2 step problem solving

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

In order to access this I need to be confident with:

Two step word problems

Two-step word problems

Here you will learn about two-step word problems, including how to solve a two-step word problem, how to represent these problems using equations, and how to assess the reasonableness of answers.

Students will first learn about two-step word problems as part of operations and algebraic thinking in 3 rd grade and will continue using this skill throughout elementary and middle school.

What are two-step word problems?

Two-step word problems are word problems or story problems that require two steps to find the answer. These two steps can involve the same operation or two different operations.

To solve a two-step word problem, you must read the problem carefully before identifying each of the two steps. After identifying the two steps, you can write an equation for each step. Then you will need to solve each equation in order before arriving at the final answer.

For example,

Sarah had \$15. She spent \$8 on a new book. Then her mom gave her \$5. How much money does she have now?

We can break down the problem into steps and write an equation.

Sarah starts with \$15, so this is the starting number.

Then she spends \$8 on a new book. This means you need to subtract \$8. This is step one .

x represents the amount of money Sarah has after buying the book.

Next, Sarah receives \$5 from her mom. This needs to be added to the remaining amount from step one. This is step two .

t represents the total of money Sarah has after her mom gave her \$5 .

Sarah has \$12 now.

You can also write an equation showing both steps and use the order of operations rules to solve.

After solving the problem, you should ask yourself: Is my answer reasonable?

You can use quick mental math or estimation to see if your answer is reasonable.

You can round the \$8 to \$10 to make the estimation easier.

\$15-\$10 + \$5 = \$10, so Sarah has about \$10 left. This means the answer of \$12 is reasonable.

Common Core State Standards

How does this relate to 3 rd grade math and 4 th grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.D.8) Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
• Grade 4 – Operations and Algebraic Thinking (4.OA.A.3) Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

[FREE] Arithmetic Worksheet (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of arithmetic. 10+ questions with answers covering a range of 4th, 5th and 6th grade topics to identify areas of strength and support!

How to solve two-step word problems

In order to solve two-step word problems:

Identify the first step and write an equation.

Identify the second step and write an equation.

Solve the equations in order.

Assess the reasonableness of your answer.

Two-step word problems examples

Example 1: two-step addition (same operation).

Hannah bought 2 bags of candy for trick-or-treaters. One bag had 78 pieces of candy and the other had 92 pieces of candy. Hannah’s friend came over and dumped another bag of candy in the bowl. This bag had 124 pieces of candy. How many pieces of candy are in the bowl altogether?

First, you need to find out the total number of pieces Hannah dumped into the bowl from her two bags.

2 Identify the second step and write an equation.

Next, you need to add the total pieces that Hannah’s friend dumped into Hannah’s total from the previous step.

3 Solve the equations in order.

There are a total of 294 pieces of candy in the bowl.

4 Assess the reasonableness of your answer.

You can use mental math or estimation to see if your answer is reasonable.

You can quickly round each number to 100, which gives you a total of 300 pieces of candy. This is close to the actual answer of 294, so the answer is reasonable.

Example 2: two-step problem (different operation)

A teacher ordered 8 new boxes of pencils for her classroom. Each box of pencils had 16 pencils. She decided to combine all of the pencils, and then split them evenly between the 4 student tables in the room. How many pencils will each table get?

First, you need to find the total number of pencils in all the boxes.

8 \times 16=x

Next, you need to divide the total number of pencils by the number of tables.

Each table will get 32 pencils.

8 \times 20= about 160 total pencils divided by 4 tables = about 40 pencils on each table. So the answer of 32 is reasonable.

Example 3: solving two-step word problems with fractions

A cookie recipe calls for \cfrac{1}{2} cup of white sugar and \cfrac{3}{4} cup of brown sugar.

The baker is making 6 batches of cookies. What is the total amount of sugar (white and brown) that will be used?

First, you need to add the amounts of sugar to find the total amount of sugar needed for 1 batch.

\cfrac{1}{2}+\cfrac{3}{4}=x

Next, you need to multiply the total amount of sugar by the number of batches being made.

x \times 6=s

7 \cfrac{1}{2} total cups of sugar will be needed for 6 batches of cookies.

\cfrac{1}{2} and \cfrac{3}{4} is a little more than 1 cup.

Since there will be 6, the answer will be more than 6 cups.

Therefore, the answer of 7 \cfrac{1}{2} cups is reasonable.

Example 4: solving two-step word problems with decimals

Chris makes \$12.50 an hour at his job at the roller skating rink. He worked 25 hours. After he got his paycheck, he spent \$65 on a new pair of shoes. How much money does he have left from his paycheck?

First, you need to find out how much money Chris’s paycheck was by multiplying the hourly rate by the number of hours he worked.

12.50 \times 25=x

Next, you need to subtract \$65 from the total paycheck amount.

Chris has \$247.50 left from his paycheck.

To estimate the amount of Chris’s paycheck, you can multiply 12 by 25. You could also multiply 12 by 30, but note this will be a high estimate.

12 \times 25=300-65 = \$235 which makes \$247.50 a reasonable answer.

Example 5: interpreting remainders in two-step division word problem

Five 5 th grade classes each have 24 students and 2 teachers attending a field trip. Each bus can hold 48 people. How many buses are needed to carry all of the students and teachers to the field trip?

First, you need to find out how many students and teachers are attending the field trip altogether. To do this, you will need to multiply 5 \times 26.

5 \times 26=x

Next, you need to divide the total number of people by the number of people each bus can carry.

x \div 48=b

Now that you have identified the steps, you can solve the equations in order.

For this problem, you will need to refer back to the question in order to interpret the remainder. It asks: How many buses are needed to carry all of the students and teachers to the field trip?

So in this word problem, the answer 2 \; R \, 34 represents 2 full buses with 34 people leftover. Since those 34 people also need a bus to ride to the field trip, you would round up the answer to 3 so that all people can attend the field trip.

3 buses are needed to carry all of the students and teachers to the field trip.

There are about 125 people attending the field trip (about 25 people \times 5 classes) and about 50 people can fit on a bus. 125 \div 50=2.5, so the answer of 3 buses is reasonable.

Example 6: interpreting remainders in two-step division word problems

Haruki had 39 books. He got 7 more books for his birthday. His bookshelf has 4 shelves. He wants to put the same number of books on each shelf and put the remaining books on top of his bookshelf. How many books will go on top of Haruki’s bookshelf?

First, you need to determine how many books Haruki has altogether.

So step 1 is to add to find the total number of books.

Next, you will need to divide the total number of books by the number of shelves Haruki has.

For this problem, you will need to refer back to the question in order to interpret the remainder. It asks: How many books will go on top of Haruki’s bookshelf?

So in this word problem, the remainder is your answer.

2 books will go on top of Haruki’s bookshelf.

Haruki has about 45 books that he wants to divide equally between 4 shelves, which means there would be about 11 books on each shelf with about 1 left over. So the answer of 2 books is reasonable.

Teaching tips for two-step word problems

• Begin with simple 2 -step word problems on math worksheets that involve familiar situations and basic operations. Gradually increase the complexity as learners gain confidence and understanding.
• Lesson plans should involve step-by-step problem-solving strategies, such as underlining important information, identifying keywords, and breaking the problem into smaller parts. Model how to solve each step before tackling the problem as a whole.
• Connect math word problems to real-life scenarios that are relevant and interesting to students. This can help them see the practical application of math and reasoning skills and increase engagement on word problem worksheets.
• Provide students with a template if needed to help them break down the problem into steps. You can also provide students with a printable answer key to check their work. If their answers do not match, they can go back to investigate and find the correct steps.
• Start with simple two-step word problems with 1 -digit numbers to allow students to focus on identifying the two steps. Then advance to 2 -digit and 3 -digit problems.

Easy mistakes to make

• Incorrect order of operations Students might perform operations in the wrong order, leading to incorrect solutions. It is imperative that students write their equations correctly to ensure the operations are performed in the correct order.
• Not checking the solution Sometimes, students may not take the time to review their solution to ensure it makes sense in the context of the problem. Checking the answer against the problem statement or using estimation to verify reasonableness can help catch errors.
• Missing a step or performing the steps in the wrong order Sometimes, students may misinterpret the problem statement, causing them to miss a step or mix up the order of steps. It’s crucial to carefully read and understand what the problem is asking for before attempting to solve it.

Related arithmetic lessons

• Skip counting
• Number sense
• Inverse operations
• Money word problems
• Calculator skills

Practice two-step word problem questions

1. Frankie has \$287 in her checking account. She spent \$56 on her phone bill and then spent \$39 at dinner. How much money is left in her account?

This is a two-step subtraction word problem, meaning each of the two steps involves subtraction.

1 st step: Subtract the amount spent on the phone bill.

2 nd step: Subtract the amount spent at dinner from what is left after step 1.

2. Elliot has 145 marbles in his collection. He lost 18 marbles and then bought 27 more. How many marbles does he have now?

154 marbles

190 marbles

136 marbles

100 marbles

This two-step word problem involves two different operations.

1 st step: Subtract the number of marbles Elliot lost.

2 nd step: Add the number of marbles Elliot bought to the total remaining marbles from step 1.

127 + 27 = 154 marbles

3. Mrs. Smith baked 24 cookies in the morning and 18 cookies in the afternoon. If she wants to pack them into bags of 6 cookies each, how many bags of cookies will she have in total?

1 st step: Find the total number of cookies Mrs. Smith baked.

2 nd step: Divide the total number of cookies by the number of cookies per bag to find the total number of bags.

42 \div 6=7 bags

4. Sarah has 15 comic books. She decides to buy 5 more comic books at the store. Each comic book costs \$3.50. If she also buys a poster for \$8, how much money will Sarah spend in total?

This two-step word problem involves two different operations. Also note that there is an extra bit of unnecessary information in the word problem (Sarah has 15 comic books).

1 st step: Determine the total cost of the comic books Sarah buys.

2 nd step: Add the amount spent on the poster to the total from step 1.

5. Libby is selling boxes of cookies for a fundraiser at her school. So far, she has sold 29 boxes of cookies for \$12 each. If she has a goal of raising \$400, how much more money does she need to earn?

1 st step: Multiply to find the amount Libby earned from selling 29 boxes of cookies.

2 nd step: Determine how much more money Libby needs to earn to reach her goal by subtracting the total from step 1 from her goal amount.

6. Georgio has 128 complete fossils in his collection. He donated 19 fossils to a local museum. He wants to arrange the rest of his fossils in a display case in his house. The display case has 7 shelves. If he wants to put the same number of fossils on each shelf, how many will go on each shelf?

1 st step: Subtract to find the number of fossils remaining after donating to the museum.

2 nd step: Determine how many fossils Georgio can put on each shelf by dividing the remaining fossils by the number of shelves.

Two-step word problems FAQs

To solve a two-step word problem, you must read the problem carefully before identifying each of the two steps. After identifying the two steps, you can write an equation. Then you will need to perform each step in the correct order before arriving at the final answer.

The next lessons are

• Properties of equality
• Types of numbers
• Rounding numbers
• Factors and multiples

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs .

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.

40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

Privacy Overview

JavaScript is not active on your browser. This webite requires JavaScript in order to function.

Two-step word problems.

Two-step word problems have three numbers which must be operated on separately, and in the right order. They take much more understanding of the problem than simple word problems because they provide the information in a less structured form.

• Addition, then Subtraction
• Addition, then Multiplication
• Addition, then Division
• Subtraction, then Addition
• Subtraction, then Multiplication
• Subtraction, then Division
• Multiplication, then Addition
• Multiplication, then Division

Two-Step Word Problems (Grade 3)

Suggested learning target.

• I can choose the correct operation to perform the first computation, and choose the correct operation to perform the second computation in order to solve two-step word problems.
• I can write equations using a letter for the unknown number.
• I can decide if my answers are reasonable using mental math and estimation strategies including rounding.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

• Two-step equations – word problems

Simply put, two-step equations – word problems are two step equations expressed using words instead of just numbers and mathematical symbols. They are just a bit more complicated than one-step equations with word problems and they demand just a bit more effort to solve. If you are not confident in your abilities to solve two-step equations with word problems, you can go to one-step equations – word problems and practice some more before continuing with this lesson. But if you feel ready, we will show you how to solve it using this example:

Hermione’s Bikes rents bikes for $10 plus $4per hour. Janice paid $30 to rent a bike. For how many hours did she rent the bike?

First thing we have to do in this assignment is to find the variable and see what its connection is with the other values. The thing we do not know is the number of hours Janice rented the bike for and we have been asked to find that out. That means the number of hours is our variable.

The cost of renting a bike is 10$ to take the bike and 4$ for every hour it spends in our possession. The final sum Janice paid is $30. Let us write that down as an equation.

4 * x + 10 = 30

Now, in order to make things neater and more clear, let us move all the numbers (except for the number 4 – we have to get rid of it in a different way) to the right side of the equation. Like this:

4 * x = 30 – 10

To simplify things further, let us perform the subtraction.

The next thing to do is to get rid of the number 4 in front of the variable. We will do that by dividing the whole equation by 4.

4 * x = 20 |:4

Now that we have calculated the value of the variable, we can tell that Janice rented that bike for 5 hours. If you want to check the result – you can. If you multiply $4 that Janice paid per hour by the 5 hours she spend with that bike and then add the $10 she had to pay regardless of the time she spent with the bike, you will get a total sum of $30 that is indeed the full sum she paid.

These word problems are called two-step because you have to perform two mathematical operations in order to solve them. In this case – addition (subtraction) and multiplication (division). To practice solving two-step equations – word problems, feel free to use the worksheets below.

Two-step equations – word problems exams for teachers

Two-step equations – word problems worksheets for students.

Basic Mathematical Operations

• Addition and subtraction
• Multiplying
• Divisibility and factors
• Order of operations
• Greatest common factor
• Least common multiple
• Squares and square roots
• Naming decimal places
• Write numbers with words
• Verbal expressions in algebra
• Rounding numbers
• Convert percents, decimals and fractions
• Simplifying numerical fractions
• Proportions and similarity
• Calculating percents
• The Pythagorean theorem
• Quadrilaterals
• Solid figures
• One-step equations
• One-step equations – word problems
• Two-step equations
• Multi-step equations
• Multiplying polynomials
• Inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Coordinates
• Graphing linear equations
• The distance formula

Linear Equations

• Writing a linear equation
• Systems of linear equations
• Linear systems – word problems

Recent Posts

• Custom Math Writing Services – How Do They Help?

Free Math Worksheets © 2024. All Rights Reserved.

Free Math Worksheets is a math related website that contains pre-algebra, algebra and geometry worksheets and tests. Now, professors and teacher can reduce cheating by selecting printable exams to the individual students. Contact Us

Free Printable Two-Step Word Problems Worksheets for 3rd Grade

Math: Discover a collection of free printable worksheets for Grade 3 students, focusing on two-step word problems to enhance their problem-solving skills and mathematical understanding.

Explore Two-Step Word Problems Worksheets by Grades

• kindergarten

Explore Other Subject Worksheets for grade 3

• Social studies
• Social emotional
• Foreign language
• Reading & Writing

Explore printable Two-Step Word Problems worksheets for 3rd Grade

Two-Step Word Problems worksheets for Grade 3 are an essential tool for teachers who want to help their students develop strong problem-solving skills in math. These worksheets provide a variety of math word problems that require students to use addition, subtraction, multiplication, and division to find the solutions. By incorporating real-life scenarios and engaging themes, these worksheets make learning fun and relatable for young learners. Teachers can use these Grade 3 math worksheets to reinforce classroom lessons, provide extra practice for struggling students, or as a homework assignment to keep math skills sharp. With a wide range of topics and difficulty levels, these Two-Step Word Problems worksheets for Grade 3 are an invaluable resource for educators looking to challenge and inspire their students in the world of mathematics.

Quizizz is an innovative platform that offers a variety of educational resources, including Two-Step Word Problems worksheets for Grade 3, to help teachers create engaging and interactive learning experiences for their students. In addition to math word problems, Quizizz also provides resources for other subjects such as science, social studies, and language arts. Teachers can easily customize and assign quizzes, worksheets, and other activities to their students, making it simple to track progress and provide targeted support. The platform also offers gamified quizzes and competitions, which can help motivate students and make learning more enjoyable. By incorporating Quizizz into their lesson plans, teachers can provide a well-rounded and dynamic educational experience for their Grade 3 students, ensuring they develop a strong foundation in math and other essential subjects.

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Math Solver

Geogebra math solver.

Get accurate solutions and step-by-step explanations for algebra and other math problems, while enhancing your problem-solving skills!

Solving Two-Step Word Problems (II) - Lesson Plan

In this lesson, students will learn how to solve two-step word problems using various strategies. they will practice solving problems involving shopping and changes in quantities. the class aims to develop problem-solving skills and mathematical thinking..

Know more about Solving Two-Step Word Problems (II) - Lesson Plan

This lesson teaches students strategies such as counting on, counting forward/backward, and the part-part-whole model.

The word problems in this lesson involve shopping scenarios and situations where quantities change.

This lesson provides opportunities for students to practice solving two-step word problems and apply their knowledge to real-world scenarios.

Your one stop solution for all grade learning needs.

Addition (Basic)

Addition (Multi-Digit)

Algebra & Pre-Algebra

Comparing Numbers

Daily Math Review

Division (Basic)

Division (Long Division)

Hundreds Charts

Measurement

Multiplication (Basic)

Multiplication (Multi-Digit)

Order of Operations

Place Value

Probability

Skip Counting

Subtraction

Telling Time

Word Problems (Daily)

More Math Worksheets

Reading Comprehension

Reading Comprehension Gr. 1

Reading Comprehension Gr. 2

Reading Comprehension Gr. 3

Reading Comprehension Gr. 4

Reading Comprehension Gr. 5

Reading Comprehension Gr. 6

Reading & Writing

Reading Worksheets

Cause & Effect

Fact & Opinion

Fix the Sentences

Graphic Organizers

Synonyms & Antonyms

Writing Prompts

Writing Story Pictures

Writing Worksheets

More ELA Worksheets

Consonant Sounds

Vowel Sounds

Consonant Blends

Consonant Digraphs

Word Families

More Phonics Worksheets

Early Literacy

Build Sentences

Sight Word Units

Sight Words (Individual)

More Early Literacy

Punctuation

Subjects and Predicates

More Grammar Worksheets

Spelling Lists

Spelling Grade 1

Spelling Grade 2

Spelling Grade 3

Spelling Grade 4

Spelling Grade 5

Spelling Grade 6

More Spelling Worksheets

Chapter Books

Charlotte's Web

Magic Tree House #1

Boxcar Children

More Literacy Units

Animal (Vertebrate) Groups

Butterfly Life Cycle

Electricity

Matter (Solid, Liquid, Gas)

Simple Machines

Space - Solar System

More Science Worksheets

Social Studies

Maps (Geography)

Maps (Map Skills)

More Social Studies

Mother's Day

Father's Day

More Holiday Worksheets

Puzzles & Brain Teasers

Brain Teasers

Logic:  Addition Squares

Mystery Graph Pictures

Number Detective

Lost in the USA

More Thinking Puzzles

Teacher Helpers

Teaching Tools

Award Certificates

More Teacher Helpers

Pre-K and Kindergarten

Alphabet (ABCs)

Numbers and Counting

Shapes (Basic)

More Kindergarten

Worksheet Generator

Word Search Generator

Multiple Choice Generator

Fill-in-the-Blanks Generator

More Generator Tools

Full Website Index

Multiple-Step Word Problems

These multiple-step word problems require students to use reasoning and critical thinking skills to determine how each problem can be solved.

Basic (Grades 2 - 3)

Logged in members can use the Super Teacher Worksheets filing cabinet to save their favorite worksheets.

Quickly access your most used files AND your custom generated worksheets!

Please login to your account or become a member and join our community today to utilize this helpful feature.

Intermediate (Grades 3 - 4)

Advanced (grades 4 - 5).

These word problems are sorted by type: addition, subtraction, multiplication, division, fractions and more.

These worksheets, sorted by grade level, cover a mix of skills from the curriculum.

S.T.W. has thousands of worksheets. Visit the full math index to find them all, sorted by topic.

Sample Images of Our Printables

PDF with answer key:

PDF no answer key:

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

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)

The 7 Best AI Tools to Help Solve Math Problems

Quick links, the test questions, wolframalpha, microsoft mathsolver.

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

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

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

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

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

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

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

It resulted in the answer:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I have problem when solving this exercise, can anyone break it into step by step for me to follow. Thank you so much

You are working on a bid to build a bridge per year for the next three years. This project requires the purchase of $1,000,000 of equipment that will be depreciated using straight-line depreciation to a zero-book value over the project's life. Ignore bonus depreciation. The equipment can be sold at the end of the project for $400,000. You will also need $250,000 in net working capital over the life of the project. The fixed costs will be $500,000 per year and the variable costs will be $3,000,000 per bridge. Your required rate of return is 20 percent for this project and your tax rate is 10 percent. What is the minimal amount, rounded to the nearest $100, you should bid per bridge?

Note: 3 years annuity factors of 20% required return = 2.106

0 users composing answers..

1 +0 answers.

Here's how to calculate the minimal amount you should bid per bridge for this three-year project:

1. Consider the Equipment:

Equipment cost: $1,000,000

Salvage value (selling price after 3 years): $400,000

Depreciation: Straight-line over 3 years

Annual depreciation expense: ($1,000,000 - $400,000) / 3 years = $200,000/year

2. Factor in Working Capital:

Net working capital needed: $250,000 (constant throughout the project)

3. Analyze Costs and Taxes:

Fixed cost per year: $500,000

Variable cost per bridge: $3,000,000

Required rate of return: 20%

Tax rate: 10%

4. Calculate After-Tax Cash Flow per Year:

We'll consider one year at a time. Let's denote the year as "Y":

Revenue from the bridge (unknown yet - this is what we're solving for): "Bid amount (X)"

Total cost: Fixed cost + Variable cost - Depreciation expense

Taxable income before depreciation: Bid amount (X) - Total cost

Depreciation tax shield: Annual depreciation expense * Tax rate = $200,000 * 10% = $20,000

Taxable income: Taxable income before depreciation - Depreciation tax shield

Taxes (at 10% rate): Taxable income * Tax rate

After-tax cash flow (Y): Bid amount (X) - Total cost - Taxes + Depreciation expense + Net working capital

5. Apply Annuity Factor for Present Value:

We're considering a three-year project, so we need to find the present value of the after-tax cash flow for each year. Since the cash flow happens at the end of each year (annuity due), we'll use an annuity factor (considering the required rate of return of 20%).

Annuity factor for 3 years at 20% interest: 2.106 (given)

6. Solve for the Minimum Bid (X):

We want the project's Net Present Value (NPV) to be zero at the minimum acceptable bid. NPV is the sum of the present values of all future cash flows.

Net Present Value (NPV) Equation:

NPV = After-tax cash flow (Year 1) + After-tax cash flow (Year 2) + After-tax cash flow (Year 3) - Initial equipment cost + Salvage value = 0

We can rewrite the above equation with the annuity factor and solve for X (the minimum bid amount).

After rearranging and substituting the after-tax cash flow formula from step 4:

X * (1 - (1 + Required return)^(-Project life)) / Required return = (Total cost - Depreciation expense + Net working capital) * Annuity factor + Equipment cost - Salvage value

Plugging in the numbers:

X * (1 - (1 + 0.2)^(-3)) / 0.2 = ($500,000 + $3,000,000 - $200,000 + $250,000) * 2.106 + $1,000,000 - $400,000

X * (1 - 0.512) / 0.2 = $3,856,000 * 2.106 + $600,000

X * 0.488 = $8,038,16 + $600,000

X = ($8,038,16 + $600,000) / 0.488 ≈ $8,647,764

Minimum Bid amount rounded to nearest $100: $8,647,800

Therefore, the minimum amount you should bid per bridge to achieve a zero Net Present Value for this three-year project is approximately $8,647,800.

3 Online Users

Sticky Topics

• Trending Now
• Foundational Courses
• Data Science
• Practice Problem
• Machine Learning
• System Design
• DevOps Tutorial

Understanding One-Step and Two-Step Equations & Inequalities

One-step and two-step equations and inequalities are foundational concepts in algebra, teaching students to solve simple equations and inequalities using basic operations. In one-step equations, only one operation is needed to isolate the variable, while in two-step equations, two operations are required. These concepts are essential for building algebraic problem-solving skills and preparing students for more complex equations and inequalities in higher grades.

Solve for 𝑥 in the equation 5+𝑥=12

Solve for 𝑦 in the equation 4𝑦−8=20

Solve the inequality 𝑥+3<10.

Solve the inequality 2𝑦−5>7.

There are 4 questions to complete.

• Talk to Expert
• Cloud Native
• June 5, 2024
• 6 minute read

Can SPIFFE Solve the "Secret Zero" Problem?

Cloud native computing moves fast, and it’s changing all the time. The Secure Production Identity Framework for Everyone (SPIFFE) has emerged as a crucial tool for establishing consistent, scalable identity security. This open standard offers a streamlined approach to ensuring secure identity verification across workloads and infrastructure. 

In this blog post, I'll discuss how SPIFFE can, in fact, solve the "secret zero" (otherwise known as the  "bottom turtle" ) problem.

What is the “secret zero” problem?

The term “secret zero” refers to the first credential (or initial secret) used to gain access to other sensitive information within a system. 

Think of it like the master key to a vault filled with numerous other keys. A bit like the master password you use to log into a password manager. But if that master key were compromised, all the secrets contained within the vault would similarly be jeopardized. 

That’s why solving the problem is so critical, especially in a zero trust architecture.

SPIFFE basics

At its heart, SPIFFE provides a common framework for identifying and authenticating application and service identities. 

This framework relies on the association of each identity with a unique SPIFFE ID, which is verified using a set of cryptographic materials.

Challenges solved by SPIFFE

Implementing SPIFFE fundamentally transforms how machine identities are managed and verified:

• Eliminates API key distribution : By using SPIFFE, you remove the need for static API keys, as well as the hassle of key storage and retrieval.
• Cloud-to-cloud authentication:  For instance, you could write a file to an AWS S3 bucket while being on a Google Cloud instance or even on-premises with a virtual machine.
• Step-by-step CI/CD security : Providing a unique identity for each step in your CI/CD pipeline streamlines auditing, so you can improve software supply chain security. 

New SPIFFE Support for Venafi Firefly Secures Workloads Across Cloud Native Environments

5 steps to building trust with spiffe.

Now, we're going to explore how a workload can be issued a SPIFFE identity, as well as the primary components of that identity. Each step builds on the last, as you will see in the reference architecture provided.

Step 1: The status quo

Here, we have a traditional application (App B) that wants to communicate with another traditional application (App X). But before it can, it must retrieve a client ID from the secrets manager. 

Once App B has that client ID, App X will validate. 

But this connection isn't  mTLS-secured.  Nor is it encrypted. Plus, App X is only validating a client ID, not verifying whether App B is actually authorized to talk to it. 

Step 2: SPIFFE ID

You can compare a SPIFFE ID to a human's name, though it looks like a website URL, with:

• Standard scheme
• Trust domain

An example SPIFFE ID could be:  spiffe://venafi.com/dc1/node10/frontend/webserver

Now, we'll return to our Applications B and X. This time, we'll send over a SPIFFE identity. 

App X can validate the connection and even authorize it. So, with the SPIFFE ID, we've solved both authorization and auditing problems. 

But it's still not fully secured.

Step 3: SPIFFE verifiable identity document (SVID)

You can think of a SPIFFE SVID like a passport, which you would use to get verified at Customs or Border Control. 

It is a cryptographically signed identity document used to verify the identity of a workload within a specific trust domain, and it aims to standardize identification and authentication for services in distributed, dynamic environments, such as microservices architecture and cloud native applications. 

Currently, JWT tokens and X.509 certificates are supported as cryptographic key material, which both cover several use cases. And these certificates typically should have short lifetimes to minimize the risk of exposure and compromise.

This is an example of an X.509 certificate:

We can see that it has a Version 3 alternative name and a scheme with a unique SPIFFE ID.

Similarly, here is a JWT token example:

Let's go back to our application. 

This time, we'll use a proper X.509 certificate with a SPIFFE SVID. When App B initiates a connection to App X, App X will authorize and validate based on the CA chain that it's aware of, in this case of where that X.509 certificate was issued. 

With this step, we've now solved authentication, in addition to authorization and auditing. Our connection is also mTLS secure.

But we're not quite there yet.

Step 4: SPIFFE workload API

How can you ensure applications receive SPIFFE SVIDs from a central system? And how can identities be issued in a trustworthy way? 

This is where the SPIFFE Workload API comes in. Like requesting a passport at the passport office, we need a way to get SPIFFE SVIDs for each workload. This is done through a standardized workload API. 

Preferably, the API only runs on locally, or at least only listens on a local host of a node, since it's an unauthenticated API. The API provides a way to retrieve SPIFFE SVIDs, and it automatically renews and rotates them, too. 

The workload API is also used to create integrations for different platforms, delivering SPIFFE SVIDs directly to apps on well-known locations. Otherwise, if you don't have integrations, the workload itself will be required to retrieve the SVID from the API.

Imagine that, wherever you run, you could have SVIDs, from AWS Lambda functions to on-prem virtual machines or even mainframes. 

In addition to the previously mentioned functionalities, the API also validates and attests the different workloads that turn on that node. And this happens out of band (otherwise, we're not really solving the secret zero problem).

Now, back to our application. 

Our architecture has extended a bit, with a SPIFFE agent on each server, as well as the workload API. This time, App B will request a SPIFFE SVID from the API, and in the background the workload API will validate the workload, then send back an assigned X.509 SPIFFE SVID to App B. Through the integration provided by the API, both applications will automatically get the SPIFFE SVID.

This time, when we initiate the connection, it'll follow all the previous steps. But we must verify and attest that the workloads are who they say they are before we can fully trust them.

That verification and attestation of SVIDs happens at two levels.

Step 5: Verification and attestation

We must first trust the node where the workload API is running. Then, we can trust the workloads themselves. To do this, we can query metadata, TPM or other verifiable data.

As this is a bit abstract, let's refer to our example. 

We've now added a SPIFFE server and node attestation, along with SPIFFE agents, workload attestations, and node attestations for both servers.

Now, when a SPIFFE agent comes up, it will gather information on the environment where it runs. It will send this proof back to the SPIFFE server, which will verify it. 

If that information is all correct, the server will issue a SPIFFE SVID for the node and send it back to the agent.

Now, Server 1 has a unique identity, and we can repeat the process for Server 2. 

Once that's done, App B can request an SVID from the agent, which will verify the workload and issue a certificate. That will be sent back to App B, and then App B can effectively authenticate with App X using a SPIFFE identity.

SPIFFE, secret zero and zero trust

That's how the five components of SPIFFE come together to secure the connection, and this is how we actually solve the true secret zero problem as it relates to zero trust networking. 

Your SPIFFE server is the central figure here, because that's where your CA is located, and where your root of trust begins.

Use cases for federation

One of the significant advancements SPIFFE brings to the table is federation, essentially enabling trust across different environments with multiple SPIFFE servers. 

This functionality can be segmented into three primary use cases:

• Internal segmentation : Different environments within the same company, such as staging versus production or PCI versus non-PCI segments, can benefit from distinct trust domains.
• Cross-company implementation : When integrating services between different companies, federation facilitates secure interactions between distinct SPIFFE deployments despite potential variations in their SPIFFE implementations.
• Facilitating external consumers : For entities without a SPIFFE setup, federation allows them to authenticate callers effortlessly, bypassing the complexity of a full SPIFFE deployment.

Industry adoption

SPIFFE's utility is gaining recognition, with notable implementations such as SPIRE—the open-source reference implementation of SPIFFE. Moreover, leading cloud providers like AWS, Google Cloud, and Azure are integrating SPIFFE into their security architectures.

Significant software tools, including Istio, cert-manager, HashiCorp Consul and Dapper, have adopted SPIFFE as the foundational identity for their workloads, further validating its importance.

SPIFFE stands out as a significant innovation in securing machine-to-machine connections.

By providing foundational identity verification, reducing reliance on API keys, and simplifying processes with automation, SPIFFE is transforming industry standards. 

With widespread adoption and ongoing enhancements, SPIFFE is poised to remain a pivotal element in cloud native security for years to come.

Want to learn more about how to protect your cloud-first, GenAI, post-quantum world? Join us in Boston for  Machine Identity Management Summit 2024 , October 1-3!

2024 Machine Identity Management Summit

Help us forge a new era of cybersecurity

Looking to vulcanize your security with an identity-first strategy? Register today and save up to $100 with exclusive deals. But hurry, this sale won't last!