1 step problem solving

Solving One-Step Equations: Explanations, Review, and Examples

• The Albert Team
• Last Updated On: March 1, 2022

One of the basic skills learned in Algebra 1 is solving one-step equations.

An equation is a mathematical sentence that shows two expressions are equal. In this article, we will focus on how to solve one-step equations including examples with all operations, working with fractions or integers, and one-step equation word problems. Let’s dive in!

What We Review

What is a one-step equation?

A one-step equation is an equation that only requires one step to solve! You can solve a one-step equation with addition, subtraction, multiplication, or division. 

Examples of one-step equations

Below are four simple examples of one-step equations:

Notice how each of the four examples above has only one operation on the left side of the equation sign.

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What does it mean to “solve” an equation?

In order to solve an equation, one must find a value of the variable that makes an equation true, or a solution . To find a solution for a one-step equation, you will use inverse operations .

Examples of inverse operations are: 

When working with inverse operations, it is important to remember that whatever you do to one side of the equation, you must also do to the other.

Solving one-step equations (basics)

Solve one-step equation with addition.

Solve for x in the following equation:

To find the solution for this equation, we must first get x by itself on the left side. Since x has 8 added to it, we must use the inverse operation of addition, subtraction . So we will subtract 8 from each side. 

Original equation

Subtract 8 from both sides

To check your answer, you can simply substitute 6 into the variable to see if the equation is true: 

Thus, x = 6 is the correct solution. 

Solve one-step equation with subtraction

Solve for y in for the following equation:

Again, we will use the inverse operation to get y by itself and solve the equation. Remember, the inverse operation of subtraction is addition . Therefore to solve, we will add 12 to both sides. 

Add 12 to both sides

To check your solution, simply substitute 19 into y :

Therefore, y = 19 is the correct solution.

Solve one-step equation with multiplication

Solve for m in the following equation:

Since 4m implies “Four times m ”, we will have to use the inverse operation of multiplication, which is division . Therefore to solve, we will simply divide each side by 4 . 

To check you answer, simply substitute 5 into m .

Therefore, m = 5 is the correct solution.

Solve one-step equation with division

Solve for z in the following equation: 

Since \dfrac{z}{3} implies “ z divided by three”, we will use the inverse operation of division, multiplication . Therefore, to solve for z , we will multiply each side by 3 .

To check you answer, simply substitute 30 into z :

Therefore, z = 30 is the correct solution.

For some more examples, check out this YouTube video from “Math with Mr. J”:

Solving one-step equations with fractions

When a one-step equation involves fractions, there are two ways that we can solve the equation. The first method treats the fraction the same as our division example above. For instance, we will solve for x in the following equation: 

Since we can see \frac{1}{5}x as the same as “ x divided by five”, we will simply use the inverse operation of division, multiplication. We can multiply each side by 5 . 

To check your answer, simply substitute 30 into x :

Therefore, x = 30 is a correct solution. 

But, what if the fraction has a numerator that is not 1 ?

Great question 🙂 We can still solve the equation in one step! When solving an equation, our goal is to isolate the variable (meaning get the variable by itself). This implies that we want the coefficient in front of the variable to be 1 . In order to do this in one step, we multiply both sides of the equation by the reciprocal of the fraction. (A reciprocal of a fraction flips the numerator and denominator of a fraction.) 

Let’s see an example: solve for t in the following equation: 

As noted above, to solve for t we will multiply both sides of the equation by the reciprocal of \dfrac{2}{3} , which is \dfrac{3}{2} :

To check your answer, simply substitute 12 into t s

Therefore, t = 12 is the correct solution.

Solving one-step equations with integers

Sometimes, a one step equation will contain an integer value on one or both sides. (Remember, an integer can be positive, negative, or zero without any fractional part). In this situation, we can use the same techniques for solving one-step equations!

Solve for d in the following equation: 

Since an equation is not solved until the variable is by itself, if we simply divide by 8 , we would still end up having -d on the left-hand side. Therefore, to solve this equation we must divide both sides by -8 as shown below

To check your answer, simply substitute -10 into d :

Therefore, d = -10 is the correct solution.

What about addition or subtraction problems with integer values? For example, solve for w in the following equation:

Since our ultimate goal is to get w by itself, we need to eliminate the -5 . To eliminate this integer value, we will add 5 to each side, as shown below:

Therefore, w = 10 is the correct solution.

Solving one-step equation word problems

We can model many real-life applications with a one-step equation. Once we have created an equation based on the information in the word problem, we simply solve the equation as we have above. 

For example, model the following situation with an equation and find a solution that makes the situation true.

Word Problem Example 1

To solve for c , we will do the inverse operation of addition and subtract 150 from each side:

To check you answer, simply plug 15 into c :

Therefore, c = 15 \text{ lbs.} is the correct solution.

Word Problem Example 2

Solution : Let t = \text{ the total amount of tickets won from the jackpot} . SInce we know the three friends split the jackpot evenly, we can model the situation with the equation: 

Then, since \dfrac{t}{3} represents “ t \text{ divided by } 3 ”, we will use the inverse operation, multiplication, to solve.

To check your answer, simply substitute 600 in for t :

Therefore, t = 600 \text{ tickets} is the correct solution.

Solving One-Step Equations: Keys to Remember

• A one-step equation is an equation that requires one step to solve
• To solve, use the inverse operation to isolate the variable by itself
• Remember whatever you do to one side, you must do to the other
• To check the solution, simply substitute the value into the variable to see if the equation is true
• When solving with negative integers, remember to eliminate the negative value when doing the inverse operation.

Read these  other helpful posts:

• Solving Two-Step Equations
• Solving Multi-Step Equations
• Forms of Linear Equations
• View ALL Algebra 1 Review Guides

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Solving One-Step Equations

Solving by subtracting.

Solving a one-step equation:

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Solving By Dividing

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Solving One-step Equations

What are one-step equations, rules for one-step algebraic equations, how to solve one-step equations, solved examples on one-step equations, practice problems on one-step equations, frequently asked questions on one-step equations.

One-step equations are algebraic equations that can be solved in one step only. They can be solved in just one step by isolating the variable using the inverse operations. An equation is a mathematical statement that shows that two mathematical expressions are equal. 

The most basic and simple algebraic equations consist of one or more variables in math. 

An equation consists of two sides, L.H.S. (Left Hand Side) and R.H.S. (Right hand Side) separated by “=” sign. Solving an equation simply means finding the value of the unknown variable.

Consider an equation. 

$x\;-\;7 = 93$

Let’s add 7 on both sides to cancel out 7 and isolate the variable.  

$x\;-\;7 + 7 = 93 + 7$

Thus, $x = 100$

Wasn’t that simple? This is what we call a “one-step equation.”

Note: Take note of inverse operations that help in solving the one-step equations.

Definition of One-step Equations

One-step equations are simple algebraic equations that can be solved in just one step.  

To solve one-step equations, we determine the value of the variable involved using different properties of equality. The operation you perform on one side of the equation must be carried out on the other side as well, in order to keep the equation balanced. One-step equations may involve integers, fractions, or decimals. 

While keeping the equation balanced on both sides, the subsequent operations are carried out to isolate the variable. By doing this, LHS continues to be equal to RHS, and eventually, the balance is maintained. There are some rules for one-step algebraic equations. They are as follows:

• One-step Equation with Addition 

If the equation involves addition, we can isolate the variable using the inverse operation, which is the subtraction operation.

If we subtract the same number from both sides of an equation, both sides will remain equal. This is the subtraction property of equality.

For example, 

$x + 5 = 2$

$\Rightarrow x + 5\;-\;5 = 2\;-\;5$ …subtract 5 from both sides

$\Rightarrow x =\;-\;3$

• One-step Equation with Subtraction

If the equation involves subtraction, we isolate the variable using the addition property of equality. If we add the same number to both sides of an equation, both sides will remain equal.

$c\;-\;1.1 = 12$

$\Rightarrow c\;-\;1.1 + 1.1 = 12 + 1.1$ …adding 1.1 on both sides

$\Rightarrow c = 13.1$

• One-step Equation with Multiplication

If the equation involves multiplication, we can isolate the variable using division. However, note that we cannot divide by 0.

If we divide both sides of an equation by the same number, both sides will remain equal.

$\frac{4x}{4}  = \frac{16}{4}$ …divide both sides by 4

$\Rightarrow x = 4$

• One-step Equation with Division

If the equation involves division, we use multiplication to solve the equation. 

If we multiply both sides of an equation by the same number, both sides will remain equal.

$\frac{x}{4} = \frac{1}{5}$ 

$\frac{x}{4} \times 4 = \frac{1}{5}\times4$ … multiply both sides by 4

$x = \frac{4}{5}$

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When you’re solving one-step algebraic equations, the goal is to get the variable on one side and all the remaining numbers on the other. We use the inverse operation or the opposite operation to the operation acting on the variable. The inverse operations pairwise are addition and subtraction and division and multiplication.

The steps to solve one-step equations are as follows:

Step 1: Note down the equation.

For example $x\;-\;5 = 10$

An equation will have a variable (x), which denotes an unknown value, and it will also have a constant, which is a number you need to add or subtract from the variable to equal a certain sum or difference.

Step 2: Isolate the variable.

You must conduct an inverse operation to cancel the constants in order to isolate a variable, on one side of the equation.

For example, in the equation, $x \;-\; 5 = 10 , 5$  is subtracted from the variable, so to isolate the variable you must cancel the 5 by adding it on both sides.

Step 3: Adding or subtracting the constant from both sides.

While solving the equation, we have to keep both sides balanced.

So, if you need to add a value to isolate the variable, you must also add that same value to the other side of the equation, also if you need to subtract a value on one side of the equation you have to subtract it from the other side.

In the equation, $x \;-\; 5 = 10$, we need to add 5 on the left as well as on the right.

$x \;-\; 5 + 5 = 10 + 5$

Thus, $x = 15$

Step 4: Verify your solution.

To verify the solution, simply plug in the value of x in the equation.

$x \;-\; 5 = 10$, the value of x as we found out is 15. 

Thus, substituting the value, the equation becomes, $15 \;-\; 5 = 10$. 

Since this equation is true, your solution is correct.

Similarly, if the variable is multiplied by a number, divide both sides of the equation by the same number to isolate the variable. (If the variable is divided by a number, multiply both sides of the equation by the same number.) 

For example, let’s solve an equation.  $\frac{x}{6} = 12$

As the variable is divided by 6, to isolate it, you need to multiply by 6.

$6(\frac{x}{6}) = 12\times6$

$\Rightarrow x   = 72$

Verifying the solution, as  $\frac{72}{6}  = 12$ , the solution is correct.

Facts about One-step Equations

• A one-step equation is an equation that requires only one step to solve.
• The most common one-step equations are linear algebraic equations.
• When solving an equation, you may keep the variable on either side of the equation. As long as in the end, the variable that you are solving is isolated on one side with a coefficient of $+1$.

In this article, we have learned how to solve one-step equations, its rules and the steps of solving the equations. Let’s now solve some examples to better understand the concept.

1. Solve this equation with a negative constant: $\;-\;8 + x = 14$

Solution: Since the constant is negative, adding it to both sides will isolate the variable.

$\Rightarrow -8 + x = 14$

$\Rightarrow -8 + x + 8 = 14 + 8$

Verifying the solution, $-8 + 22 = 14$. The solution is correct.

2. Solve this equation with a negative coefficient:  $-4x = 32$

Solution:  

Since the variable is multiplied by $-4$, to isolate the variable, you must divide each side by $-4$. 

Remember that dividing a positive number by a negative number equals a negative quotient.

 $-4x = 32$

$\frac{-4x}{-4} = \frac{32}{-4}$

$\Rightarrow x = \;-\; 8$

Verifying the solution, $-4(\;-8) = 32$. The solution is correct.

3. Jennifer weighed herself on the scale and found her weight to be 120 lbs. Then, she held the cat and stepped on the scale and found the combined weight to be 132 lbs.

Create an equation that models the situation and solve the equation to find c, the cat’s weight.

Solution: To model the following situation, we will create an equation to show the combined           weights of Jennifer and cat.        

 $c + 120 = 132$

To solve for c, we will do the inverse operation of addition and subtract 120 from each side:

$c + 120 \;-\; 120 = 132 \;-\; 120$

$\Rightarrow c = 12$ lbs.

To verify answer we will plug in the values in the equation

 $12 + 120 = 132$

Therefore, $c = 12$ lbs is the correct solution.

4. Solve the equation: $10 = \frac{p}{10}$

Solution: Since the variable is divided by 10, we have to isolate it by multiplying it by 10.

 $10 = \frac{p}{10}$

$\Rightarrow 10\times10 = \frac{p}{10}\times10$

$\Rightarrow P = 100$

Verifying the solution, $10 = \frac{100}{10}$ . The solution is correct.

Attend this quiz & Test your knowledge.

What is the value of x in the equation $5x = 40$?

Identify the equation, where the value of $z = 7$., identify the equation that does not have a solution at $w = 5$., what equation matches this situation maria had some pencils then bought 4 more. now she has a total of 10 pencils., what is the first step to solve this equation: $2x = 18$..

What are the various methods for solving equations?

There are three methods used to solve systems of equations. They are as follows:

•   Graphing
•   Substitution
•   Elimination

What is the difference between one-step equations and two-step equations?

One-step equations require only one step, one inverse operation to be solved and have only one operation. Two-step equations require two inverse operations to solve and have two operations.For eg., $x^2 = 64$ is a one-step equation, while $3x + 2 = 14$ is a two-step equation.

What are linear equations in one variable?

A linear equation in one variable is an equation in which the degree of the variable is 1. These are also known as first degree equations, because the highest exponent on the variable is 1. All linear equations eventually can be written in the form $ax + b = c$, where a, b, and c are real numbers and $a \neq 0$.

What is a one-solution equation?

Some equations have exactly one solution. In these equations, there is only one value for the variable that makes the equation true. You can tell that an equation has one solution if you solve the equation and get a variable equal to a number.

For eg., $5x = 25$ has a variable term, $5x$ on one side of the equation and a constant term, 25, on the other side of the equation. So, it has one solution. Let’s solve the equation to see why.

 $5x = 25$  . Divide both sides by 5.

  $x = 5$

So, $5x = 25$  has one solution, $x = 5$.

What are equations with no solutions?

If a linear equation has the same variable term but different constant values on opposite sides of the equation, it has no solutions.

For eg., $2x + 4 = 1 + 2x$ has the same variable term, $2x$, but different constant terms, 4 and 1, on opposite sides of the equation. So, it has no solutions. Let’s solve the equation to see why.

$2x + 4 = 1 + 2x$     

Subtract 2x from both sides.

$4 = 1$The statement $4 = 1$ is false. So, $2x + 4 = 1 + 2x$ has no solutions.

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How to Solve Word Problems Involving the One-Step Equation

In this article, we will guide you through the process of understanding and solving word problems that involve one-step equations.

A step-by-step guide to solving word problems involving the one-step equation

One-step equations are equations that can be solved in a single step. They are equations of the form ax = b or x/a = b, where a and b are numbers and x is the variable we are solving for.

Word problems that involve one-step equations often ask us to solve for a missing variable given some other information.

Here is a step-by-step guide to solving word problems involving one-step equations:

Step 1: Read the problem carefully and identify the unknown variable.

It is important to understand what the problem is asking and what we are trying to find. Identify the variable that is unknown and needs to be solved.

Step 2: Translate the problem into an equation.

Use the given information in the problem to set up an equation that can be solved for the unknown variable. Make sure to use the correct mathematical operation for the given problem. For example, if the problem involves addition, subtraction, multiplication, or division, use the corresponding mathematical symbol to write the equation.

Step 3: Solve the equation.

Solve the equation by isolating the unknown variable on one side of the equation. If the variable is on the left side, move it to the right side by performing the opposite mathematical operation. If the variable is on the right side, move it to the left side by performing the opposite mathematical operation. Perform the same operation on both sides of the equation to maintain balance.

Step 4: Check the solution.

Plug the solution obtained in Step 3 into the original equation and check whether the equation holds true. If the equation holds true, the solution is correct.

Step 5: Write the final answer.

Write the final answer in a sentence or in the appropriate units, depending on what the problem is asking for.

Note that the steps may need to be adapted depending on the specific problem. However, following this guide can help make the process of solving one-step equation word problems more straightforward.

Word Problems Involving the One-Step Equation – Example 1

How many packs of popcorn (p) can you buy for $52 if one package costs $4? Solution: List keywords and phrases in the problem to write the equation. Each package costs $4. The total money is $52. How many packs of popcorn is p. The model is a multiplication equation. \(4.p=52\) Solve it by the inverse operation. \(4.p=52→p=52÷4=13\) You can buy 13 packs of popcorn for $52.

Word Problems Involving the One-Step Equation – Example 2

At a restaurant, Maria and her three friends decided to divide the bill evenly. If each person paid $21 then what was the total bill (b)? Solution: List keywords and phrases in the problem to write the equation. There were 4 people. Each person paid $21. What was the total bill b. The model is a division equation. \(\frac{b}{4}=21\) Solve it by the inverse operation. \(\frac{b}{4}=21→b=21×4=84\) The total bill was $84.

by: Effortless Math Team about 1 year ago (category: Articles )

Effortless Math Team

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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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One-Step Equation Word Problems Worksheets

This page encompasses a compilation of printable worksheets on one-step equation word problems that are custom-made for learners of grade 6, grade 7, and grade 8. One-step equation word problems with coefficients involving integers, fractions and decimals are incorporated here. Use the real-life word problems to assist students improve their analytical thinking. Procure some of these worksheets for free!

Addition and Subtraction: Integers

Read each of the 15 word problems spread over three pdf worksheets. Get students to frame one-step equations and apply either addition or subtraction operations to find the value of the unknown.

• Download the set

Multiplication and Division: Integers

Read each word problem and interpret them into one-step equations. To isolate the variable from the equation, multiply or divide an integer on both sides. Use the answer key to verify your responses.

Addition and Subtraction: Fractions and Decimals

Frame equations with fractions and decimals. Solve the equation and find the solution. This series of printable word problems helps 6th grade, 7th grade, and 8th grade students grasp the basics of one-step equations.

Multiplication and Division: Fractions and Decimals

Use multiplication or division operations to solve the obtained equation. Test a student's comprehension in forming one-step equations with word problems that feature an assortment of fractions and decimals.

Multiple Choice Questions

Read the word problems that comprise a mix of integers, fractions and decimal numbers attentively. Choose the one-step equation that best matches the scenarios given in these pdf worksheets.

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» Equation Word Problems

» One-Step Equation: Addition and Subtraction

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• 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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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills.  students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

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A step in pension reform

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Solving a RM1.3 trillion pension liability problem

Interactive: reforming malaysia’s ballooning civil service pensions, 'defined contribution' scheme for new civil servants puts country in sustainable fiscal position, says economist.

THE restructuring of the Employees Provident Fund’s (EPF) members’ accounts with the introduction of account 3 (Akaun Fleksibel) is a middle ground approach between meeting the contributors’ short-term needs and long-term financial security.

The initiative reflects EPF’s greater goal of addressing Malaysia’s social security coverage gap, which is still wide among Malaysia’s workforce.

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