14°F in the morning. If the temperature dropped 7°F, what is the temperature now? RESULTS BOX: |
RESULTS BOX: |
RESULTS BOX: |
RESULTS BOX: |
39°C. The freezing point of alcohol is 114°C. How much warmer is the melting point of mercury than the freezing point of alcohol? RESULTS BOX: |
Last modified on August 3rd, 2023
We use integers in our everyday life for counting money and measuring the speed of sound and light, height and weight of objects, temperature and pressure of the atmosphere, and depth of a sea. Solving word problems related to the above measurements will help us relate better to the concept with applications.
A submarine was located 600 feet below sea level. Calculate its new position if it ascends 250 feet from its position.
Given, The initial position of the submarine is 600 feet It ascends 250 feet Thus, the new position of the submarine is (600 – 250) feet = 350 feet Thus, the new depth of the submarine is 550 feet
Today’s weather report suggested that the temperature of New York City increased from 10-degree celsius to 20-degree celsius. What is the rise in temperature?
Given, Initial temperature = -10 degree celsius Final temperature = 20 degrees celsius Increase in temperature = 20 – (-10) = 30 degrees celsius Thus, the rise in temperature of New York City is 30 degree Celsius.
Demitri owes her mother \$5.00. He earns \$12.00 doing chores. How much is left with him after he gives his mother what she owes?
Given, Amount to be given to his mother = \$5.00 He earns = \$12.00 Thus, he is left with = \$(12.00 – 5.00) = \$7.00 Thus, Demitri is left with \$7.00.
The local movie theater reported losses of \$475 each day for 3 days. What was the total loss incurred for the 3 days?
Given, Loss incurred by the movie theater per day = \$475 The loss incurred in 3 days = \$(3 × 475) = \$1,425 Thus the total loss incurred by the movie theater in 3 days is \$1,425
The Second World War began in the year 1939 and ended in 1945. How long did it last?
Given, The Second World War began in 1939 and ended in 1945 Thus, it lasted (1945 – 1939) = 6 years
Mt. Everest is 29,028 feet above sea level, and the Dead Sea is 1,312 feet below sea level. Find the difference between the 2 elevations.
Given The height of Mt. Everest is 29,028 feet above sea level The height of the Dead Sea is 1,312 feet below sea level Thus, the difference between the 2 elevations = (29,028 + 1,312) feet = 30,340 feet
Dakota withdrew a total of \$600 over 4 days. If he withdrew the same amount daily, find the amount he withdrew each day.
Given, Total amount Dakota withdrew = \$600 Number of days he withdrew = 4 days Amount withdrawn each day = \$(600 ÷ 4) = \$150 Thus, Dakota withdrew \$150 per day.
Andrew had \$12,000 in his account. He once withdrew \$2,000 and then deposited $6,000. What is the current balance in the account?
Given, Initial balance Andrew had = \$12,000 Amount withdrawn = \$2,000 Amount deposited = \$6,000 Thus, the current balance = \$(12,000 – 2,000 + 6,000) = \$16,000 Thus, the current balance Andrew has currently in his account is \$16,000
A kite rises 100 feet from the ground and then falls back 40 feet. What is the current height of the kite from the ground?
Given, The kite rises = 100 feet and then falls back = 40 feet Thus, the current height of the kite from the ground is (100 – 40) feet = 60 feet
If it is 3°C outside and the temperature drops 15° C in the next 6 hours, how cold will it be after 6 hours?
Initially, the outside temperature was 2°C Then, the temperature drops to = (2 – 15)°C = -13°C Thus, the temperature after 6 hours will be -13°C
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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.
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.
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.
Are you ready to tackle integer word problems? Here are a few steps:
In dealing with integer word problems, practice is critical. The more problems you tackle, the more proficient you become. Happy problem-solving!
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.
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 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.
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.
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.
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.
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 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.
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.
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 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.
Solving multi-step word problems can seem daunting, but a systematic approach simplifies the task. Below are vital strategies:
Remember, practice significantly improves problem-solving skills and the ability to tackle complex multi-step word problems involving integers. Happy problem-solving!
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.
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.
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!
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.
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.
Let’s walk through the solutions together to help you understand how these problems are solved.
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.
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.
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.
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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 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 :
Let | x = | boxes sold in the first week |
x + 4 = | boxes sold in the second week | |
2(x + 4) = | boxes sold in the third week |
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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Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.
If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.
We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.
Coordinate graph paper can be very useful when studying integers. Coordinate geometry is a practical application of integers and can give students practice with using integers while learning another related skill. Coordinate graph paper can be found on the Graph Paper page:
Coordinate Graph Paper
Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.
For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.
By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.
Two-color counters are fantastic manipulatives for teaching and learning about integer addition. Two-color counters are usually plastic chips that come with yellow on one side and red on the other side. They might be available in other colors, so you'll have to substitute your own colors in the following description.
Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side; then you mash them all together, take out the zeros (if any) and behold, you have your answer! Need further elaboration? Read on!
The correct side means using red to model negative numbers and yellow to model positive numbers. You would model —5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward although, you'll want to caution your students to be less exuberant than usual, so none of the chips get flipped. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You can do this because —1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't affect the answer. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question. If you have no chips left at the end, the answer is zero!
Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.
The worksheets in this section include addition and subtraction on the same page. Students will have to pay close attention to the signs and apply their knowledge of integer addition and subtraction to each question. The use of counters or number lines could be helpful to some students.
These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.
In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).
Multiplying integers is very similar to multiplication facts except students need to learn the rules for the negative and positive signs. In short, they are:
In words, multiplying two positives or two negatives together results in a positive product, and multiplying a negative and a positive in either order results in a negative product. So, -8 × 8, 8 × (-8), -8 × (-8) and 8 × 8 all result in an absolute value of 64, but in two cases, the answer is positive (64) and in two cases the answer is negative (-64).
Should you wish to develop some "real-world" examples of integer multiplication, it might be a stretch due to the abstract nature of negative numbers. Sure, you could come up with some scenario about owing a debt and removing the debt in previous months, but this may only result in confusion. For now students can learn the rules of multiplying integers and worry about the analogies later!
Luckily (for your students), the rules of dividing integers are the same as the rules for multiplying:
Dividing a positive by a positive integer or a negative by a negative integer will result in a positive integer. Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.
This section includes worksheets with both multiplying and dividing integers on the same page. As long as students know their facts and the integer rules for multiplying and dividing, their sole worry will be to pay attention to the operation signs.
In this section, the integers math worksheets include all of the operations. Students will need to pay attention to the operations and the signs and use mental math or another strategy to arrive at the correct answers. It should go without saying that students need to know their basic addition, subtraction, multiplication and division facts and rules regarding operations with integers before they should complete any of these worksheets independently. Of course, the worksheets can be used as a source of questions for lessons, tests or other learning activities.
Order of operations with integers can be found on the Order of Operations page:
Order of Operations with Integers
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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.
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.
Adding Integers 1. Rearrange the terms so that integers with the same sign are next to each other. 2. Add integers with like signs together. 3. Subtract the absolute values of integers with different signs. 4. The sign of the solution will be the sign of the larger integer.
Subtracting Integers 1. Rewrite the problem by changing the second term to its additive inverse. 2. Add the values.
Multiplying Integers 1. Multiply the absolute values of the integers. 2. If the two factors have the same sign, the product is positive. 3. If the two factors have different signs, the product is negative.
The price of one share of stock fell 4 dollars each day for 8 days. How much value did one share of the stock lose?
The stock price fell, so it is represented by -4. This happened for 8 days. -4 x 8 = -32 The stock’s value dropped by $32.
Practice Integer Word Problems
Integer – whole numbers and their opposites {…, -3, -2, -1, 0, 1, 2, 3, …}
Negative integer – any integer less than zero.
Positive integer – any integer greater than zero.
Absolute value – the positive distance that a number is from 0 on a number line.
Zero Pair – a yellow counter and a red counter that represents zero.
Opposites – two integers that are the same distance from 0 on a number line but in opposite directions, like -5 and 5.
Additive inverses – two integers that are opposites.
Pre-requisite Skills Multi-Step Word Problems II Order of Operations
Related Skills Evaluating Algebraic Expressions Evaluate Algebraic Expressions Add Linear Expressions Subtract Linear Expressions Solve Complex Equations
Last Updated: September 2, 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 32,343 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.
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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Learning objectives.
By the end of this section, you will be able to:
Before you get started, take this readiness quiz.
Translate “6 less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.26 .
Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 2.16 .
Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 2.27 .
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?
Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.
Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.
Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.
Restate the problem in one sentence with all the important information. | |
Translate into an equation. |
Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.
Solve the equation. | |
Multiply both sides by 2. | |
Simplify. |
Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that p = 36 , p = 36 , which means “the original price” was $36.
Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”
If this were a homework exercise, our work might look like this:
Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?
Let the original price. | |
18 is one-half the original price. | |
Multiply both sides by 2. | |
Simplify. | |
Check. Is $36 a reasonable price for a purse? | |
Yes. | |
Is 18 one half of 36? | |
The original price of the purse was $36. |
Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?
Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?
Let’s try this approach with another example.
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?
the problem. | |
what we are looking for. | How many boys were in the study group? |
Choose a variable to represent the number of boys. | Let the number of boys. |
Restate the problem in one sentence with all the important information. | |
Translate into an equation. | |
the equation. | |
Subtract 3 from each side. | |
Simplify. | |
Divide each side by 2. | |
Simplify. | |
First, is our answer reasonable? Yes, having 4 boys in a study group seems OK. The problem says the number of girls was 3 more than twice the number of boys. If there are four boys, does that make eleven girls? Twice 4 boys is 8. Three more than 8 is 11. | |
the question. | There were 4 boys in the study group. |
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?
Solve Number Problems
Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
The difference of a number and six is 13. Find the number.
the problem. Are all the words familiar? | |
what we are looking for. | the number |
Choose a variable to represent the number. | Let the number. |
Remember to look for clue words like "difference... of... and..." | |
Restate the problem as one sentence. | |
Translate into an equation. | |
the equation. | |
Simplify. | |
The difference of 19 and 6 is 13. It checks! | |
the question. | The number is 19. |
The difference of a number and eight is 17. Find the number.
The difference of a number and eleven is −7 . −7 . Find the number.
The sum of twice a number and seven is 15. Find the number.
the problem. | |
what we are looking for. | the number |
Choose a variable to represent the number. | Let the number. |
Restate the problem as one sentence. | |
Translate into an equation. | |
the equation. | |
Subtract 7 from each side and simplify. | |
Divide each side by 2 and simplify. | |
Is the sum of twice 4 and 7 equal to 15? | |
the question. | The number is 4. |
Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
The sum of four times a number and two is 14. Find the number.
The sum of three times a number and seven is 25. Find the number.
Some number word problems ask us 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. In order to avoid using more than 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 21. Find the numbers.
the problem. | ||
what we are looking for. | We are looking for two numbers. | |
We have two numbers to name and need a name for each. | ||
Choose a variable to represent the first number. | Let number. | |
What do we know about the second number? | One number is five more than another. | |
number | ||
Restate the problem as one sentence with all the important information. | The sum of the 1 number and the 2 number is 21. | |
Translate into an equation. | ||
Substitute the variable expressions. | ||
the equation. | ||
Combine like terms. | ||
Subtract 5 from both sides and simplify. | ||
Divide by 2 and simplify. | ||
Find the second number, too. | ||
Do these numbers check in the problem? | ||
Is one number 5 more than the other? | ||
Is thirteen 5 more than 8? Yes. | ||
Is the sum of the two numbers 21? | ||
the question. | The numbers are 8 and 13. |
One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
the problem. | ||
what we are looking for. | We are looking for two numbers. | |
Choose a variable. | Let number. | |
One number is 4 less than the other. | number | |
Write as one sentence. | The sum of the 2 numbers is negative 14. | |
Translate into an equation. | ||
the equation. | ||
Combine like terms. | ||
Add 4 to each side and simplify. | ||
Simplify. | ||
Is −9 four less than −5? | ||
Is their sum −14? | ||
the question. | The numbers are −5 and −9. |
The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.
The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.
One number is ten more than twice another. Their sum is one. Find the numbers.
the problem. | ||
what you are looking for. | We are looking for two numbers. | |
Choose a variable. | Let number. | |
One number is 10 more than twice another. | number | |
Restate as one sentence. | Their sum is one. | |
The sum of the two numbers is 1. | ||
Translate into an equation. | ||
the equation. | ||
Combine like terms. | ||
Subtract 10 from each side. | ||
Divide each side by 3. | ||
Is ten more than twice −3 equal to 4? | ||
Is their sum 1? | ||
the question. | The numbers are −3 and 4. |
One number is eight more than twice another. Their sum is negative four. Find the numbers.
One number is three more than three times another. Their sum is −5 . −5 . Find the numbers.
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .
The sum of two consecutive integers is 47. Find the numbers.
the problem. | ||
what you are looking for. | two consecutive integers | |
each number. | Let integer. | |
next consecutive integer | ||
Restate as one sentence. | The sum of the integers is 47. | |
Translate into an equation. | ||
the equation. | ||
Combine like terms. | ||
Subtract 1 from each side. | ||
Divide each side by 2. | ||
the question. | The two consecutive integers are 23 and 24. |
The sum of two consecutive integers is 95 . 95 . Find the numbers.
The sum of two consecutive integers is −31 . −31 . Find the numbers.
Find three consecutive integers whose sum is −42 . −42 .
the problem. | ||
what we are looking for. | three consecutive integers | |
each of the three numbers. | Let integer. | |
2 consecutive integer | ||
3 consecutive integer | ||
Restate as one sentence. | The sum of the three integers is −42. | |
Translate into an equation. | ||
the equation. | ||
Combine like terms. | ||
Subtract 3 from each side. | ||
Divide each side by 3. | ||
the question. | The three consecutive integers are −13, −14, and −15. |
Find three consecutive integers whose sum is −96 . −96 .
Find three consecutive integers whose sum is −36 . −36 .
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
Notice each integer is 2 more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The next one would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.
Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.
Find three consecutive even integers whose sum is 84.
the problem. | |
what we are looking for. | three consecutive even integers |
the integers. | Let even integer. consecutive even integer consecutive even integer |
Restate as one sentence. | The sume of the three even integers is 84. |
Translate into an equation. | |
the equation. | |
Combine like terms. | |
Subtract 6 from each side. | |
Divide each side by 3. | |
the question. | The three consecutive integers are 26, 28, and 30. |
Find three consecutive even integers whose sum is 102.
Find three consecutive even integers whose sum is −24 . −24 .
A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
the problem. | ||
what we are looking for. | How much does the husband earn? | |
. | ||
Choose a variable to represent the amount the husband earns. | Let the amount the husband earns. | |
The wife earns $16,000 less than twice that. | the amount the wife earns. | |
Together the husband and wife earn $110,000. | ||
Restate the problem in one sentence with all the important information. | ||
Translate into an equation. | ||
the equation. | h + 2h − 16,000 = 110,000 | |
Combine like terms. | 3h − 16,000 = 110,000 | |
Add 16,000 to both sides and simplify. | 3h = 126,000 | |
Divide each side by 3. | h = 42,000 | |
$42,000 amount husband earns | ||
2h − 16,000 amount wife earns | ||
2(42,000) − 16,000 | ||
84,000 − 16,000 | ||
68,000 | ||
If the wife earns $68,000 and the husband earns $42,000 is the total $110,000? Yes! | ||
the question. | The husband earns $42,000 a year. |
According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?
U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?
Practice makes perfect.
Use the Approach Word Problems with a Positive Attitude
In the following exercises, prepare the lists described.
List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?
Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?
Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.
There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.
Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?
Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?
Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?
Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?
Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?
In the following exercises, solve each number word problem.
The sum of a number and eight is 12. Find the number.
The sum of a number and nine is 17. Find the number.
The difference of a number and 12 is three. Find the number.
The difference of a number and eight is four. Find the number.
The sum of three times a number and eight is 23. Find the number.
The sum of twice a number and six is 14. Find the number.
The difference of twice a number and seven is 17. Find the number.
The difference of four times a number and seven is 21. Find the number.
Three times the sum of a number and nine is 12. Find the number.
Six times the sum of a number and eight is 30. Find the number.
One number is six more than the other. Their sum is 42. Find the numbers.
One number is five more than the other. Their sum is 33. Find the numbers.
The sum of two numbers is 20. One number is four less than the other. Find the numbers.
The sum of two numbers is 27. One number is seven less than the other. Find the numbers.
The sum of two numbers is −45 . −45 . One number is nine more than the other. Find the numbers.
The sum of two numbers is −61 . −61 . One number is 35 more than the other. Find the numbers.
The sum of two numbers is −316 . −316 . One number is 94 less than the other. Find the numbers.
The sum of two numbers is −284 . −284 . One number is 62 less than the other. Find the numbers.
One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.
One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.
One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.
One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.
One number is one more than twice another. Their sum is −5 . −5 . Find the numbers.
One number is six more than five times another. Their sum is six. Find the numbers.
The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.
The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.
The sum of two consecutive integers is 77. Find the integers.
The sum of two consecutive integers is 89. Find the integers.
The sum of two consecutive integers is −23 . −23 . Find the integers.
The sum of two consecutive integers is −37 . −37 . Find the integers.
The sum of three consecutive integers is 78. Find the integers.
The sum of three consecutive integers is 60. Find the integers.
Find three consecutive integers whose sum is −3 . −3 .
Find three consecutive even integers whose sum is 258.
Find three consecutive even integers whose sum is 222.
Find three consecutive odd integers whose sum is 171.
Find three consecutive odd integers whose sum is 291.
Find three consecutive even integers whose sum is −36 . −36 .
Find three consecutive even integers whose sum is −84 . −84 .
Find three consecutive odd integers whose sum is −213 . −213 .
Find three consecutive odd integers whose sum is −267 . −267 .
Sale Price Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?
Sale Price Travis bought a pair of boots on sale for $25 off the original price. He paid $60 for the boots. What was the original price of the boots?
Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.
Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.
Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?
Price before Sales Tax Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?
What has been your past experience solving word problems?
When you start to solve a word problem, how do you decide what to let the variable represent?
What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.
What are consecutive even integers? Name three consecutive even integers between −50 −50 and −40 . −40 .
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.
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Adding integers practice problems with answers.
The ten (10) practice questions below focus on adding integers . I’m hoping that it will aid in developing your integer addition skills. You get better at something the more often you do it. Have fun!
Below is a quick summary for the rules of adding integers.
Problem 1: Add the integers: [latex]2 + 7[/latex]
[latex]9[/latex]
Explanation: The two integers are both positive that means they have the same sign. It implies that we should add their absolute values and copy the common sign which is positive.
Problem 2: Add the integers: [latex]\left( { – 13} \right) + 9[/latex]
[latex]-4[/latex]
Explanation: The two integers have different signs. That means we are going to subtract their absolute values. The integer [latex]-13[/latex] has a larger absolute value than [latex]9[/latex] which means the final answer will have a negative sign.
Problem 3: Add the integers: [latex]\left( { – 7} \right) + \left( { – 8} \right)[/latex]
[latex]-15[/latex]
Explanation: Since both of the integers are negative, they have the same sign. It suggests that we should sum their absolute values and take the common, negative sign.
Problem 4: Add the integers: [latex]23 + \left( { – 6} \right)[/latex]
[latex]17[/latex]
Explanation: The integers [latex]23[/latex] and [latex]-6[/latex] have different signs which implies that we are going to subtract their absolute values. The sign of the final answer will depend on the sign of the integer with the larger absolute value which in this case is positive coming from [latex]23[/latex].
Problem 5: Add the integers: [latex]\left( { – 15} \right) + 15[/latex]
[latex]0[/latex]
Explanation: Given that the signs of the integers [latex]15[/latex] and [latex]-15[/latex] differ, we must subtract their absolute values. Since the result after subtraction is [latex]0[/latex], the sign is neither positive nor negative.
Problem 6: Add the integers: [latex]27 + \left( { – 32} \right)[/latex]
[latex]-5[/latex]
Explanation: Because the signs of [latex]27[/latex] and [latex]-32[/latex] are not the same, we must subtract their absolute values.
The final answer will have a negative sign since the absolute value of [latex]-32[/latex] is greater than the absolute value of [latex]27[/latex], therefore [latex]27 + \left( { – 32} \right) = – 5[/latex].
Problem 7: Add the integers: [latex]\left( { – 1} \right) + \left( { – 2} \right) + \left( { – 3} \right)[/latex]
[latex]-6[/latex]
Explanation: In general, when adding more than two integers we do it two at a time. However, we can do it all at once because the signs of the integers are all the same which is negative. We simply add the absolute values of the integers then copy the common negative sign.
Problem 8: Add the integers: [latex]10 + \left( { – 16} \right) + 7[/latex]
[latex]1[/latex]
Explanation: There are more than two integers to add so we are going to add them two at a time.
[latex]10 + \left( { – 16} \right) + 7[/latex] [latex]\\[/latex]
[latex]\left[ {10 + \left( { – 16} \right)} \right] + 7[/latex] [latex]\\[/latex]
[latex]\left[ { – 6} \right] + 7[/latex] [latex]\\[/latex]
[latex]1[/latex] [latex]\checkmark[/latex]
Problem 9: Add the integers: [latex]\left( { – 7} \right) + \left( { – 2} \right) + 5[/latex]
Explanation: We will add the integers two at a time because there are more than two to add.
[latex]\left( { – 7} \right) + \left( { – 2} \right) + 5[/latex] [latex]\\[/latex]
[latex]\left[ {\left( { – 7} \right) + \left( { – 2} \right)} \right] + 5[/latex] [latex]\\[/latex]
[latex]\left[ { – 9} \right] + 5[/latex] [latex]\\[/latex]
[latex] – 4[/latex] [latex]\checkmark[/latex]
Problem 10: Add the integers: [latex]12 + \left( { – 9} \right) + 12 + \left( { – 13} \right)[/latex]
[latex]2[/latex]
Explanation: This time, we have to add four integers. Let’s apply everything we know. Once more, we’ll add two integers at a time. Be careful every step of the way.
[latex]12 + \left( { – 9} \right) + 12 + \left( { – 13} \right)[/latex] [latex]\\[/latex]
[latex]\left[ {12 + \left( { – 9} \right)} \right] + 12 + \left( { – 13} \right)[/latex] [latex]\\[/latex]
[latex]3 + 12 + \left( { – 13} \right)[/latex] [latex]\\[/latex]
[latex]\left[ {3 + 12} \right] + \left( { – 13} \right)[/latex] [latex]\\[/latex]
[latex]15 + \left( { – 13} \right)[/latex] [latex]\\[/latex]
[latex]2[/latex] [latex]\checkmark[/latex]
You might also like these tutorials:
x^{\msquare} | \log_{\msquare} | \sqrt{\square} | \nthroot[\msquare]{\square} | \le | \ge | \frac{\msquare}{\msquare} | \cdot | \div | x^{\circ} | \pi | |||||||||||
\left(\square\right)^{'} | \frac{d}{dx} | \frac{\partial}{\partial x} | \int | \int_{\msquare}^{\msquare} | \lim | \sum | \infty | \theta | (f\:\circ\:g) | f(x) |
▭\:\longdivision{▭} | \times \twostack{▭}{▭} | + \twostack{▭}{▭} | - \twostack{▭}{▭} | \left( | \right) | \times | \square\frac{\square}{\square} |
x^{\msquare} | \log_{\msquare} | \sqrt{\square} | \nthroot[\msquare]{\square} | \le | \ge | \frac{\msquare}{\msquare} | \cdot | \div | x^{\circ} | \pi | |||||||||||
\left(\square\right)^{'} | \frac{d}{dx} | \frac{\partial}{\partial x} | \int | \int_{\msquare}^{\msquare} | \lim | \sum | \infty | \theta | (f\:\circ\:g) | f(x) |
- \twostack{▭}{▭} | \lt | 7 | 8 | 9 | \div | AC |
+ \twostack{▭}{▭} | \gt | 4 | 5 | 6 | \times | \square\frac{\square}{\square} |
\times \twostack{▭}{▭} | \left( | 1 | 2 | 3 | - | x |
▭\:\longdivision{▭} | \right) | . | 0 | = | + | y |
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September 6, 2024
Adam Nickel for Quanta Magazine
Mathematics started with numbers — clear, concrete, intuitive. Over the last two centuries, however, it has become a far more abstract enterprise. One of the first major steps down this road was taken in the late 18th and early 19th centuries. It involved a field called group theory, and it changed math — theoretical and applied — as we know it.
Groups generalize essential properties of the whole numbers. They have transformed geometry, algebra and analysis, the mathematical study of smoothly changing functions. They’re used to encrypt messages and study the shapes of viruses . Physicists rely on them to unify the fundamental forces of nature: At high energies, group theory can be used to show that electromagnetism and the forces that hold atomic nuclei together and cause radioactivity are all manifestations of a single underlying force.
The term “group” in a mathematical context was coined in 1830 by Évariste Galois, a French prodigy, just 18 years old at the time. (Two years later, he would be killed in a duel, having already changed the course of mathematical history.) But he didn’t discover groups single-handed. “It’s not like a bunch of mathematicians got together one day and said, ‘Let’s create an abstract structure just for a laugh,’” said Sarah Hart, a group theorist at Gresham College in London. “It emerged gradually, over maybe 50 years in the 19th century, that these were the right rules to require. They give you the most flexibility and generality, while still allowing you to prove things.”
Évariste Galois helped lay the foundations for group theory as a teenager.
Public domain
A group is a set, or collection of objects, together with an operation that takes in two objects and outputs a third. Arguably the simplest example is the integers and the operation of addition. Groups must satisfy four rules.
To understand the significance of these four properties, it helps to consider a noteworthy omission. When adding two numbers together, you can change the order without affecting the outcome: 3 + 5 is the same as 5 + 3. This property is called commutativity. But there’s no requirement that groups be commutative. By making this property optional, mathematicians have been able to explore a rich variety of structures.
For an example of a noncommutative group, consider an equilateral triangle with labeled corners. If you rotate the triangle a third of the way around or flip it along its vertical axis, the only thing that will change about the image is the locations of the labels. There are six of these transformations that leave the shape otherwise unchanged, called symmetries of the triangle. They form a group called D 6 . (More generally, D 2 n is a group formed by the symmetries of a regular shape with n sides, so D 8 is the group of symmetries of a square.)
5W Infographics/Mark Belan for Quanta Magazine
To “add” two symmetries, just do one after another. You will quickly find that D 6 is not commutative: Flipping then rotating leaves the labels in different places than they will be if you rotate then flip.
D 6 is one of only two possible groups with six elements. For the other example of a six-element group, take the numbers {0, 1, 2, 3, 4, 5} as the set. For the operation, add two numbers in the usual way and then divide by 6, ignoring the quotient but retaining the remainder. So 3 and 5 deliver 2, since 8 leaves a remainder of 2 when divided by 6. This is called addition modulo 6, and the group is called Z 6 . In general, Z n is a group with n elements obtained from the numbers {0, 1, 2, 3, …, n − 1} together with addition modulo n . Unlike D 6 , Z 6 is commutative, because 3 + 5 = 5 + 3, and so forth.
Z 6 and D 6 have different structures. Not only is one commutative and the other not, but you can generate any element of Z 6 using just one of its elements, the number 1: Start with 1, then keep adding 1. In D 6 , no element has this property. Figuring out the possible structures of groups has been one of the central projects of algebra over the last century.
To do this, mathematicians try to identify smaller groups contained within a group, called subgroups. These must retain the operation used for the full group. For example, the even integers form a subgroup within the integers. An even integer plus an even integer always results in another even integer. On the other hand, the odd numbers are not a subgroup, since if you add two odd numbers, you’ll get an even number. The identity element always forms a subgroup by itself, called the trivial subgroup.
Figuring out what subgroups a group contains is one way to understand its structure. For example, the subgroups of Z 6 are {0}, {0, 2, 4} and {0, 3} — the trivial subgroup, the multiples of 2, and the multiples of 3. In the group D 6 , rotations form a subgroup, but reflections don’t. That’s because two reflections performed in sequence produce a rotation, not a reflection, just as adding two odd numbers results in an even one.
Certain types of subgroups called “normal” subgroups are especially helpful to mathematicians. In a commutative group, all subgroups are normal, but this isn’t always true more generally. These subgroups retain some of the most useful properties of commutativity, without forcing the entire group to be commutative. If a list of normal subgroups can be identified, groups can be broken up into components much the way integers can be broken up into products of primes. Groups that have no normal subgroups are called simple groups and cannot be broken down any further, just as prime numbers can’t be factored. The group Z n is simple only when n is prime — the multiples of 2 and 3, for instance, form normal subgroups in Z 6 .
However, simple groups are not always so simple. “It’s the biggest misnomer in mathematics,” Hart said. In 1892, the mathematician Otto Hölder proposed that researchers assemble a complete list of all possible finite simple groups. (Infinite groups such as the integers form their own field of study.)
It turns out that almost all finite simple groups either look like Z n (for prime values of n ) or fall into one of two other families. And there are 26 exceptions, called sporadic groups. Pinning them down, and showing that there are no other possibilities, took over a century.
The largest sporadic group, aptly called the monster group, was discovered in 1973. It has more than 8 × 10 54 elements and represents geometric rotations in a space with nearly 200,000 dimensions. “It’s just crazy that this thing could be found by humans,” Hart said.
By the 1980s, the bulk of the work Hölder had called for appeared to have been completed, but it was tough to show that there were no more sporadic groups lingering out there. The classification was further delayed when, in 1989, the community found gaps in one 800-page proof from the early 1980s. A new proof was finally published in 2004, finishing off the classification.
Many structures in modern math — rings, fields and vector spaces, for example — are created when more structure is added to groups. In rings, you can multiply as well as add and subtract; in fields, you can also divide. But underneath all of these more intricate structures is that same original group idea, with its four axioms. “The richness that’s possible within this structure, with these four rules, is mind-blowing,” Hart said.
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12. The temperature was -3o C last night. It is now -4o C. What was the change in temperature? 13. While watching a football game, Lin Chow decided to list yardage gained as positive integers and yardage lost as negative integers. After these plays, Lin recorded 14, -7, and 9.
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.
Problems with Solutions. Problem 1: Find two consecutive integers whose sum is equal 129. Solution to Problem 1: Let x and x + 1 (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 129 to write the equation. x + (x + 1) = 129. Solve for x to obtain. x = 64.
In the following exercises, evaluate. 35 − a when a = −4. (−2r) 2 when r = 3. 3m − 2n when m = 6, n = −8. −|−y| when y = 17. In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible. the difference of −7 and −4. the quotient of 25 and the sum of m and n.
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.
Challenge Exercises Integer Word Problems. Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Each answer should be given as a positive or ...
We use integers in our everyday life for counting money and measuring the speed of sound and light, height and weight of objects, temperature and pressure of the atmosphere, and depth of a sea. ... Solving word problems related to the above measurements will help us relate better to the concept with applications. A submarine was located 600 ...
Step 1: Assign variables: Let x = red marbles. Sentence: Initially, blue marbles = red marbles = x, then John took out 5 blue marbles. Step 2: Solve the equation. x = 2 (x -5) Answer: There are 10 red marbles in the bag. The following videos give more examples of integer word problems.
They will also solve open ended integer word problems (i.e., more than one correct answer is possible.) Problems involve both positive and negative integers. Students will then use variables to represent integers, and work up to translating a statement that is written in words to an expression that is written using integers.
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.
Solving a consecutive integer word problem How to translate and solve a word problem dealing with consecutive even integers. Example: Find three consecutive even integers such that 6 times the sum of the first and the third is 24 greater than 11 times the second.
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.
This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers. If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand.
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 ...
For example: 5 + (-1) = 4. 4. Use the commutative property when a is negative and b is positive. Do the addition as follows: -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. 5.
According to step #1, we have to change the subtraction sign to an addition sign. According to step #2, we have to take the opposite of 4, which is -4. Therefore the problem becomes: 3 + (-4) Using the rules for addition, the answer is -1. Here are a few other examples: Example 2: -2 - 8 = -2 + (-8) = -10.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7.
Practising such problems regularly could help improve mathematical problem-solving skills, critical thinking abilities and make students more comfortable with the concept of integers. Integrating real-world contexts in math problems makes learning more engaging and relatable. Learn more about Integer word problems. https://brainly.in/question ...
Solve. 1. Mrs. Bautista has a bank balance of -42 dollars at the start of the month. After she deposits 6 dollars, what is the new balance? $. 2. A hike starts at an elevation 30 meters below sea level and ends at a point 9500 meters higher than the starting point. How high would you be at the end of the hike?
We simply add the absolute values of the integers then copy the common negative sign. There are more than two integers to add so we are going to add them two at a time. We will add the integers two at a time because there are more than two to add. \left [ {\left ( { - 7} \right) + \left ( { - 2} \right)} \right] + 5.
Figure 1.3.1. The number line shows the location of positive and negative numbers. You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and − 2 are the same distance from zero, each one is called the opposite of the other.
Learn how to use integers to represent positive and negative numbers, and how to perform operations with them. This unit covers the concepts of addition, subtraction, multiplication, and division of integers, as well as the properties of these operations. You will also practice solving word problems involving integers and applying them to real-world situations.
Here's how to make the most of it: Begin by typing your algebraic expression into the above input field, or scanning the problem with your camera. After entering the equation, click the 'Go' button to generate instant solutions. The calculator provides detailed step-by-step solutions, aiding in understanding the underlying concepts.
A group is a set, or collection of objects, together with an operation that takes in two objects and outputs a third. Arguably the simplest example is the integers and the operation of addition. Groups must satisfy four rules. The first is called closure: Add any two integers and you'll get another integer.