problem solving involving addition of integers

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How to Add and Subtract Integers: Word Problems

Mastering the art of tackling word problems involving the addition and subtraction of integers is a vital skill in the mathematical universe. Integers, which include positive, negative, and zero, are more nuanced than their natural number counterparts. To craft a highly detailed, comprehensive, and sophisticated guide, let's dive into the labyrinth of integers, unraveling the mystery of word problems step-by-step.

A Step-by-step Guide to Solve Integers Addition and Subtraction: Word Problems

Here is a step-by-step guide to solving word problems of integers addition and subtraction:

Step 1: Decipher the Problem

The journey begins with an intensive reading of the word problem. Identify the integers involved, noting their signs (\(+\) or \(-\)), and the operations stated or implied (addition or subtraction). Understand the context and constraints of the problem to guide your strategy.

Step 2: Pinpoint the Unknowns

Next, determine what the problem demands you to find. This could be an unknown quantity or a relationship between different quantities. Assign variables to these unknowns, typically ‘\(x\)’, ‘\(y\)’, or ‘\(z\)’.

Step 3: Translate into Mathematical Language

Now, morph the word problem into an equivalent mathematical expression or equation. Expressions such as “increased by” or “more than” often signify addition, while “decreased by” or “less than” hint towards subtraction. This translation serves as a bridge between the narrative of the problem and the mathematical steps to solve it.

Step 4: Construct the Equation(s)

Based on your translation, construct an equation or a system of equations that encapsulate the conditions outlined in the problem. Be vigilant of the signs of the integers; a positive integer added to a negative integer can be treated as subtraction and vice versa.

Step 5: Resolve the Equation(s)

Once your equation(s) are set, deploy your arithmetic and algebraic skills to solve them. Remember the basic rules of integer arithmetic, such as the fact that subtracting a negative integer is equivalent to adding a positive one.

Step 6: Validate Your Solution

Substitute your solution back into the original equation(s) to verify its correctness. If it stands the test, your solution is accurate. If it fails, reexamine your steps to identify potential missteps or miscalculations.

Step 7: Answer the Query

The final act is responding to the original question asked in the problem. Ensure your answer aligns with the question and is phrased appropriately, incorporating units if necessary.

by: Effortless Math Team about 9 months ago (category: Articles )

Effortless Math Team

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1.2: Adding and Subtracting Integers

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Learning Objectives

• Add and subtract signed integers.
• Translate English sentences involving addition and subtraction into mathematical statements.
• Calculate the distance between two numbers on a number line.

Addition and Subtraction \((+, -)\)

Visualize adding \(3 + 2\) on the number line by moving from zero three units to the right then another two units to the right, as illustrated below:

The illustration shows that \(3 + 2 = 5\). Similarly, visualize adding two negative numbers \((−3) + (−2)\) by first moving from the origin three units to the left and then moving another two units to the left.

In this example, the illustration shows \((−3) + (−2) = −5\), which leads to the following two properties of real numbers.

\[\begin{align*} & \color{Cerulean}{positive\ number\;} \color{Black}{+\;} \color{Cerulean}{positive\ number\;} \color{Black}{=\;} \color{Cerulean}{positive\ number} \\ & \color{Cerulean}{negative\ number\;} \color{Black}{+\;} \color{Cerulean}{negative\ number\;} \color{Black}{=\;} \color{Cerulean}{negative\ number} \end{align*}\]

Next, we will explore addition of numbers with unlike signs. To add \(3 + (−7)\), first move from the origin three units to the right, then move seven units to the left as shown:

In this case, we can see that adding a negative number is equivalent to subtraction:

\(3+(-7)=3-7=-4\)

It is tempting to say that a positive number plus a negative number is negative, but that is not always true: \(7+(−3)=7−3=4\). The result of adding numbers with unlike signs may be positive or negative. The sign of the result is the same as the sign of the number with the greatest distance from the origin. For example, the following results depend on the sign of the number \(12\) because it is farther from zero than \(5\):

\[\begin{align*} &12+(-5)=7 \\ &-12+5=-7 \end{align*} \]

Example \(\PageIndex{1}\)

Simplify: \(14+(−25)\).

Here \(−25\) is the greater distance from the origin. Therefore, the result is negative.

\[\begin{align*} 14+(-25) &= 14-25 \\ &= -11 \end{align*}\]

Properties of Addition

Given any real numbers \(a\) , \(b\), and \(c\), we have the following properties of addition:

• Additive Identity Property : \[a+0=0+a=a\]
• Additive Inverse Property :\[a+(−a)=(−a)+a=0\]
• Associative Property :\[(a+b)+c=a+(b+c)\]
• Commutative Property :\[a+b=b+a\]

Below are some examples of these properties in action.

Example \(\PageIndex{2}\)

b. \(10+(−10)\)

a. Adding zero to any real number results in the same real number.

\[5+0=5 \nonumber\]

b. Adding opposites results in zero.

a. \(5\); b. \(0\)

Example \(\PageIndex{3}\)

a. \((3+7)+4\)

b. \(3+(7+4)\)

Parentheses group the operations that are to be performed first.

a. \[\begin{align*} (\color{Cerulean}{3+7} \color{Black}{)}+4 &= \color{Cerulean}{10}\ \color{Black}{+\ 4} \\ &= 14 \end{align*} \]

b. \[\begin{align*} 3+(\color{Cerulean}{7+4} \color{Black}{)} &= 3+ \color{Cerulean}{10} \\ &= 14 \end{align*} \]

These two examples both result in \(14\): changing the grouping of the numbers does not change the result.

\((\color{Cerulean}{3+7} \color{Black}{)} +4=3+(\color{Cerulean}{7+4} \color{Black}{)}=14\)

a. \(14\); b. \(14\)

At this point, we highlight the fact that addition is commutative: the order in which we add does not matter and yields the same result.

\[\begin{align*} 2+9 &= 9+2 \\ 11 &= 11 \end{align*} \]

On the other hand, subtraction is not commutative.

\[\begin{align*} 2-9 &\neq 9-2 \\ -7 &\neq 7 \end{align*} \]

We will use these properties, along with the double-negative property for real numbers, to perform more involved sequential operations. To simplify things, we will make it a general rule to first replace all sequential operations with either addition or subtraction and then perform each operation in order from left to right.

Example \(\PageIndex{4}\)

Simplify: \(4−(−10)+(−5)\).

Replace the sequential operations and then perform them from left to right.

\[\begin{align*} 4-(-10)+(-5) &= 4+10-5 && \color{Cerulean}{Replace\ -(-)\ with\ addition\ (+).} \\ & && \color{Cerulean}{Replace\ +(-)\ with\ addition\ (-).} \\ &= 14-5 \\ &=9 \end{align*} \]

Example \(\PageIndex{5}\)

Simplify: \(−3+(−8)−(−7)\).

\[\begin{align*} -3+(-8)-(-7) &= -3-8+7 && \color{Cerulean}{Replace\ +(-)\ with\ (-).} \\ & && \color{Cerulean}{Replace\ -(-)\ with\ (+).} \\ &= -11+7 \\ &=-4 \end{align*} \]

Example \(\PageIndex{6}\)

Simplify: \(12−(−9)+(−6)\).

(click to see video)

Often we find the need to translate English sentences involving addition and subtraction to mathematical statements. Listed below are some key words that translate to the given operation.

Example \(\PageIndex{7}\)

What is the difference of \(7\) and \(−3\)?

The key word “difference” implies that we should subtract the numbers.

\[\begin{align*} 7-(-3) &= 7+3 \\ &=10 \end{align*} \]

The difference of \(7\) and \(−3\) is \(10\).

Example \(\PageIndex{8}\)

What is the sum of the first five positive integers?

The initial key word to focus on is “sum”; this means that we will be adding the five numbers. The first five positive integers are \(\{1, 2, 3, 4, 5\}\). Recall that \(0\) is neither positive nor negative.

\(1+2+3+4+5=15\)

The sum of the first five positive integers is \(15\).

Example \(\PageIndex{9}\)

What is \(10\) subtracted from the sum of \(8\) and \(6\)?

We know that subtraction is not commutative; therefore, we must take care to subtract in the correct order. First, add \(8\) and \(6\) and then subtract \(10\) as follows:

It is important to notice that the phrase “\(10\) subtracted from” does not translate to a mathematical statement in the order it appears. In other words, \(10−(8+6)\) would be an incorrect translation and leads to an incorrect answer. After translating the sentence, perform the operations.

\[\begin{align*} (8+6)-10 &= 14-10 \\ &= 4 \end{align*} \]

Ten subtracted from the sum of \(8\) and \(6\) is \(4\).

Distance on a Number Line

One application of the absolute value is to find the distance between any two points on a number line. For real numbers \(a\) and \(b\), the distance formula for a number line is given as,

\(d=|b-a|\)

Example \(\PageIndex{10}\)

Determine the distance between \(2\) and \(7\) on a number line.

On the graph we see that the distance between the two given integers is \(5\) units.

Using the distance formula we obtain the same result.

\[\begin{align*} d &= |7-2| \\ &=|5| \\ &=5 \end{align*} \]

\(5\) units

Example \(\PageIndex{11}\)

Determine the distance between \(−4\) and \(7\) on a number line.

Use the distance formula for a number line \(d=|b−a|\), where \(a=−4\) and \(b=7\).

\[\begin{align*} d &= |7-(-4)| \\ &= |7+4| \\ &= |11| \\ &= 11 \end{align*} \]

\(11\) units

It turns out that it does not matter which points are used for \(a\) and\(b\); the absolute value always ensures a positive result.

Exercise \(\PageIndex{1}\)

Determine the distance between \(−12\) and \(−9\) on the number line.

Video Solution

Key Takeaways

• A positive number added to a positive number is positive. A negative number added to a negative number is negative.
• The sign of a positive number added to a negative number is the same as the sign of the number with the greatest distance from the origin.
• Addition is commutative and subtraction is not.
• When simplifying, it is a best practice to first replace sequential operations and then work the operations of addition and subtraction from left to right.
• The distance between any two numbers on a number line is the absolute value of their difference. In other words, given any real numbers a and b , use the formula \(d=|b−a|\) to calculate the distance d between them.

Exercise \(\PageIndex{2}\)

Add and subtract.

• \(24+(−18)\)
• \(9+(−11)\)
• \(−31+5\)
• \(−12+15\)
• \(−30+(−8)\)
• \(−50+(−25)\)
• \(−7+(−7)\)
• \(−13−(−13)\)
• \(8−12+5\)
• \(−3−7+4\)
• \(−1−2−3−4\)
• \(6−(−5)+(−10)−14\)
• \(−5+(−3)−(−7)\)
• \(2−7+(−9)\)
• \(−30+20−8−(−18)\)
• \(10−(−12)+(−8)−20\)
• \(5−(−2)+(−6)\)
• \(−3+(−17)−(−13)\)
• \(−10+(−12)−(−20)\)
• \(−13+(−5)−(−25)\)
• \(20−(−4)−(−5)\)
• \(17+(−12)−(−2)\)

3: −26

5: −38

7: −14

11: −10

13: −1

19: −2

Exercise \(\PageIndex{3}\)

Translate each sentence to a mathematical statement and then simplify.

• Find the sum of \(3\), \(7\), and \(−8\).
• Find the sum of \(−12\), \(−5\), and \(7\).
• Determine the sum of the first ten positive integers.
• Determine the sum of the integers in the set \(\{−2, −1, 0, 1, 2\}\).
• Find the difference of \(10\) and \(6\).
• Find the difference of \(10\) and \(−6\).
• Find the difference of \(−16\) and \(−5\).
• Find the difference of \(−19\) and \(7\).
• Subtract \(12\) from \(10\).
• Subtract \(−10\) from \(−20\).
• Subtract \(5\) from \(−31\).
• Subtract \(−3\) from \(27\).
• Two less than \(8\).
• Five less than \(−10\).
• Subtract \(8\) from the sum of \(4\) and \(7\).
• Subtract \(-5\) from the sum of \(10\) and \(−3\).
• Subtract \(2\) from the difference of \(8\) and \(5\).
• Subtract \(6\) from the difference of \(−1\) and \(7\).
• Mandy made a \($200\) deposit into her checking account on Tuesday. She then wrote \(4\) checks for \($50.00\), \($125.00\), \($60.00\), and \($45.00\). How much more than her deposit did she spend?
• The quarterback ran the ball three times in last Sunday’s football game. He gained \(7\) yards on one run but lost \(3\) yards and \(8\) yards on the other two. What was his total yardage running for the game?
• The revenue for a local photographer for the month is \($1,200\). His costs include a studio rental of \($600\), props costing \($105\), materials fees of \($135\), and a make-up artist who charges \($120\). What is his total profit for the month?
• An airplane flying at \(30,000\) feet lost \(2,500\) feet in altitude and then rose \(1,200\) feet. What is the new altitude of the plane?
• The temperature was \(22°\) at \(6:00\) p.m. and dropped \(26°\) by midnight. What was the temperature at midnight?
• A nurse has \(30\) milliliters of saline solution but needs \(75\) milliliters of the solution. How much more does she need?
• The width of a rectangle is \(2\) inches less than its length. If the length measures \(16\) inches, determine the width.
• The base of a triangle is \(3\) feet shorter than its height. If the height measures \(5\) feet, find the length of the base.

7: \(−11\)

9: \(−2\)

11: \(−36\)

19: \($80\)

21: \($240\)

23: \(−4°\)

25: \(14\) inches

Exercise \(\PageIndex{4}\)

Find the distance between the given numbers on a number line.

• \(−3\) and \(12\)
• \(8\) and \(−13\)
• \(−25\) and \(−10\)
• \(−100\) and \(−130\)
• \(−7\) and \(−20\)
• \(0\) and \(−33\)
• \(-10\) and \(10\)
• \(−36\) and \(36\)
• The coldest temperature on earth, \(−129°\)F, was recorded in 1983 at Vostok Station, Antarctica. The hottest temperature on earth, \(136°\)F, was recorded in 1922 at Al ’Aziziyah, Libya. Calculate earth’s temperature range.
• The daily high temperature was recorded as \(91°\)F and the low was recorded as \(63°\)F. What was the temperature range for the day?
• A student earned \(67\) points on his lowest test and \(87\) points on his best. Calculate his test score range.
• On a busy day, a certain website may have \(12,500\) hits. On a slow day, it may have as few as \(750\) hits. Calculate the range of the number of hits.

1: \(15\) units

3: \(15\) units

5: \(13\) units

7: \(20\) units

9: \(265°\)F

11: \(20\) points

Exercise \(\PageIndex{5}\)

Discussion Board Topics

• Share an example of adding signed numbers in a real-world application.
• Demonstrate the associative property of addition with any three real numbers.
• Show that subtraction is not commutative

Math Problems and Solutions on Integers

Problems related to integer numbers in mathematics are presented along with their solutions.

• Math Article
• 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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Word Problems - Integer Mixed Operations (addition and subtraction)

Description:  This packet helps students practice solving word problems that require addition and subtraction with integers. Understand when to add and when to subtract positive and negative numbers can be difficult.  So, these word problems are useful in assessing how well students understand addition and subtraction operations when working with integers.  Each page contains 5 problems. Each page also has a speed and accuracy guide, to help students see how fast and how accurately they should be doing these problems. After doing all 10 problems, students should be more comfortable doing these problems and have a clear understanding of how to solve them. 

Abraham is spending a day at the hotel pool. He enjoys jumping from the very cool pool (62 degrees) to the very hot Jacuzzi (113 degrees). What temperature change is Abraham experiencing as he moves from pool to Jacuzzi?

Zack just got a grade of 93 on a report. However, because he turned it in late, his teacher is going to deduct 15 points. What will Zack’s final score on the report be?

Practice problems require knowledge of how to add and subtract whole numbers.

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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.

☛ Check Grade wise Integers Word Problems Worksheets

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• Introduction to applications of integer operations
• The melting point of dry ice is -109.4 degrees Fahrenheit. The boiling point of dry ice is - 109.3 degrees Fahrenheit. How many degrees is the boiling point above the melting point?
• The lowest point in Africa is Lake Assal, located at 156 meters below sea level. The lowest point in North America is Death Valley, located at 86 meters below sea level. Calculate the difference in depth between Lake Assal and Death Valley.
• A rocket countdown is at -11 s. In how many seconds will the rocket be 22 seconds into the flight?
• At a particular school, there are 138 students in grade 10, 126 students in grade 11 and 92 students in grade 12. If all senior students are sent to the hall for an assembly, how many students are there in the hall?
• An auto mall was opened and it has 154 cars in stock. 38 cars were sold on the first day and 42 cars were sold on the second day. How many cars were left in stock after the first two days?
• Thomas was planning for a house party. He had 24 cans of coke at home, and he bought another 38 cans of root beer from the supermarket. Later on, he returned 17 cans of root beer. How many cans of pop did he have for the party?

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Topic Notes

Basic concepts.

• Divisibility rules
• Prime factorization
• Introduction to Exponents

Related Concepts

• Adding and subtracting decimals
• Order of operations (PEMDAS)
• Adding fractions with like denominators
• Subtracting fractions with like denominators

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APPLYING ADDITION AND SUBTRACTION OF INTEGERS

Solving a multistep problem.

You can use what you know about adding and subtracting integers to solve a multistep problem.

Example : 

A seal is swimming in the ocean 5 feet below sea level. It dives down 12 feet to catch some fish. Then, the seal swims 8 feet up towards the surface with its catch. What is the seal’s final elevation relative to sea level ?

Step 1 : 

The seal starts at 5 feet below the surface, so its initial position is -5 ft.

Write an expression.

Starts  - Dives down + Swims up

-5  -  12  +  8

Step 2 : 

Add or subtract from left to right to find the value of the expression.

-5  -  12  +  8  =  -17  +  8

The seal’s final elevation is 9 feet below sea level.

Applying Properties to Solve Problems

You can use properties of addition to solve problems involving integers.

Jack has a checking account. On Monday he writes a $160 check for groceries. Then he deposits $125. Finally he writes another check for $40. What was the total change in the amount in Jack’s account ?

Analyze Information :

When Jack deposits money, he adds that amount to the account. When he writes a check, that money is deducted from the account.

Formulate a Plan : 

Use a positive integer for the amount Jack added to the account. Use negative integers for the checks he wrote. Find the sum.

-160 + 125 + (-40)

Solve : 

Add the amounts to find the total change in the account. Use properties of addition to simplify calculations.

Commutative Property :

-160 + 125 + (-40)  =  -160 + (-40) + 125

Associative Property :

=  -200 + 125

=  -75

The amount in the account decreased by $75.

Justify and Evaluate :

Jack's account has $75 less than it did before Monday. This is reasonable because he wrote checks for $200 but only deposited $125.

Solved Problems

Problem 1 : 

Anna is in a cave 40 feet below the cave entrance. She descends 13 feet, then ascends 18 feet. Find her new position relative to the cave entrance.

Solution : 

Anna starts at 40 feet below cave entrance, so her initial position is -40 ft.

Starts  - Descends + Ascends

-40  -  13  +  18

-40  -  13  +  18  =  -53  +  18

=  -35

Anna's new position is  35 feet below the cave entrance.

Problem 2 : 

Tomas works as an underwater photographer. He starts at a position that is 15 feet below sea level. He rises 9 feet, then descends 12 feet to take a photo of a coral reef. Write and evaluate an expression to find his position relative to sea level when he took the photo.

Tomas starts at 15 feet below sea level, so his initial position is -15 ft.

Starts  + Rises - Descends

-15  +  9  -  12

-15  +  9  -  12  =  -6  -  12

=  -18

When Tomas takes photo, his position is 18 feet below the sea level. 

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Solving word problems both rely the development of reading and language skills. Addition means “putting together” groups of objects and finding how many they are in total while subtraction tells “how many are left” or “how many more or less”.

Steps in Problem Solving:

• Identify the Problem. Understand what is asked.
• Encircle important numbers.
• Underline the keywords. Analyze if it is for addition or subtraction .
• Solve the problem.
• Present the answer.

Addition: in all, sum, total, more than, plus, altogether, increased by add

Subtraction: fewer, left, less than, take away, minus, difference, remain, decreased

There are 6 surfboards and the surfer bought another 8 pieces. How many surfboards are there in all?

There were 8 beach balls but 5 of them were damaged. How many beach balls were left?

8 – 5 = 3

Solving Word Problems Involving Addition and Subtraction of Numbers within 120 Worksheets

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Subtracting integers word problems

Here are four great examples about subtracting integers word problems.

Problem #1:

The record high temperature for Massachusetts is 104 degrees Fahrenheit. The record low is -18 degrees Fahrenheit. What is the difference between high and low? Solution The problem has 1 important component. It is the phrase " difference between high and low ." Difference between high and low is the same as subtracting the smaller number from the bigger number.

104 - -18 = 104 + 18 = 122

The difference between high and low is 122 degrees Fahrenheit.

More subtracting integers word problems 

Problem #2:

Sylvia checked the balance in her bank account early in the morning and saw that she had 98 dollars. In the afternoon, she noticed that there was an overdraft of 6 dollars in the account for unpaid bills. How much were the unpaid bills? Solution The problem has 1 important component. It is the word " Overdraft ." Overdraft occurs when the bank gives you money to cover amount you did not have at the time the bills were paid. In this case, what you did not have is 6 dollars. Since the bank gave it to you, you owe it to the bank. Therefore, you owe 6 dollars to the bank and this can be expressed as -6.

The amount of unpaid bills is the difference between what you had that morning (98) and what you have this afternoon (-6). 98 - -6 = 98 + 6 = 104 dollars

The amount of unpaid bills is 104 dollars

Problem #3:

You are 5 dollars in debt. You borrow 12 dollars more. What is the total amount of your debt? Solution The problem has 2 important components shown in bold below. You are 5 dollars in debt . You borrow 12 dollars more . What is the total amount of your debt? 5 dollars in debt can be represented by -5 Borrow 12 more can be represented by - 12 We get -5 - 12 = -5 + -12 = -17 The amount of your debt is 17 dollars.

Problem #4:

An airplane takes off and then climbs 2500 feet. After 20 minutes, the airplane descends 150 feet.  What is the airplane's current height? Solution The problem has 2 important components shown in bold below. An airplane takes off and then climbs 2500 feet . After 20 minutes, the airplane descends 150 feet .  What is the airplane's current height? Climbs 2500 feet can be represented with +2500 Descends 150 feet can be represented with - 150 We get +2500 - 150 = +2500 + -150 = 2350

The airplane is now at a height of 2350 feet.

Take a look also at the problem below

Adding integers word problems

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