?id=142763&theme=oea&iframe_resize_id=mom50

The sum of twice a number and seven is fifteen. Find the number.

Show Solution

https://ohm.lumenlearning.com/multiembedq.php?id=142770&theme=oea&iframe_resize_id=mom60

Watch the following video to see another example of how to solve a number problem.

Solving for Two or More Numbers

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142775&theme=oea&iframe_resize_id=mom70

Watch the following video to see another example of how to find two numbers given the relationship between the two.

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142806&theme=oea&iframe_resize_id=mom80

One number is ten more than twice another. Their sum is one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142811&theme=oea&iframe_resize_id=mom90

Solving for Consecutive Integers

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other.  Some examples of consecutive integers are:

[latex]\begin{array}{c} \hfill \text{…}1, 2, 3, 4, 5, 6\text{,…}\hfill \end{array}[/latex] [latex]\text{…}-10,-9,-8,-7\text{,…}[/latex] [latex]\text{…}150,151,152,153\text{,…}[/latex]

If we are looking for several consecutive numbers, it is important to first identify what they look like with variables before we set up the equation.  Notice that each number is one more than the number preceding it. So if we define the first integer as [latex]n[/latex], the next consecutive integer is [latex]n+1[/latex]. The one after that is one more than [latex]n+1[/latex], so it is [latex]n+1+1[/latex], or [latex]n+2[/latex].

[latex]\begin{array}{cccc}n\hfill & & & \text{1st integer}\hfill \\ n+1\hfill & & & \text{2nd consecutive integer}\hfill \\ n+2\hfill & & & \text{3rd consecutive integer}\hfill \end{array}[/latex]

For example, let’s say I want to know the next consecutive integer after [latex]4[/latex]. In mathematical terms, we would add [latex]1[/latex] to [latex]4[/latex] to get [latex]5[/latex]. We can generalize this idea as follows: the consecutive integer of any number, [latex]x[/latex], is [latex]x+1[/latex]. If we continue this pattern, we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.

We apply the idea of consecutive integers to solving a word problem in the following example.

The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.

The sum of three consecutive integers is [latex]93[/latex]. What are the integers?

• Read and understand:  We are looking for three numbers, and we know they are consecutive integers.
• Constants and Variables: [latex]93[/latex] is a constant. The first integer we will call [latex]x[/latex]. Second integer: [latex]x+1[/latex] Third integer: [latex]x+2[/latex]
• Translate:  The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. “ is 93 ” translates to “[latex]=93[/latex]” since “ is ” is associated with equals.
• Write an equation:  [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]

[latex]x+x+1+x+2=93[/latex]

Combine like terms, simplify, and solve.

[latex]\begin{array}{r}x+x+1+x+2=93\\3x+3 = 93\\\underline{-3\,\,\,\,\,-3}\\3x=90\\\frac{3x}{3}=\frac{90}{3}\\x=30\end{array}[/latex]

• Check and Interpret: Okay, we have found a value for [latex]x[/latex]. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables:

The first integer we will call [latex]x[/latex], [latex]x=30[/latex] Second integer: [latex]x+1[/latex] so [latex]30+1=31[/latex] Third integer: [latex]x+2[/latex] so [latex]30+2=32[/latex] The three consecutive integers whose sum is [latex]93[/latex] are [latex]30\text{, }31\text{, and }32[/latex]

Find three consecutive integers whose sum is [latex]42[/latex].

Watch this video for another example of how to find three consecutive integers given their sum.

• Ex: Linear Equation Application with One Variable - Number Problem. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/juslHscrh8s . License : CC BY: Attribution
• Ex: Write and Solve an Equation for Consecutive Natural Numbers with a Given Sum. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/Bo67B0L9hGs . License : CC BY: Attribution
• Write and Solve a Linear Equations to Solve a Number Problem (1) Mathispower4u . Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/izIIqOztUyI . License : CC BY: Attribution
• Question ID 142763, 142770, 142775, 142806, 142811, 142816, 142817. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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• Jovan Fonte
• Matin Naseri
• Sharky Kesa
• Anuj Shikarkhane

An integer is a number that does not have a fractional part. The set of integers is

\[\mathbb{Z}=\{\cdots -4, -3, -2, -1, 0, 1, 2, 3, 4 \dots\}. \]

The notation \(\mathbb{Z}\) for the set of integers comes from the German word Zahlen , which means "numbers". Integers strictly larger than zero are positive integers and integers strictly less than zero are negative integers .

For example, \(2\), \(67\), \(0\), and \(-13\) are all integers (2 and 67 are positive integers and -13 is a negative integer). The values \(\frac{4}{7}\), \(10.7\), \(\frac{34}{7}\), \(\sqrt{2}\), and \(\pi\) are not integers.

Number Line

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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• High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,...

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The 7 Best AI Tools to Help Solve Math Problems

How do you make seven even? Use these tools to solve the big math problems in life.

Quick Links

The test questions, wolframalpha, microsoft mathsolver.

While OpenAI's ChatGPT is one of the most widely known AI tools, there are numerous other platforms that students can use to improve their math skills.

I tested seven AI tools on two common math problems so you know what to expect from each platform and how to use each of them.

I used two math problems to test each tool and standardize the inputs.

• Solve for b: (2 / (b - 3)) - (6 / (2b + 1)) = 4
• Simplify the expression: (4 / 12) + (9 / 8) x (15 / 3) - (26 / 10)

These two problems give each AI tool a chance to show reasoning, problem-solving, accuracy, and how it can guide a learner through the process.

Thetawise provides more than simple answers; you can also opt to have the AI model tutor you by sharing a detailed step-by-step breakdown of the solution. Using the platform is fairly straightforward, given that all you need to do is navigate to the platform and key in the math problem at hand. Alternatively, you can even upload a photo of the math problem onto the platform, and the AI will analyze the image and provide you with an answer.

The AI platform gave us a step-by-step breakdown of the problem:

It resulted in the answer:

While the answer is correct, the tool also provides further options for students to generate a more detailed breakdown of the steps or ask more specific questions.

WolframAlpha is an AI tool capable of solving advanced arithmetic, calculus, and algebra equations. While WolframAlpha's free version provides you with a direct answer, the paid version of the tool generates step-by-step solutions. If you want to make the best use of WolframAlpha's capabilities, you can sign up for the Pro version, which costs $5 per month for the annual plan if you're a student.

As expected, Wolfram Alpha solved both problems, showcasing its ability to handle different problems and provide precise answers quickly.

Julius works pretty similarly to the other AI tools on this list. That said, the highlight of this platform is that it has a built-in community forum, which users can use to discuss their prompts, results, or even issues they might be facing with the platform. Its active user base helps you quickly exchange ideas and receive feedback or advice. The platform's default version uses a combination of GPT-4 and Calude-3, based on whichever model best suits the prompt you input.

We tested the platform's accuracy by submitting the same problems that we did with the other AI tools. When submitting your prompt, you have the option of typing your question or uploading an image or a Google Sheet.

Julius provided correct solutions and offered options to help users verify the solution.

One of the oldest AI platforms, Microsoft's MathSolver is a great option if you want a tool capable of providing free step-by-step solutions to calculus, algebra, and other math problems. Here's how it fared when we submitted our math problems.

Microsoft's MathSolver provided the correct answers, and you can view the steps to the solution, take a quiz, solve similar problems, and more. This can be a great way to practice and perfect your understanding of different concepts.

Symbolab allows you to practice your math skills via quizzes, track your progress, and provide solutions to mathematical problems of different types, including calculus, fractions, trigonometry, and more. You can also use the Digital Notebook feature to keep track of any math problems you solve and share them with your friends. Another highlight of this platform is that educators can use the tool to create a virtual classroom, generate assessments, and share feedback, among other things.

The platform not only displays the answer but also lets you view a breakdown of the steps involved in solving the problem. You can also share the answers and steps via email or social media or print them for reference.

Anthropic launched its Claude 3 AI models in March 2024. Anthropic stated that Claude Opus, the most advanced Claude 3 model, outperforms comparable AI tools on most benchmarks for AI systems, including basic mathematics, undergraduate-level expert knowledge, and graduate-level expert reasoning. To test the platform's accuracy and ease of use, we submitted our two math problems. Here's how the platform performed:

While Claude initially got the answer wrong, probing it and requesting further clarification led to a correct solution.

Remember that we used the free version of Claude to solve this problem; subscribing to Opus (its more advanced model) is recommended if you want to take advantage of Claude's more advanced problem-solving capabilities.

Given that Claude got the previous problem wrong, our second, more basic fraction-based problem will indicate if the AI's performance was an anomaly or part of a consistent pattern.

As you can see, Claude correctly solved this problem and provided a detailed step-by-step breakdown of how it arrived at the answer.

GPT-4 can solve problems with far greater accuracy than its predecessor, GPT-3.5. If you're using the free version of ChatGPT, you'll likely only have access to GPT 3.5 and GPT-4o . However, for $20 per month, you can subscribe to the Plus model, which gives you access to GPT-4 and allows you to input five times the number of messages per day compared to the free version. That said, let's check how it performs with math problems.

In both cases, GPT-4o provided the correct answer with a detailed breakdown of the steps. While the platform is free, unlike other models, it does not have a quiz feature or a community forum.

These AI tools offer unique features and capabilities that make them a good option for math problems. Ultimately, the best way to pick a tool is by testing different models to determine which platform best fits your preferences and learning needs.