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problem-solving

Definition of problem-solving

Examples of problem-solving in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'problem-solving.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

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“Problem-solving.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/problem-solving. Accessed 26 Apr. 2024.

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• problem-solving

adjective as in analytic

Strongest matches

analytical , investigative

Weak matches

inquiring , rational , sound , systematic

adjective as in analytical

analytic , cogent , detailed , diagnostic , interpretive , investigative , penetrating , rational , scientific , systematic , thorough

conclusive , discrete , dissecting , explanatory , expository , inquiring , inquisitive , judicious , logical , organized , perceptive , perspicuous , precise , questioning , ratiocinative , reasonably , searching , solid , sound , studious , subtle , testing , valid

adjective as in analytic/analytical

cogent , conclusive , detailed , diagnostic , discrete , dissecting , explanatory , expository , inquiring , inquisitive , interpretive , investigative , judicious , logical , organized , penetrating , perceptive , perspicuous , precise , questioning , ratiocinative , rational , reasonable , scientific , searching , solid , sound , studious , subtle , systematic , testing , thorough , valid , well-grounded

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Example sentences.

“These are problem-solving products but that incorporate technology in a really subtle, unobtrusive way,” she says.

And it is a “problem-solving populism” that marries the twin impulses of populism and progressivism.

“We want a Republican Party that returns to problem-solving mode,” he said.

Problem-solving entails accepting realities, splitting differences, and moving forward.

It teaches female factory workers technical and life skills, such as literacy, communication and problem-solving.

Problem solving with class discussion is absolutely essential, and should occupy at least one third of the entire time.

In teaching by the problem-solving method Professor Lancelot 22 makes use of three types of problems.

Sequential Problem Solving is written for those with a whole brain thinking style.

Thus problem solving involves both the physical world and the interpersonal world.

Sequential Problem Solving begins with the mechanics of learning and the role of memorization in learning.

Related Words

Words related to problem-solving are not direct synonyms, but are associated with the word problem-solving . Browse related words to learn more about word associations.

adjective as in logical

• investigative

adjective as in examining and determining

• explanatory
• inquisitive
• interpretive
• penetrating
• perspicuous
• questioning
• ratiocinative
• well-grounded

adjective as in examining

Viewing 5 / 11 related words

On this page you'll find 87 synonyms, antonyms, and words related to problem-solving, such as: analytical, investigative, inquiring, rational, sound, and systematic.

From Roget's 21st Century Thesaurus, Third Edition Copyright © 2013 by the Philip Lief Group.

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

• The Art of Effective Problem Solving: A Step-by-Step Guide
• Learn Lean Sigma
• Problem Solving

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Table of Contents

Problem solving methodologies.

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

• Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
• Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
• Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
• Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
• Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
• Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
• Prevent recurrence: Put in place measures to keep the problem from recurring.
• Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

• Determine the issue: Define the issue clearly, including its impact on the customer.
• Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
• Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
• Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

• What exactly is the issue?
• What are the problem’s symptoms or consequences?
• Who or what is impacted by the issue?
• When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

• What causes the problem?
• Why does it continue?
• Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

• Can the solution be implemented within the available resources, time, and budget?
• What are the possible barriers to implementing the solution?
• Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

• Novick, L.R. and Bassok, M., 2005.  Problem Solving . Cambridge University Press.

Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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Article • 7 min read

What Is Problem Solving?

By the Mind Tools Content Team

We all spend a lot of our time solving problems, both at work and in our personal lives.

Some problems are small, and we can quickly sort them out ourselves. But others are complex challenges that take collaboration, creativity, and a considerable amount of effort to solve.

At work, the types of problems we face depend largely on the organizations we're in and the jobs we do. A manager in a cleaning company, for example, might spend their day untangling staffing issues, resolving client complaints, and sorting out problems with equipment and supplies. An aircraft designer, on the other hand, might be grappling with a problem about aerodynamics, or trying to work out why a new safety feature isn't working. Meanwhile, a politician might be exploring solutions to racial injustice or climate change.

But whatever issues we face, there are some common ways to tackle them effectively. And we can all boost our confidence and ability to succeed by building a strong set of problem-solving skills.

Mind Tools offers a large collection of resources to help you do just that!

How Well Do You Solve Problems?

Start by taking an honest look at your existing skills. What's your current approach to solving problems, and how well is it working? Our quiz, How Good Is Your Problem Solving? lets you analyze your abilities, and signposts ways to address any areas of weakness.

Define Every Problem

The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem – not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the real issue might be a general lack of training, or an unreasonable workload across the team.

Tools like 5 Whys , Appreciation and Root Cause Analysis get you asking the right questions, and help you to work through the layers of a problem to uncover what's really going on.

However, defining a problem doesn't mean deciding how to solve it straightaway. It's important to look at the issue from a variety of perspectives. If you commit yourself too early, you can end up with a short-sighted solution. The CATWOE checklist provides a powerful reminder to look at many elements that may contribute to the problem, keeping you open to a variety of possible solutions.

Understanding Complexity

As you define your problem, you'll often discover just how complicated it is. There are likely several interrelated issues involved. That's why it's important to have ways to visualize, simplify and make sense of this tangled mess!

Affinity Diagrams are great for organizing many different pieces of information into common themes, and for understanding the relationships between them.

Another popular tool is the Cause-and-Effect Diagram . To generate viable solutions, you need a solid understanding of what's causing the problem.

When your problem occurs within a business process, creating a Flow Chart , Swim Lane Diagram or a Systems Diagram will help you to see how various activities and inputs fit together. This may well highlight a missing element or bottleneck that's causing your problem.

Quite often, what seems to be a single problem turns out to be a whole series of problems. The Drill Down technique prompts you to split your problem into smaller, more manageable parts.

General Problem-Solving Tools

When you understand the problem in front of you, you’re ready to start solving it. With your definition to guide you, you can generate several possible solutions, choose the best one, then put it into action. That's the four-step approach at the heart of good problem solving.

There are various problem-solving styles to use. For example:

• Constructive Controversy is a way of widening perspectives and energizing discussions.
• Inductive Reasoning makes the most of people’s experiences and know-how, and can speed up solution finding.
• Means-End Analysis can bring extra clarity to your thinking, and kick-start the process of implementing solutions.

Specific Problem-Solving Systems

Some particularly complicated or important problems call for a more comprehensive process. Again, Mind Tools has a range of approaches to try, including:

• Simplex , which involves an eight-stage process: problem finding, fact finding, defining the problem, idea finding, selecting and evaluating, planning, selling the idea, and acting. These steps build upon the basic, four-step process described above, and they create a cycle of problem finding and solving that will continually improve your organization.
• Appreciative Inquiry , which is a uniquely positive way of solving problems by examining what's working well in the areas surrounding them.
• Soft Systems Methodology , which takes you through four stages to uncover more details about what's creating your problem, and then define actions that will improve the situation.

Further Problem-Solving Strategies

Good problem solving requires a number of other skills – all of which are covered by Mind Tools.

For example, we have a large section of resources to improve your Creativity , so that you come up with a range of possible solutions.

By strengthening your Decision Making , you'll be better at evaluating the options, selecting the best ones, then choosing how to implement them.

And our Project Management collection has valuable advice for strengthening the whole problem-solving process. The resources there will help you to make effective changes – and then keep them working long term.

Problems are an inescapable part of life, both in and out of work. So we can all benefit from having strong problem-solving skills.

It's important to understand your current approach to problem solving, and to know where and how to improve.

Define every problem you encounter – and understand its complexity, rather than trying to solve it too soon.

There's a range of general problem-solving approaches, helping you to generate possible answers, choose the best ones, and then implement your solution.

Some complicated or serious problems require more specific problem-solving systems, especially when they relate to business processes.

By boosting your creativity, decision-making and project-management skills, you’ll become even better at solving all the problems you face.

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Creative Problem Solving

Finding Innovative Solutions to Challenges

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What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
• How to Resolve Employee Conflict at Work [Steps, Tips, Examples]
• How to Write Inspiring Core Values? 5 Steps with Examples
• 30 Employee Feedback Examples (Positive & Negative)

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Word Problems Involving the Mean of a Data Set

As we delve deeper into the world of statistics and probability, one specific term becomes very important to us: The mean. You've probably encountered this concept before -- even if you weren't aware of it. "Mean" essentially means the same thing as "average," and this number becomes very useful in a number of different fields. A baseball player's "batting average" tells you how skilled he or she is. The "average" return of an investment helps us understand how profitable it really is. As we can see, the mean is very useful in real-world applications. Let's tackle some word problems that show us how we can apply means to these real-world situations.

What is the mean?

We can find the mean of any data set with two easy steps:

• Add all the data points together
• Divide by the number of data points

Solving word problems involving means

Now let's put our knowledge to good use and solve a few word problems that involve means:

Consider the following data set: 4 , 8 , 20 , 25 , 32

If we add one more number ( x ) to this data set, we are left with a mean of 15. What is the value of this additional number?

First, let's try to add all of our numbers together: 4 + 8 + 20 + 25 + 32 + x

We don't know the value of x, so we can't complete this addition. But let's continue with the operation anyway:

If we add one more value (x) to our current data set, we get 6 total data points. This means that we need to divide the above sum by six:

4 + 8 + 20 + 25 + 32 + x + 6

We know that the end result is a mean of 15. So how can we solve this equation? Easy: We simplify it.

4 + 8 + 20 + 25 + 32 + x + 6 = 15

89 + x / 6 = 15

Now, all we need to do is multiply each side of the equation by 6:

6 × ( 89 + x ) / ( 6 ) = 15 × 6

89 + x = 90

The answer is 1.

We can even use our knowledge of means to help us set higher goals!

What happens if we know that we need an average of 90% or higher to get an "A" in math class?

What if our past test scores were 80%, 85%, 88%, and 95%?

We have one more test for the term. What score do we need to get an "A" average?

Right away, we know that our end result needs to be at least 90.

We also know that the total number of tests is 5.

With these values, we can find our minimum test score for an A average (x).

80 + 85 + 88 + 95 + x / 5

We can simplify this as 348 + x / 5

Now, all we need to do is multiply each side of the equation by 5:

5 × ( 348 + x ) / ( 5 ) = 90 × 5

Or: 348 + x = 450

Uh oh, looks like that value is 102. As we know, the highest possible score on a test is 100%. This means that it's theoretically impossible for us to get an A average this term -- unless, of course, we speak to our teacher about the possibility of doing additional work for extra credit. This highlights the importance of calculating means in order to plan effectively for the future.

Exercises such as this also show us how even a few low scores can seriously affect the overall average or mean.

The especially low score of 80% is 10 percentage points below our target grade, and this skews the data toward a lower mean. We can consider this value an outlier.

Topics related to the Word Problems Involving the Mean of a Data Set

Center and Variation of Data

Word Problems

Interpreting Data

Flashcards covering the Word Problems Involving the Mean of a Data Set

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Common Core: High School - Statistics and Probability Flashcards

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Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

Tutoring helps your student gain confidence when solving word problems involving means

Tutoring gives your student the opportunity to go over a few more examples of word problems involving means. Sometimes, all it takes to reach for confidence in these concepts is a little extra time outside of class. Tutors can even cater to your student's learning style during these sessions, whether they prefer verbal or visual cues. Speak with Varsity Tutors' Educational Directors today to learn more about the possibilities of tutoring. We'll match your student with a suitable tutor.

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What Is Problem Solving?

You will often see beach clean-up drives being publicized in coastal cities. There are already dustbins available on the beaches,…

You will often see beach clean-up drives being publicized in coastal cities. There are already dustbins available on the beaches, so why do people need to organize these drives? It’s evident that despite advertising and posting anti-littering messages, some of us don’t follow the rules.

Temporary food stalls and shops make it even more difficult to keep the beaches clean. Since people can’t ask the shopkeepers to relocate or prevent every single person from littering, the clean-up drive is needed.  This is an ideal example of problem-solving psychology in humans. ( 230-fifth.com ) So, what is problem-solving? Let’s find out.

What Is Problem-Solving?

At its simplest, the meaning of problem-solving is the process of defining a problem, determining its cause, and implementing a solution. The definition of problem-solving is rooted in the fact that as humans, we exert control over our environment through solutions. We move forward in life when we solve problems and make decisions. 

We can better define the problem-solving process through a series of important steps.

Identify The Problem: 

This step isn’t as simple as it sounds. Most times, we mistakenly identify the consequences of a problem rather than the problem itself. It’s important that we’re careful to identify the actual problem and not just its symptoms. 

Define The Problem: 

Once the problem has been identified correctly, you should define it. This step can help clarify what needs to be addressed and for what purpose.

Form A Strategy: 

Develop a strategy to solve your problem. Defining an approach will provide direction and clarity on the next steps. 

Organize The Information:  

Organizing information systematically will help you determine whether something is missing. The more information you have, the easier it’ll become for you to arrive at a solution.  

Allocate Resources:  

We may not always be armed with the necessary resources to solve a problem. Before you commit to implementing a solution for a problem, you should determine the availability of different resources—money, time and other costs.

Track Progress: 

The true meaning of problem-solving is to work towards an objective. If you measure your progress, you can evaluate whether you’re on track. You could revise your strategies if you don’t notice the desired level of progress. 

Evaluate The Results:  

After you spot a solution, evaluate the results to determine whether it’s the best possible solution. For example, you can evaluate the success of a fitness routine after several weeks of exercise.

Meaning Of Problem-Solving Skill

Now that we’ve established the definition of problem-solving psychology in humans, let’s look at how we utilize our problem-solving skills.  These skills help you determine the source of a problem and how to effectively determine the solution. Problem-solving skills aren’t innate and can be mastered over time. Here are some important skills that are beneficial for finding solutions.

Communication

Communication is a critical skill when you have to work in teams.  If you and your colleagues have to work on a project together, you’ll have to collaborate with each other. In case of differences of opinion, you should be able to listen attentively and respond respectfully in order to successfully arrive at a solution.

As a problem-solver, you need to be able to research and identify underlying causes. You should never treat a problem lightly. In-depth study is imperative because often people identify only the symptoms and not the actual problem.

Once you have researched and identified the factors causing a problem, start working towards developing solutions. Your analytical skills can help you differentiate between effective and ineffective solutions.

Decision-Making

You’ll have to make a decision after you’ve identified the source and methods of solving a problem. If you’ve done your research and applied your analytical skills effectively, it’ll become easier for you to take a call or a decision.

Organizations really value decisive problem-solvers. Harappa Education’s   Defining Problems course will guide you on the path to developing a problem-solving mindset. Learn how to identify the different types of problems using the Types of Problems framework. Additionally, the SMART framework, which is a five-point tool, will teach you to create specific and actionable objectives to address problem statements and arrive at solutions. 

Explore topics & skills such as Problem Solving Skills , PICK Chart , How to Solve Problems & Barriers to Problem Solving from our Harappa Diaries blog section and develop your skills.

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Word Problems on Arithmetic Mean

Here we will learn to solve the three important types of word problems on arithmetic mean (average). The questions are mainly based on average (arithmetic mean), weighted average and average speed.

How to solve average (arithmetic mean) word problems?

To solve various problems we need to follow the uses of the formula for calculating average (arithmetic mean)

Average = (Sums of the observations)/(Number of observations)

Follow the explanation to solve the word problems on arithmetic mean (average):

1.  The heights of five runners are 160 cm, 137 cm, 149 cm, 153 cm and 161 cm respectively. Find the mean height per runner.

Mean height = Sum of the heights of the runners/number of runners

= (160 + 137 + 149 + 153 + 161)/5 cm

Hence, the mean height is 152 cm.

2. Find the mean of the first five prime numbers.

The first five prime numbers are 2, 3, 5, 7 and 11.

Mean = Sum of the first five prime numbers/number of prime numbers

= (2 + 3 + 5 + 7 + 11)/5

Hence, their mean is 5.6

3. Find the mean of the first six multiples of 4.

The first six multiples of 4 are 4, 8, 12, 16, 20 and 24.

Mean = Sum of the first six multiples of 4/number of multiples

= (4 + 8 + 12 + 16 + 20 + 24)/6

Hence, their mean is 14.

4. Find the arithmetic mean of the first 7 natural numbers.

The first 7 natural numbers are 1, 2, 3, 4, 5, 6 and 7.

Hence, their mean is 4.

5.  If the mean of 9, 8, 10, x, 12 is 15, find the value of x.

Mean of the given numbers = (9 + 8 + 10 + x + 12)/5 = (39 + x)/5

According to the problem, mean = 15 (given).

Therefore, (39 + x)/5 = 15

⇒ 39 + x = 15 × 5

⇒ 39 + x = 75

⇒ 39 - 39 + x = 75 - 39

Hence, x = 36.

More examples on the worked-out word problems on arithmetic mean:

6. If the mean of five observations x, x + 4, x + 6, x + 8 and x + 12 is 16, find the value of x.

Solution: Mean of the given observations

= x + (x + 4) + (x + 6) + (x + 8) + (x + 12)/5

= (5x + 30)/5

According to the problem, mean = 16 (given).

Therefore, (5x + 30)/5 = 16

⇒ 5x + 30 = 16 × 5

⇒ 5x + 30 = 80

⇒ 5x + 30 - 30 = 80 - 30

Hence, x = 10.

148 + 153 + 146 + 147 + 154

7. The mean of 40 numbers was found to be 38. Later on, it was detected that a number 56 was misread as 36. Find the correct mean of given numbers.

Calculated mean of 40 numbers = 38.

Therefore, calculated sum of these numbers = (38 × 40) = 1520.

Correct sum of these numbers

= [1520 - (wrong item) + (correct item)]

= (1520 - 36 + 56)

Therefore, the correct mean = 1540/40 = 38.5.

8. The mean of the heights of 6 boys is 152 cm. If the individual heights of five of them are 151 cm, 153 cm, 155 cm, 149 cm and 154 cm, find the height of the sixth boy.

Mean height of 6 boys = 152 cm.

Sum of the heights of 6 boys = (152 × 6) = 912 cm

Sum of the heights of 5 boys = (151 + 153 + 155 + 149 + 154) cm = 762 cm.

Height of the sixth boy

= (sum of the heights of 6 boys) - (sum of the heights of 5 boys)

= (912 - 762) cm = 150 cm.

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Word Problems: When to Use Mean, Median, and Mode - Expii

Word problems: when to use mean, median, and mode, explanations (4).

(Video) Word Problems: When to Use Mean, Median, and Mode

by Anusha Rahman

This video by Anusha Rahman goes over three examples of when to use mean, median, and mode in word problems.

Example 1: Jessie’s math teacher said that their final class grade is based on the average of all exam grades. Jessie has gotten a 93,87,71, and 97 on her math exams. What measure of central tendency will she use to calculate the average? What is the average?

We know that when we see the word average , we're talking about the mean ! So, we will be using mean to solve this problem.

Mean=93+87+71+974 =3484 =87

So, we know that Jessie's average score on her math exams was 87!

Example 2: Meghan wants to get an accurate picture of household salary in her neighborhood. The salaries for people in her neighborhood are: 87,000;94,000;103,000;97,000; and 55,000. What measure of central tendency will most accurate describe the data?

At first glance, we can see that there is an outlier at 55,000. At outlier is going to mess with the mean, and mode (the frequency) is not going to give us a lot of information. So, we will use the median, or the middle value.

Step 1: Put the salaries in numerical order, from least to greatest.

55,000,87,000,94,000,97,000,103,000

Step 2: Cross of the numbers evenly from each side until you're left with a middle number.

In this case, the median is 94,000.

Example 3: At her birthday party, Charlotte asks everyone to go around and say their favorite Taylor Swift song. In order, people respond with:

All Too Well, Shake it Off, Black Space, New Romantics, All Too Well, New Romantics, All Too Well, and Mr Perfectly Fine

Charlotte wants to know which song is the most popular. What measure of central tendency should she use?

Because the question is asking for the most popular song, we know we are looking for the mode ! We also know that the mean and median cannot be used for nominal or qualitative data.

Our first step is to make a frequency table with the data.

SongFrequencyAll Too Well3Shake It Off1Blank Space1New Romantics2Mr. Perfectly Fine1

From this frequency table, we can see that All Too Well has the highest frequency, so that is the mode of the data set, and it is the most popular song!

When solving word problems with central tendency, it's important to understand the pros and cons of each type . Let's break them down.

PRO: asymmetrical data (with an outlier)

CON: doesn't work with qualitative or nominal data

PRO: symmetrical data (no outlier)

PRO: big data sets

CON: highly affected by outliers

PRO: working with qualitative or nominal data

PRO: very easy to look at and read graphs/charts

CON: very limited in what it tells you

Related Lessons

(video) understanding and calculating measures of central tendency.

by Nurse Killam

This video by Nurse Killam explains when to use mean, median, and mode.

She starts with the definition of a normal distribution . All of the measures of central tendency work best with a normal distribution.

The mean is the average of a data set. We find the mean by adding all of the values, and dividing the sum by the number of values. The mean gets more accurate with more values.

Mean Benefits

• Best measure for symmetrical distributions
• Influenced by all data
• Most reliable
• Good for interval and ratio data

Mean Limitations

• Works best with no outliers

The median is the middle value of a set. We find this by ordering the data, then finding the literal middle number. Even sized sets will have two numbers that share the middle. The median of these sets is the average of those two numbers.

Median Benefits

• Good for asymmetrical data
• Works for ordinal, interval, and ratio data

Median Limitations

• Does not account for extreme scores
• Not algebraically defined
• Not appropriate for nominal data

The mode is the value that occurs the most in a set. We find this by ordering the set and finding the number that repeats the most.

Mode Benefits

• The only appropriate measure for nominal data
• Can also be used for ordinal, interval, and ratio data

Mode Limitations

• Cannot be used if all scores are different
• There can be several modes
• Cannot be used for further calculations

Using the Measures of Central Tendency: Media, Median, and Mode

When you're faced with word problems where you're asking to look into the distribution of a dataset, it can feel intimidating to figure out whether to use mean, median, or mode. The good news is that because each measure is different, they each have their own advantages and disadvantages!

Image source: by Anusha Rahman

Let's go through some example problems.

Which measure of central tendency is the most effective way to analyze the following dataset? 4,7,9,10,11,18,2,20,15,350,14

When to Use Mean, Median, and Mode

Three extremely important measures of central tendency (part of statistics ) are mean , median , and mode . You can find the mean, median, and mode of most data sets. Sometimes, it is more advantageous to only focus on one or two of these measures of central tendency, depending on your data set. Let's find out when we should use the mean, median, or mode to accurately describe a data set.

The mean is the average value in a data set. We can calculate the mean using the following formula:

mean=sum of all valuestotal number of values

Advantages:

The mean is the best measure for symmetrical data distributions. It is affected by all of the data points, and the more data points you know, the more accurate the mean will be. The mean is helpful when working with interval and ratio data.

Limitations:

Outliers can really throw off the mean. For example, say we have the data set {1,4,5,6,6,7,8,8,9}. The average of this data set is 1+4+5+6+6+7+8+8+99=549=6. But, what if we had an outlier in the set? What if our new set was {1,4,5,6,6,7,8,8,9,4946}? When our new mean would be 1+4+5+6+6+7+8+8+9+494610=500010=500. See how much an outlier can throw off the mean?

The median is the value separating the first half of the data set with the second half. To find the median of a data set, first put the data set in order. Then, if there are an odd number of values, cross an equal amount of values off from the beginning of the list and the end of the list. The remaining data point in the middle of your list is the median! If there are an even number of values in your set, the median will be the average of the two middle values.

The median is helpful when describing asymmetrical data , and it also works for ordinal, ratio, or interval data. It is easy to find, especially if your data is in order. Also, it is not usually too affected by outliers.

Limitations

However, the fact that the median is not usually affected by outliers means that we can't use it to account for these extreme values. Also, the median is not helpful when we are working with nominal or qualitative data. Think about it ... how can you order a list of colors and find the middle color?

The mode of a data set is the value that occurs most frequently. We can find the mode by putting the data in order, and then identifying which value(s) repeat the most. Frequency tables , histograms , and frequency distributions can be helpful in identifying the mode.

Finding the mode can be especially helpful when we are working with qualitative data or nominal data , but it can also be used when we have ordinal, ratio, or interval data. It's easy to spot on a graph, frequency table, or even an ordered set of data. It is not usually affected by outliers.

There can be many modes, so finding the mode may not say much about how the data is distributed. On the other hand, if all the data points only occur once, there is no real mode. The mode is unstable, and can be easily influenced by certain changes in the data. Adding just one new data point can change the mode!

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Definition of problem adjective from the Oxford Advanced Learner's Dictionary

• She was a problem child , always in trouble with the police.
• a new approach to problem drinking
• One out of every five people is a problem drinker.

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By New York Times Games

Welcome to The Wordle Review. Be warned: This page contains spoilers for today’s puzzle. Solve Wordle first , or scroll at your own risk.

Note the date before you comment. To avoid spoiling the game for others, make sure you are posting a comment about Wordle 1,039.

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The difficulty of each puzzle is determined by averaging the number of guesses provided by a small panel of testers who are paid to solve each puzzle in advance to help us catch any issues and inconsistencies.

Today’s average difficulty is 6 guesses out of 6, or very challenging.

For more in-depth analysis, visit our friend, WordleBot .

Today’s word is ROVER, a noun. According to Webster’s New World College Dictionary, it refers to a wandering someone or something.

Our Featured Artist

Grace J. Kim is a Korean-Canadian illustrator based in New York. Her drawings depict characters in everyday moments and situations related to current events. Ms. Kim always adds a serene and utopian touch, hoping that she can share moments of peacefulness with her audience. She has collaborated with various clients, among them Apple, The New Yorker, The Metropolitan Museum of Art, The Washington Post, Bloomberg Business and Bloomsbury.

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An earlier version of this article misstated today’s Wordle answer. Today’s word is ROVER, not LOVER.

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Posted Apr 25, 2024

At 5:00 PM UTC

Our latest episode of Decoder is about the brand-new TikTok ban — and how years of congressional inaction on a federal privacy law helped lead us to this moment of apparent national panic about algorithmic social media.

This is a thorny discussion, and to help break it all down, I invited Verge senior policy reporter Lauren Feiner on the show. Lauren has been closely covering efforts to ban TikTok for years now, and she’s also watched Congress fail to pass meaningful privacy regulation for even longer. We’ll go over how we got here, what this means for both TikTok and efforts to pass new privacy legislation, and what might happen next. 

This self-transforming Megatron is as badass as it is expensive

What happens after your country runs on 99 percent renewable electricity, blizzcon 2024 has been canceled, microsoft’s surface and xbox hardware revenues take a big hit in q3, the fossil-size hole in wear os, more from this stream tiktok ban: all the news on attempts to ban the video platform, anyone want to buy tiktok, biden signs tiktok ‘ban’ bill into law, starting the clock for bytedance to divest it, senate passes tiktok ban bill, sending it to president biden’s desk, the great conundrum of campaigning on tiktok.