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50 Third Grade Math Word Problems of the Day

The area of Happytown is 42 square miles.

50 Third grade math problems.

Opening your daily math lesson with a Math Word Problem of the Day is an excellent way to set the stage for learning. We all know that word problems are difficult for young learners to grasp, even when the mathematical operation portion of the problem is basic. Incorporate these third grade math word problems one day at a time at the start of your math block to build confidence, critical thinking skills, and a learning community. Students will get used to reading slowly for meaning, while also identifying key information. Encourage students to write out equations and draw pictures to explain their thinking, since this helps them see the light when they are stuck!

Topics covered include addition, subtraction, multiplication, division, mixed operation, fractions, area and perimeter, and measurement. All you need to do is post one of these third grade math word problems on your whiteboard or projector screen, then let kids take it from there.

Want this entire set of word problems in one easy document? Get your free PowerPoint bundle by submitting your email here .

50 Third Grade Math Word Problems

1. gage went shopping and bought 19 cupcakes, 18 apple pies, and 47 glazed donuts. how many items did he buy altogether.

Addition third grade math word problem

2. A shelter is taking care of 384 cats. 176 more arrive. How many cats are in the shelter now?

Addition third grade math word problem

3. Gabby has 42 books on her bookshelf. Her dad gives her 23 more for her birthday. How many books does Gabby have now?

Addition third grade math word problem

4. There were 823 people attending a soccer game after 37 people left. How many people were at the game before the people left?

Addition third grade math word problem

5. Mr. Washington has 44 pencils. He found 37 pencils inside of the cubbies and opened a new pack of 60 pencils. How many pencils does Mr. Washington have in all?

Addition third grade math word problem

6. Jeff plays Minecraft. On Monday he played for 67 minutes. On Tuesday he played for 32 minutes, and on Wednesday he played for 43 minutes. How many total minutes did Jeff play during the week?

Jeff plays Minecraft. On Monday he played for 67 minutes. On Tuesday he played for 32 minutes, and on Wednesday he played for 43 minutes. How many total minutes did Jeff play during the week?

7. Smith Elementary School has 286 boys and 241 girls. What is the total number of students that attend Smith Elementary?

Addition third grade math word problem

8. Ashley rides her bike to school every day. The ride from her house to her school takes 21 minutes. If she has already been biking for 17 minutes, how much longer does she have to ride before she arrives at school?

Subtraction third grade math word problem

9. Jason baked 93 cookies to sell at the school bake sale. He plans to take home whatever cookies he doesn’t sell. If he sold 77 cookies, how many cookies will he take home?

Subtraction third grade math word problem

10. A school collects money for T-shirts. Third graders collected $327. Fourth graders collected 138 fewer dollars. How much money did fourth graders collect?

A school collects money for T-shirts. Third graders collected $327. Fourth graders collected 138 fewer dollars. How much money did fourth graders collect?

11. Hayden has 610 stickers. There are 250 in a white box and some in a yellow box. How many are in the yellow box?

Subtraction third grade math word problem

12. The table shows students’ class points. Which two students’ scores have a difference of 15 points?

The table shows students’ class points. Which two students’ scores have a difference of 15 points?

13. Ms. Brady has 356 tomato plants. She sells 91 of the plants and gives 49 of the plants away. How many tomato plants does Ms. Brady have left?

Subtraction third grade math word problem

14. David’s mom buys apples for his class. There are 5 rows of 4 green apples. There is 1 row of 4 red apples. Fill in the blanks to complete the expressions.

Multiplication third grade math word problem

15. A box of chocolates has 14 rows. There are 6 pieces of chocolate in each row. How many pieces of chocolate are in the box?

Multiplication third grade math word problem

16. Christy and Jan are playing a card game. Christy has 4 cards and Jan has 4 times as many cards. How many cards do they have together?

Multiplication third grade math word problem

17. Emerson is in charge of collecting the basketballs at the end of gym class. There are 6 bins, and 7 basketballs can fit in each bin. How many basketballs are there if each bin is full?

Multiplication third grade math word problem

18. Luz had two pages of homework. Each page had five problems on it. How many total problems did she have to complete?

Multiplication third grade math word problem

19. Keyanna was drawing on scrap paper. She could fit seven drawings on each page. If she has three pieces of paper, how many drawings can she make?

Multiplication third grade math word problem

20. A bakery sells one sugar cookie for $2.00. How much will a dozen sugar cookies cost?

Multiplication third grade math word problem

21. Mrs. Smith is getting her classroom ready for the first day of school. She has 25 students in her class. She wants 5 students to sit at every table. How many tables will she need?

Division third grade math word problem

22. Alex is organizing his school supplies. He has 141 index cards. He wants to split them into 3 piles.  How many cards will each pile get?

grade 3 math problem solving questions

23. Mrs. Blackley’s class plays a review game. The class earns 5 points every time they answer a question correctly. On Monday, the class earns 50 points playing the game. How many questions did the class answer correctly?

Division third grade math word problem

24. Maria has 56 gummy bears. She will give all the gummy bears to 8 of her friends. Each friend will get the same number. How many gummy bears will each friend receive?

Division third grade math word problem

25. Daniel has 63 coins and 9 piggy banks. If each piggy bank has the same number of coins, how many coins are in each piggy bank?

Division third grade math word problem

26. Leslie has 32 dolls in 4 baskets. Each basket has the same number of dolls. How many dolls are in each basket?

Division third grade math word problem

27. Levi has 10 fish in his fish tank. 4 fish are goldfish and the rest are minnows. What fraction of the fish are goldfish?

Fractions third grade math word problem

28. Melanie has 8 colored pencils. 3 are green, 2 are yellow, and 3 are blue. What fraction of the pencils are blue?

Fractions third grade math word problem

29. Hannah has a bag of M&M’s. There are 24 M&M’s in the bag. 8 are red, 6 are green, 7 are yellow, and 3 are brown. What fraction of the M&M’s are brown?

Fractions third grade math word problem

30. Mr. Brown needs to order pizza for 18 students. He wants each student to have ¼ of a pizza. How many pizzas should he order?

Fractions third grade math word problem

31. The class took a survey about favorite animals. ¼ of students chose lions as their favorite animal, and ½ of the students chose elephants. The other students chose either sharks or whales. Partition a circle graph to show these results.

Fractions third grade math word problem

32. John wanted a chocolate cake for his birthday. After blowing out the candles, he cut the cake. Out of his 12 party guests, only 6 ate cake. John let his friend Jackson take half the remaining cake home. What fraction of the cake was left over?

Fractions third grade math word problem

33. The area of a rectangle is 72 square units. One side has a length of 9 units. What is the other side length?

Geometry third grade math word problem

34. Monico drew a shape. It was a quadrilateral, and all sides were the same length. Which shape did Monico draw?

Geometry third grade math word problem

35. Gloria is buying a new rug for her bedroom floor. If the floor measures 12 feet x 12 feet, which size rug does she need?

Geometry third grade math word problem

36. The area of Happytown is 42 square miles. If the length is 7 miles, what is the perimeter of the town?

Geometry third grade math word problem

37. Aiden read 2 pages in his chapter book every day for 7 days. The book has 32 total pages. How many pages does Aiden have left to read?

Third grade math word problem

38. Sophia has a twenty dollar bill. She buys six pop-it fidget toys for $2 each. How much money does Sophia have left?

Third grade math word problem

39. There are 10 people at a cookout. Each person will eat 2 hot dogs. There are 8 hot dogs in a package. How many packages are needed?

Third grade math word problem

40. Mrs. Potter bought 160 index cards. She gave the first row 55 index cards, then the second row 72 index cards. How many does she have left to give to the last row?

Third grade math word problem

41. Samantha has 38 markers. She gives 29 markers to her classmates. Then she opens a new pack of 15 markers. How many markers does she have now?

Samantha has 38 markers. She gives 29 markers to her classmates. Then she opens a new pack of 15 markers. How many markers does she have now?

42. There were 18 volleyballs in a bag and 13 volleyballs in a basket. 10 of these were used at recess. How many volleyballs were not used?

Third grade math word problem

43. Mrs. Horton has 1 gallon of milk. She poured out 3 cups of milk. How many cups of milk does she have left?

Third grade math word problem

44. Mark needs to buy yarn for 5 friends to do a science activity. Each friend needs 2 feet of green yarn and 1 foot of purple yarn. How many yards of yarn does Mark need to buy?

Third grade math word problem

45. Nadine’s new jump rope is 4 inches longer than her old jump rope. Her old jump rope was 32 inches long. How long is Nadine’s new jump rope?

Third grade math word problem

46. Dervin lined paper clips up in two rows. Each row was 18 feet long. How many feet of paper clips did Dervin have if he added the two rows together?

Third grade math word problem

47. London started looking for her missing dog at 2:10 pm. It took her 43 minutes to find him. What time did London find her dog?

Third grade math word problem

48. Damon took a train from his house to downtown. The train left the station at 1:08 pm and traveled 33 minutes before arriving downtown. What time did the train arrive?

Third grade math word problem

49. Hannah worked on her homework for 37 minutes. If she started at 7:14 pm, what time did she finish her homework?

Third grade math word problem

50. Kevin started cleaning his room at 6:03 pm. If he finally finished at 6:40 pm, how long did Ned spend cleaning his room?

Third grade math word problem

If you like these third grade math word problems, check out our third grade hub for even more resources.

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You Might Also Like

Check Out These 50 Fourth Grade Math Word Problems of the Day

Check Out These 50 Fourth Grade Math Word Problems of the Day

Mr. Smith’s class collected coins in a big jar for 6 months. Continue Reading

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grade 3 math problem solving questions

35 math problems for 3 rd graders

In 3rd grade, students are introduced to important math concepts that become the building blocks for future mathematical learning. Let’s practice those concepts with this list of math problems for 3rd graders.

headshot of author, Katie Wickliff

Author Katie Wickliff

grade 3 math problem solving questions

Published September 25, 2023

grade 3 math problem solving questions

  • Key takeaways
  • In 3rd grade math, students develop a deeper understanding of multiplication and division, fractions, and measurements.
  • Word problems for 3rd graders require more advanced comprehension skills.
  • Place Value: Understanding place value helps us read and write numbers correctly and compare their values.
  • Look for opportunities for your child to practice their math at home–worksheets, math games , or a math practice app will help strengthen new mathematical knowledge. While consistency is key, math practice for 3rd graders should be engaging and fun.

Table of contents

  • Subtraction
  • Multiplication
  • Measurement
  • Multi-step word problems

Answer sheet

In addition to reviewing and strengthening the skills learned in previous years, 3rd graders learn multiplication and division, explore fractions, calculate measurements, and more. Also, word problems for 3rd graders are often multi-step, requiring students to understand what operations to perform and how to perform them correctly. 

This list of 3rd grade math problems is intended to reinforce concepts students have already been taught. The list is divided into sections: addition, subtraction, multiplication, division, fractions, measurement, and multi-step problems. These sections allow students to specifically focus their practice, but feel free to skip around and choose the 3rd grade math practice problems that best suit your child’s needs. 

35 Third Grade Math Problems

grade 3 math problem solving questions

Section 1: Addition

Doodle-Blog-NumberIcons_1

At the school carnival, Hattie won 25 tickets at ring toss and 40 tickets at balloon darts. How many tickets did she win altogether?

Doodle-Blog-NumberIcons_2

There are 15 baby chicks at the barn. 12 more chicks are born. How many baby chicks are at the barn now?

Doodle-Blog-NumberIcons_3

How can you represent the number line with an equation?

grade 3 math problem solving questions

Using the number line, solve 7+ _ = 22

grade 3 math problem solving questions

Amir practices basketball three days a week. On Tuesday, he practices basketball for 60 minutes. On Wednesday, he practices for 37 minutes. On Thursday, he practices for 15 minutes. Which expression shows how to find the total number of minutes Amir practiced? Circle the correct answer

Section 2: Subtraction

Using the number line, solve 17-6=

grade 3 math problem solving questions

There are 224 Lions fans and 212 Tigers fans at the football game. How many more Lions fans than Tigers fans are at the game?

An animal shelter has 27 kittens available for adoption. On Friday, 8 kittens are adopted. How many kittens are left at the shelter?

Gemma has 130 beads for bracelets. She gives 75 beads to Piper and 6 beads to Claire. Which expression shows how to find the total number of beads Gemma has left? Circle the correct answer.

Using the number line, solve 30- _ = 14

grade 3 math problem solving questions

Section 3: Multiplication

Using the picture below, write a multiplication equation to find the number of shapes

grade 3 math problem solving questions

Francis and Jane are playing a card game. Francis has 7 cards. Jane has 3 times as many cards as Francis. How many cards does Jane have?

grade 3 math problem solving questions

Using the number line to show hops, complete the multiplication sentence

grade 3 math problem solving questions

Solve the equation

grade 3 math problem solving questions

Mariela has 8 pages of math homework. There are 4 problems on each page. How many total problems does Mariela have for math homework?

Section 4: Division

Connor has 80 crayons that he will put into crayon boxes. Each crayon box can hold 8 crayons. How many crayon boxes does he need?

Using the number line to show hops, complete the division sentence

grade 3 math problem solving questions

Natalie divided 12 marbles into equal groups of 3. Draw a representation to show equal grouping.

Craig’s mom brought 12 cookies to the soccer game. She shared them equally among Craig and 3 teammates. Which expression will solve the problem correctly? Circle the answer.

Section 5: Fractions

Using the number line, work out what A, B and C represent.

grade 3 math problem solving questions

Plot ¼ on this number line:

grade 3 math problem solving questions

A cheese pizza has 8 slices. Marco ate 2 of the slices. His dad at the rest.  What fraction of the pizza did Marco eat? What fraction of the pizza did his dad eat?

There are 15 flowers in a garden. 10 flowers are yellow and the rest are red. What is the fraction of red flowers in the garden?

Using the number line, find the value of K and L. Which is bigger?

grade 3 math problem solving questions

Section 6: Measurement

A school bus is 12 feet long and a car is 4 feet shorter than the bus. How long is the car?

John is 64 inches tall, Mabel is 59 inches tall, and Phoebe is 57 inches tall. How much taller is John than Phoebe?

Use the ruler provided to measure to the nearest ¼ inch:

grade 3 math problem solving questions

Find the area of the square below

grade 3 math problem solving questions

Find the perimeter of the rectangle below

grade 3 math problem solving questions

Section 7: Multi-step word problems

Sarah has two $10 bills. Her sister has one $5 bill. They combine their money to buy a game that costs $22 dollars. How much change will they receive?

Dad baked 36 cookies for the bake sale. Jonas baked 24 cookies for the bake sale. His brother ate 5 cookies before the bake sale started. How many cookies do they have left to sell?

The school bus seats 75 students in all. On Friday, the school bus was full. At the first stop, 7 students got off. At the second stop, 15 students got off. How many children were left on the bus?

George has 25 water balloons. He wants to divide the water balloons equally between himself and 2 friends. How many water balloons will each get? Will there be any water balloons left over?

On Monday, 5 inches of snow fell. On Tuesday, 3 inches of snow fell. On Wednesday, 7 inches of snow fell. The school will have a snow day if it snows 21 inches or more. How many more inches would it have to snow for the school to have a snow day?

Addition answers

grade 3 math problem solving questions

25+40= 65. Hattie won 65 tickets

15+12= 27. 27 baby chicks are at the barn

7+ 15= 22. This should be represented by 15 “hops” on the number line

Subtraction answers

17-6=11. This should be represented by 11 “hops” on the number line

224-212= 12. There are 12 more Lions fans than Tiger fans

27-8= 19. There are 19 kittens left at the shelter.

Multiplication answers

2×3=6; 3×2=6

7×3=21. Jane has 21 cards.

grade 3 math problem solving questions

8×4= 32

Division answers

grade 3 math problem solving questions

40÷8= 5. It will take Max 5 days to pay Bennett.

Drawing should show 3 groups of 4 marbles each.

Fractions answers

The ¼ should be plotted on the point between 0 and ½

Marco ate 2/8 (or ¼) of the pizza. His dad ate 6/8 (or ¾) of the pizza

5/15 (or ⅓)

K= ⅜, L= 6/8 (or ¾). L is larger.

Measurement answers

12-4=8. The car is 8 feet long.

64-57=7. John is 7 inches taller than Phoebe

8×8=64. The area is 64 yd2

8+8+2+2=20 cm

Multi-step word answer answers

$10+$10= $20. 

$20+$5=$25.

$25-$22= $3

 Sarah and her sister will receive $3 in change

They have 55 cookies left to sell

53 students are left on the bus

George and his friends each get 6 water balloons, with one left over.

  • 5 in+ 3 in+ 7 in= 15 in

21 in- 15 in= 6 in

It must snow 6 more inches to get a snow day. 

grade 3 math problem solving questions

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grade 3 math problem solving questions

Lesson credits

grade 3 math problem solving questions

Katie Wickliff

Katie holds a master’s degree in Education from the University of Colorado and a bachelor’s degree in both Journalism and English from The University of Iowa. She has over 15 years of education experience as a K-12 classroom teacher and Orton-Gillingham certified tutor. Most importantly, Katie is the mother of two elementary students, ages 8 and 11. She is passionate about math education and firmly believes that the right tools and support will help every student reach their full potential.

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grade 3 math problem solving questions

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3rd Grade Addition and Subtraction

Word problems.

Welcome to our 3rd Grade Addition and Subtraction Word Problems page. Here is our selection of free addition and subtraction word problem worksheets, which will help your child practice solving a range of addition and subtraction problems using 3-digit and 4-digit numbers.

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  • 3 Digit Addition & Subtraction Word Problems
  • 4 Digit Addition & Subtraction Word Problems
  • 2-Step Addition & Subtraction Word Problems
  • Easier Addition & Subtraction Worksheets
  • More related resources
  • Addition & Subtraction Word Problems up to 1000 Online Quiz

3-digit and 4-digit Word Problems

Each sheet consists of a range of 5 addition and subtraction word problems with numbers up to thousands.

There is a space on each sheet for working out, so that your child can write out the problem and solve it.

We have split the worksheets up into 3-digit word problems and 4-digit word problems.

Using these sheets will help your child to:

  • add and subtract numbers with 3-digit and 4-digit numbers;
  • solve addition and subtraction word problems with and without regrouping.
  • recognise the language used in addition and subtraction - sum, total, difference, how many more, etc.
  • solve problems involving more than 1 step.

3rd Grade Addition and Subtraction Word Problems

3-digit addition and subtraction word problems.

There are two versions of each sheet.

The first version (version A) contains problems where no regrouping is needed.

The second version (version B) contains similar problems but regrouping is needed to solve them.

Answers have been provided with each sheet, along with the calculation needed to solve the problem.

The following sheets are all 1-step addition or subtraction word problems.

  • 3-Digit Addition and Subtraction Word Problems Sheet 1A (no regrouping)
  • PDF version
  • 3-Digit Addition and Subtraction Word Problems Sheet 1B
  • 3-Digit Addition and Subtraction Word Problems Sheet 2A (no regrouping)
  • 3-Digit Addition and Subtraction Word Problems Sheet 2B

4-digit Addition and Subtraction Word Problems

The word problems all involve either addition or subtraction (both with and without regrouping).

  • 4-Digit Addition and Subtraction Word Problems Sheet 1
  • 4-Digit Addition and Subtraction Word Problems Sheet 2

Harder 2-Step Addition and Subtraction Word Problems

These problems are trickier than those above.

Each problem involved two steps to solve, and may involve just addition, just subtraction or both operators.

There are 4 problems on each sheet, and more space to work out the solutions.

  • 2-Step Addition Subtraction Problems up to 3-digits Sheet 1
  • 2-Step Addition Subtraction Problems up to 4-digits Sheet 1

Looking for some easier worksheets?

Take a look at our Subtraction word problems for first graders.

On this page, your child will learn to work out basic subtraction word problems with numbers to 20.

  • 2nd Grade Addition and Subtraction Word Problems (up to 100)

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

More 3rd Grade Addition and Subtraction Worksheets

Here you will find some more of our Addition and Subtraction Worksheet generators for creating your own bespoke addition and subtraction worksheets.

  • Addition Subtraction Fact Worksheets (randomly generated)
  • Column Addition and Subtraction Worksheets (randomly generated)

More 3rd Grade Math Word Problems

Using the 3rd Grade Math worksheets will help your child to:

  • apply their addition, subtraction, and multiplication skills;
  • develop their knowledge of fractions;
  • apply their knowledge of rounding and place value;
  • solve a range of 'real life' problems.

These sheets involve solving one or two more challenging longer problems.

  • 3rd Grade Math Problems

These sheets involve solving many 'real-life' problems involving data.

  • Math Word Problems for kids 3rd Grade

These sheets involve solving 3-digit and 4-digit addition word problems.

  • Addition Word Problems 3rd Grade (3- and 4-digits)

These sheets involve solving 3-digit and 4-digit subtraction problems.

  • Subtraction Word Problems 3rd Grade

These sheets involve solving a range of multiplciation problems.

  • Multiplication Word Problem Worksheets 3rd Grade

These sheets involve solving a range of division problems.

  • Division Worksheets Grade 3 Word Problems

Addition and Subtraction Word Problems to 1000 Online Quiz

Our quizzes have been created using Google Forms.

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This quick quiz tests your knowledge and skill at solving addition and subtraction word problems within 1000.

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Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

What is solving an equation?

Common Core State Standards

How does this relate to 8 th grade and high school math?

  • Grade 8 – Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • High school – Reasoning with Equations and Inequalities (HSA.REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

  • Combine like terms .
  • Simplify the equation by using the opposite operation to both sides.
  • Isolate the variable on one side of the equation.

Solving equations examples

Example 1: solve equations involving like terms.

Solve for x.

Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2 Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3 Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

Example 2: solve equations with variables on both sides

Combine the v terms on the same side of the equation. To do this, add 8v to both sides.

7v+8v=8-8v+8v

After combining like terms, you are left with the equation 15v=8.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

Divide both sides of the equation by 15 to solve for v. This step will isolate v to one side of the equation and allow you to solve.

15v \div 15=8 \div 15

The final solution to the equation 7v=8-8v is \cfrac{8}{15} \, .

Example 3: solve equations with the distributive property

Combine like terms by using the distributive property.

The 3 outside the parentheses needs to be multiplied by both terms inside the parentheses. This is called the distributive property.

\begin{aligned}& 3 \times c=3 c \\\\ & 3 \times(-5)=-15 \\\\ &3 c-15-4=2\end{aligned}

Once the 3 is distributed on the left side, rewrite the equation and combine like terms. In this case, the like terms are the constants on the left, –15 and –4. Subtract –4 from –15 to get –19.

Simplify the equation by using the opposite operation on both sides.

The goal is to isolate the variable, c, on one side of the equation. By adding 19 to both sides, you move the constant term to the other side.

\begin{aligned}& 3 c-19+19=2+19 \\\\ & 3 c=21\end{aligned}

Isolate the variable to one side of the equation.

To solve for c, you want to get c by itself.

Dividing both sides by 3 accomplishes this.

On the left side, \cfrac{3c}{3} simplifies to c, and on the right, \cfrac{21}{3} simplifies to 7.

The final solution is c=7.

As an additional step, you can plug 7 back into the original equation to check your work.

Example 4: solve linear equations

Combine like terms by simplifying.

Using steps to solve, you know that the goal is to isolate x to one side of the equation. In order to do this, you must begin by subtracting from both sides of the equation.

\begin{aligned} & 2x+5=15 \\\\ & 2x+5-5=15-5 \\\\ & 2x=10 \end{aligned}

Continue to simplify the equation by using the opposite operation on both sides.

Continuing with steps to solve, you must divide both sides of the equation by 2 to isolate x to one side.

\begin{aligned} & 2x \div 2=10 \div 2 \\\\ & x= 5 \end{aligned}

Isolate the variable to one side of the equation and check your work.

Plugging in 5 for x in the original equation and making sure both sides are equal is an easy way to check your work. If the equation is not equal, you must check your steps.

\begin{aligned}& 2(5)+5=15 \\\\ & 10+5=15 \\\\ & 15=15\end{aligned}

Example 5: solve equations by factoring

Solve the following equation by factoring.

Combine like terms by factoring the equation by grouping.

Multiply the coefficient of the quadratic term by the constant term.

2 x (-20) = -40

Look for two numbers that multiply to give you –40 and add up to the coefficient of 3. In this case, the numbers are 8 and –5 because 8 x -5=–40, and 8+–5=3.

Split the middle term using those two numbers, 8 and –5. Rewrite the middle term using the numbers 8 and –5.

2x^2+8x-5x-20=0

Group the terms in pairs and factor out the common factors.

2x^2+8x-5x-20=2x(x + 4)-5(x+4)=0

Now, you’ve factored the equation and are left with the following simpler equations 2x-5 and x+4.

This step relies on understanding the zero product property, which states that if two numbers multiply to give zero, then at least one of those numbers must equal zero.

Let’s relate this back to the factored equation (2x-5)(x+4)=0

Because of this property, either (2x-5)=0 or (x+4)=0

Isolate the variable for each equation and solve.

When solving these simpler equations, remember that you must apply each step to both sides of the equation to maintain balance.

\begin{aligned}& 2 x-5=0 \\\\ & 2 x-5+5=0+5 \\\\ & 2 x=5 \\\\ & 2 x \div 2=5 \div 2 \\\\ & x=\cfrac{5}{2} \end{aligned}

\begin{aligned}& x+4=0 \\\\ & x+4-4=0-4 \\\\ & x=-4\end{aligned}

The solution to this equation is x=\cfrac{5}{2} and x=-4.

Example 6: solve quadratic equations

Solve the following quadratic equation.

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

To factorize a quadratic expression like this, you need to find two numbers that multiply to give -5 (the constant term) and add to give +2 (the coefficient of the x term).

The two numbers that satisfy this are -1 and +5.

So you can split the middle term 2x into -1x+5x: x^2-1x+5x-5-1x+5x

Now you can take out common factors x(x-1)+5(x-1).

And since you have a common factor of (x-1), you can simplify to (x+5)(x-1).

The numbers -1 and 5 allow you to split the middle term into two terms that give you common factors, allowing you to simplify into the form (x+5)(x-1).

Let’s relate this back to the factored equation (x+5)(x-1)=0.

Because of this property, either (x+5)=0 or (x-1)=0.

Now, you can solve the simple equations resulting from the zero product property.

\begin{aligned}& x+5=0 \\\\ & x+5-5=0-5 \\\\ & x=-5 \\\\\\ & x-1=0 \\\\ & x-1+1=0+1 \\\\ & x=1\end{aligned}

The solutions to this quadratic equation are x=1 and x=-5.

Teaching tips for solving equations

  • Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.
  • Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.
  • Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.
  • Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

Easy mistakes to make

  • Forgetting to distribute or combine like terms One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.
  • Misapplying the distributive property Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.
  • Failing to perform the same operation on both sides It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.
  • Making calculation errors Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.
  • Ignoring fractions or misapplying operations When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Solve equations with fractions
  • Linear equations
  • Writing linear equations
  • Substitution
  • Identity math
  • One step equation

Practice solving equations questions

1. Solve 4x-2=14.

GCSE Quiz False

Add 2 to both sides.

Divide both sides by 4.

2. Solve 3x-8=x+6.

Add 8 to both sides.

Subtract x from both sides.

Divide both sides by 2.

3. Solve 3(x+3)=2(x-2).

Expanding the parentheses.

Subtract 9 from both sides.

Subtract 2x from both sides.

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

Multiply by 6 (the lowest common denominator) and simplify.

Expand the parentheses.

Subtract 4 from both sides.

Subtract 3x from both sides.

5. Solve \cfrac{3 x^{2}}{2}=24.

Multiply both sides by 2.

Divide both sides by 3.

Square root both sides.

6. Solve by factoring:

Use factoring to find simpler equations.

Set each set of parentheses equal to zero and solve.

x=3 or x=10

Solving equations FAQs

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

The next lessons are

  • Inequalities
  • Types of graph
  • Coordinate plane

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