grade 3 problem solving math

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.

Author Katie Wickliff

Published September 25, 2023

• 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. 

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Sample questions, 35 third grade math problems, section 1: addition.

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

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

How can you represent the number line with an equation?

Using the number line, solve 7+ _ = 22

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=

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

Section 3: Multiplication

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

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?

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

Solve the equation

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

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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

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.

Plot ¼ on this number line:

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?

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:

Find the area of the square below

Find the perimeter of the rectangle below

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?

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Addition answers

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.

8×4= 32

Division answers

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. 

Lesson credits

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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50 Third Grade Math Word Problems of the Day

The area of Happytown is 42 square miles.

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.

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

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

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

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?

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?

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

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?

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?

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?

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?

12. 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?

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.

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?

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?

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?

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

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?

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

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?

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?

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?

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?

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?

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

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?

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

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?

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?

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.

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?

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

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

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?

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

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?

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

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?

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?

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?

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?

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

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?

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?

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?

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?

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?

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

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?

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

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Examples of 3rd Grade Math Word Problems with Solutions

In this post, we are going to look at a selection of 3rd grade math word problems in Smartick.

We’ll begin by explaining what a word problem is.

What is a Word Problem?

A word problem is a question or approach, that cannot be answered using one specific operation, but rather a combination of previous knowledge whether mathematical or not.

Solving a math problem is cognitively complex since it involves understanding the situation presented in the word problem, as well as comprehending the message and words used in the word problem. Therefore, at first, to face a math problem is to face a problem of linguistic understanding . Once the word problem is understood, you must choose which, or what, procedures you are going to use to reach a solution (mathematical understanding).

This type of activity helps a child to connect mathematics to the world around them and by doing this, they are able to apply what they have learned to everyday life. To solve a problem, it is not only necessary to master the specific knowledge related to the problem, but also plan and perform some procedures to reach the solution.

3rd Grade Math Word Problems

To assess mathematical competence in 3rd grade math word problems, there are some cognitive processes to keep in mind that fall into these three groups:

• Know and produce . It refers to the knowledge of the basic mathematical language, essential mathematical properties and facts, and the repetition of calculation algorithms practiced, linking them to familiar math processes and problems that include basic operations. This is required for even the most basic exercises.
• Apply and analyze . Involves knowing how to use different mathematical tools and establishing relationships between diverse situations in relatively well-known contexts, selecting problem-solving strategies, performing calculations, and obtaining solutions. This is necessary for exercises that are slightly more difficult.
• Reason and reflect . It requires the ability to think logically and systematically, and involves analyzing the results obtained in solving a problem and reflecting upon the process used. It refers to the interpretation of the results according to the context, communication of the method used, and the result obtained appropriately. It implies a higher degree of competitive acquisition.

Before starting practicing, I would recommend that you try to solve the 3rd grade math word problems on your own, without looking at the solutions. Give it a try!

Word Problem 1

After reading the word problem we need to make sure that we understand the information given.

• The lasagna will be cut into equally sized pieces.
• Catherine and Noah will eat different sized portions.
• Noah will eat 1/4 more than Catherine.

Since we know that Catherine will eat a portion of 1 1/2 (1 full piece and one 1/2 piece) we just need to find out how large Noah’s portion will be. Let’s look at the fraction, how many times can 1/4 go into 1/2? Two times because there are 2 halves to a whole and 4 quarters to a whole, therefore 1/2 = 2/4. Now that we know Catherine will eat 1 2/4 and Noah will eat 1/4 more, 2/4 + 1/4 = 3/4.

Noah will eat 1 3/4 pieces of lasagna .

Word Problem 2

We have been asked to change only one number in the word problem so that the number of cards that Alexander has at the end of the first part of the game is 6. We are told that he has won 17 cards during the second half and finishes the game with 28 cards. In order to know how many cards he had in the first part we should subtract what he has won during the second part from the total number of cards he ended the game with, in other words, 28 – 17 = 11. Since we have been asked to make sure the answer is 6 and can only change one data, we have two valid options.

• Change the number of cards he won during the second part, if he won 17 and we add 5, 17 +5 = 22. Then when we subtract that number from the total, 28 – 22 = 6 , it gives us the number we need to solve the word problem. Alexander had 6 cards at the end of the first part of the game.
• Change the number of cards that he had at the end of the game, subtracting 5 from the 28 cards, 28 – 5 = 23 . Then subtract the number of cards he won during the second part, 17, from the total, 23 – 17 = 6 . Alexander had 6 cards at the end of the first part of the game.

Sometimes math problems have more than one solution.

Word Problem 3

This problem is very easy because the table shown has all of the information.

• The first thing we need to know is the total amount of sandwiches , of all varieties, served yesterday, both day and night. Put simply, we need to add 16 + 54 = 70 sandwiches .
• Next, we need to fill in the empty space for the Fisherman’s sandwich . In order to know how many were served in the evening , we need to subtract the number of sandwiches served during the day from the total .
• 54 – 27 = 27 Fisherman’s sandwiches we served yesterday evening in the cafeteria where Catherine works.

Word Problem 4

First of all, we want to find out the total number of children that are on the bus. To do this, we just need to complete a very simple operation. We know that there are 6 children on the bus and the bus makes 4 trips, so we can do 6 x 4 = 24 . Now we know that there are a total of 24 children on the bus during the day . Next, we need to determine how many children are dropped off at each school. The word problem tells us that the same number of children are dropped off at each school , there are two schools, so let’s divide by 2!

24 / 2 = 12 , the bus takes 12 children to each school.

If you would like to continue practicing more 3rd grade math word problems and other topics from primary mathematics, adapted to your level, register with Smartick and try it free!

Learn More:

• Examples of Second Grade Word Problems with Solutions
• How to Solve Addition Problems
• Fraction Word Problems: Addition, Subtraction, and Mixed Numbers
• Grade 5 Math Word Problems with Smartick
• Working on Combined Operations through Word Problems
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Printable 3rd Grade Math Word Problem Worksheets

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" class="arrow-title-img">3rd Grade math word problems with answers - Steps for math problem solving in Grade 3

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Importance of math word problems online practice for 3rd graders

Get more contents on third grade..., math problem solving for grade 3 with solutions and explanations.

We have given our grade 3 math word problems with answers to strengthen your kid’s math concept skills by relating them to everyday life. Because problem-solving requires over one step, we have planned short and straightforward steps in problem-solving in math Grade 3 . These steps will go a long way to enhance your kid’s speed and accuracy in solving many math problems in class and in the real world.

Besides this, our math word problems for grade 3 are straightforward and familiar to our everyday life, thus offering your kids remarkable ways to practice and apply math concepts to everyday situations by providing high-quality resources for math p roblem solving for grade 3 with solutions and explanations .

• Adding three numbers up to four digits word problems Grade 3
• Adding three numbers up to three digits word problems 3rd Grade
• Adding two numbers up to four digits word problems Grade 3
• Adding two numbers up to three digits word problems 3rd grade
• Dividing larger numbers by one digit numbers word problems grade 3
• Dividing three digits numbers by one digit numbers word problems grade 3
• Dividing two digits numbers by one digit numbers word problems grade 3
• Fractions Worksheets And Online Exercises With Answers
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In this first place, you’ll realize that math problem solving will encourage your kids to believe in their ability to always think mathematically, when finding solutions in the real world.

Also, apart from being an enjoyable experience for kids, our grade 3 math problem solving online will provide your kids will exciting strategies vital for solving all grade 3 math word problems always.

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Daily Math Word Problems - Grade 3 (Worksheets)

Updated:  20 Jul 2019

A set of 20 problem solving questions suited to grade 3 students.

Non-Editable:  PDF

Pages:  6 Pages

• Curriculum Curriculum:  CCSS, TEKS
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CCSS.MATH.CONTENT.3.OA.A.3

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

CCSS.MATH.CONTENT.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 roundi...

Math 3.4(A)

Solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction;

Math 3.4(K)

Solve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.

This set of problem solving questions has been designed to support teachers when teaching students about problem solving in mathematics.

It provides students with the opportunity to work through 20 math word problems , identifying the important information and how they can work it out using a variety of methods.

An answer sheet has been included.

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Daily math problems - grade 3.

A 63 slide editable PowerPoint Template for problem solving in mathematics.

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A worksheet to use to practice place value to the thousands place.

Identifying and Naming Angles – 4th Grade Math Worksheet

Identify acute, right, obtuse, straight, reflex and revolution angles with this cut-and-paste sorting worksheet.

Acute, Right and Obtuse Angles – 4th Grade Math Worksheet

Identify acute, right and obtuse angles with this cut-and-paste sorting worksheet.

Location Math Investigation - Blackbeard's Bounty

A mathematics investigation about location, embedded in a real-world context.

Multiplying and Dividing By 1,000 Poster

A set of posters to display in the classroom when learning to multiply and divide by 1,000.

Multiplying by 1,000 Worksheet

A worksheet for students to complete when learning how to multiply by 1,000.

Dividing by 1,000 Worksheet

A worksheet for students to complete when learning to divide by 1,000.

2-D Shape Bingo

Engage your students while consolidate learning about 2D shapes, their names and properties with 2D Shape Bingo!

Multiplication Facts PowerPoint - One Times Tables

A 28-slide PowerPoint to use when learning about multiplication.

Multiplication Facts PowerPoint - Two Times Tables

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Multiplication Word Problem Worksheets 3rd Grade

Welcome to our Multiplication Word Problem Worksheets for 3rd Grade. Here you will find our range of printable multiplication problems which will help your child apply and practice their multiplication and times tables skills to solve a range of 'real life' problems.

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Multiplication Word Problems up to 10x10 Online Quiz

Multiplication word problem worksheets, 3rd grade multiplication problems.

Here you will find a range of problem solving worksheets involving multiplication.

Each sheet involves solving a range of written multiplication problems.

There are 3 levels of difficulty for each worksheet below: A,B and C.

Worksheet A is the easiest level, suitable for children at the beginning of their grade.

Worksheet B is a medium level worksheets for children who are working at the expected level in their grade.

Worksheet C is set at a harder level, suitable for children who are more able mathematicians.

The problems in each worksheet are similar in wording, but the numbers involved become trickier as the level gets harder.

To encourage careful checking and thinking skills, each sheet includes one 'trick' question which is not a multiplication problem. Children need to spot this word problem, and work out which operation they need to solve it.

Using these sheets will help your child to:

• apply their multiplication and times tables skills at a 3rd grade level;
• apply their times table knowledge to work out related facts;
• recognise multiplication problems, and try to spot 'trick' problems;
• solve a range of 'real life' problems.

Some of the sheets have a UK version with spelling set for the UK.

3rd Grade Multiplication Word Problem Worksheets

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Series 3 Sheet 1 Set

• Series 3 Sheet 1A (easier)
• Series 3 Sheet 1B (medium)
• Series 3 Sheet 1C (hard)

• PDF version Series 3.1 (6 sheets)

Series 3 Sheet 2 Set

• Series 3 Sheet 2A (easier)
• Series 3 Sheet 2B (medium)
• Series 3 Sheet 2C (hard)
• PDF version Series 3.2 (6 sheets)
• PDF version Series 3.2 UK version (6 sheets)

Series 3 Sheet 3 Set

• Series 3 Sheet 3A (easier)
• Series 3 Sheet 3B (medium)
• Series 3 Sheet 3C (hard)
• PDF version Series 3.3 (6 sheets)

Series 3 Sheet 4 Set

• Series 3 Sheet 4A (easier)
• Series 3 Sheet 4B (medium)
• Series 3 Sheet 4C (hard)
• PDF version Series 3.4 (6 sheets)

Looking for some easier Multiplication Problems?

In our 2nd Grade Multiplication word problem area, you will find a range of multiplication word problems aimed at 2nd graders.

The following areas are covered:

• basic multiplication fact sheets;
• multiplication facts to 5x5;
• word problems involving the 2, 5 and 10 times tables.
• Multiplication Word Problems 2nd Grade

Looking for some harder Multiplication Problems?

In our 4th Grade Multiplication word problem area, you will find a range of multiplication word problems aimed at 4th graders.

• multiplication related facts to 10x10 e.g. 6 x 70;
• problems needing written multiplication methods to solve (e.g. HTU x U; TU x TU)
• Multiplication Word Problems 4th Grade

More Recommended Math Worksheets

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

Looking for more 3rd Grade Word Problems?

Here are a range of problems solving sheets for 3rd graders. Most of the sheets contain 'real-life' problems which involve real data.

Using the sheets 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;

All the 3rd grade sheets in this section support Elementary math benchmarks.

• Math Word Problems for kids 3rd Grade
• 3rd Grade Addition and Subtraction Word Problems (3- and 4-digits)

Finding all Possibilities Problems

This is our finding all possibilities area where all the worksheets involve finding many different answers to the problem posed.

The sheets here encourage systematic working and logical thinking.

The problems are different in that, there is typically only one problem per sheet, but the problem may take quite a while to solve!

• Finding all Possibilities problems
• Math Logic Problems

This is our logic problems area where all the worksheets involve using reasoning and logical thinking skills.

The sheets here are designed to get children thinking logically and puzzling the problems out.

There are a range of different logic problems from 1st through 5th grade!

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This quick quiz tests your knowledge and skill at solving multiplication word problems up to 10x10.

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75 Word Problems for 3rd Graders: Ideas For Addition, Subtraction, And More

December 17, 2023 //  by  Seda Unlucay

Develop your 3rd Graders’ math skills with these captivating word problems! Math can be daunting for any individual, so why not add some colorful visuals to engage your kids as you introduce them to the art of problem-solving?

These multi-step word problems incorporate addition , subtraction , multiplication , and division as well as time , money, and fractions – to make the perfect playground for your learners to test their mathematical skills!

1. Jennifer picked 72 cherries and Kim picked 45 cherries. They used 24 cherries to make pies for their school bake sale. How many cherries do they have left?

2. kim had 19 candies and then she bought 23 more candies. she wants to share them between herself and 6 friends. how many candies will each friend get, 3. andrew has 147 marbles. 35 marbles are orange and 52 are purple. the rest of the marbles are yellow. how many yellow marbles are there, 4. sandra and her friend, brenda went shopping. they each bought 10 new dolls. sandra returned 3 of her new dolls to the store. how many dolls do sandra and brenda still have, 5. lauren has 600 pencils. she wants to place them in 10 equal groups. how many pencils will be in each group, 6. stanley and eddy each bought 12 slices of pizza. for dinner, they each ate 2 slices. how many slices do they have now, 7. jim planted 30 rows of 15 tulips. 137 of them are yellow and the rest are red. how many red tulips are there, 8. megan has 8 quarters, 4 dimes, and 7 nickels for bus fare. if a bus ticket costs $1.15 how much money will she have left, 9. sam has 63 stamps from asia, 59 stamps from europe, and 162 stamps from africa in his collection. how many more stamps does he have from africa than asia and europe combined, 10. angie used 3 red ornaments, 5 blue ornaments, and 7 green ornaments to decorate a christmas tree. she has 12 ornaments left. how many ornaments did she have to begin with, 11. jenny and her friends bought 3 boxes of cupcakes. there were 16 cupcakes in each case. james ate 3 cupcakes, stewart ate 5 cupcakes and kim ate 13 cupcakes. how many cupcakes are left, 12. tom finished a 354-piece jigsaw puzzle and stella finished a 567-piece jigsaw puzzle. how many fewer pieces did tom’s puzzle have, 13. stephanie has $217 and derek has $138 to spend. they spend some money and now they have $112 left. how much money did they spend, 14. cassandra ran 15 miles each day for 8 days. then she ran 12 miles each day for two weeks. how many miles did she run altogether, 15. andy’s pencil case weighs 32 grams. his notebook weighs 45 grams more than his pencil case. what is the total weight of his pencil case and notebook, 16. daniel bought 4 packs of chewing gum. there are 9 pieces of gum in each pack. he wants to share the gum evenly with 3 people. how many pieces of gum will each person get, 17. jen climbed 48 steps up to a diving board. she climbed down 5 steps to talk to a friend. then she climbed 12 steps to get to the top. how many steps does the diving board have, 18. there are 78 balls on the playground. 22 are soccer balls and 18 are basketballs. the rest are tennis balls. how many tennis balls are there, 19. tommy made 63 cookies for the bake sale. lindsay made 35 cookies. they sold 22 cookies altogether. how many cookies do they have left, 20. adam found 235 pennies on the playground. he spent 98 pennies. then he found 123 more. how many pennies does he have now, 21. lisa saw 86 animals at the zoo. she saw 54 monkeys, 17 parrots, and some elephants. how many elephants did she see, 22. julia has a crayon collection of 156 crayons. she gave some to her friend emily. now she has 72 crayons left. how many crayons did she give emily, 23. sandy had $225 to buy bracelets. the store was selling 2 packs of bracelets for $5. how many packs can sandy afford to buy, 24. brandon earned $12 an hour mowing the lawn and $15 an hour babysitting. he worked 20 hours mowing the lawn and 18 hours babysitting. how much money did he earn in total, 25. gavin bought 14 pencils. he had $48 before he bought the pencils. after he bought the pencils, he had $20 left. how much did each pencil cost, 26. tina won 160 teddy bears at the carnival. she gave 8 to each of her friends. then she had 32 left. how many friends did she give teddy bears to, 27. billy sold half of his trading cards and then he bought 132 more. now he has 325 trading cards. how many did he have to start with, 28. lacy’s school is going on a field trip. there are 24 kids in each class. there are 8 classes. if 30 kids can fit on a bus, how many buses will they need for their field trip, 29. stephanie had 5 dozen cupcakes. she gave 27 cupcakes to her friend. how many cupcakes does she have left, 30. angela has 1345 stickers. dan has 845 stickers. how many more stickers does angela have than dan, 31. ms. smith went grocery shopping. her groceries cost $82.96. she had coupons worth $22.50. if she paid the clerk $90 for her groceries, how much change would she get, 32. serena had $77. then she bought movie tickets for 8 friends for $4 each. now she wants to buy popsicles that cost $3 each. how many popsicles can she buy, 33. sam had $34. then he received $19 for his birthday. how much money does he need if he wants to buy a bike that costs $98, 34. miranda bought 4 necklaces that cost $13 each. then she gave $16 to her little brother. she started with $105. how much money does she have now, 35. anthony earns $15 every week he does chores. he does chores for 6 weeks. now he wants to buy a gaming console for $114. how much more money does he need, 36. how many sides do 3 triangles, 8 squares and 4 rectangles have, 37. emily saw some chickens on the farm. she counted 56 wings in total. how many chickens did she see, 38. ben picked 18 berries. zane picked 6 times as many berries as ben. how many berries did zane pick, 39. gavin bought 70 apples. he bought twice as many apples as tim. how many apples did tim buy, 40. anita planted 10 rows of carrots with 7 carrots in each row. how many carrots did she plant, 41. a dozen donuts cost $5.50. how much do 7 dozen donuts cost, 42. jennifer made 23 cups of potato salad for the school picnic. at the end of the picnic, there were 4 and a quarter cups of potato salad left. how many cups of potato salad were eaten, 43. emily wants to buy a jar of marbles that costs $2. she has 7 quarters, 5 dimes, and 3 nickels to spend. how much change will she get back, 44. use the fewest bills and coins to make $25.33., 45. mary spent $325 on decorations for her birthday party. she spent $123 on food. how much more money did she spend on decorations than food, 46. there are 74 students in third grade. 23 are in ms. smith’s class, 19 are in ms. park’s class and the rest are in ms. anderson’s class. how many more students are in ms. anderson’s class than in ms. smith’s class, 47. how many days are there in 4 full weeks, 48. rob is 3 years older than his friend andy. andy is twice as old as rob’s brother who is 4 years old. how old is rob, 49. sandy can bake 36 cookies in 30 minutes. how many cookies can she bake in 8 hours, 50. mandy is reading a book with 313 pages in it. she read 54 pages on saturday. then she read 72 more pages on monday afternoon. how many pages does mandy have left to read, 51. there are 35 students in stanley’s class. ⅕ of them bike to school. how many students don’t bike to school, 52. there are 250 animals in a zoo. ⅗ of them are herbivores. how many animals are herbivores, 53. danny is reading a book with 120 pages. he has already read ⅓ of it. how many more pages does he have left to read, 54. jen had $36 to spend. she spent a ¼ on candies and ⅓ on stickers. how much money does she have left, 55. sam brought 80 cupcakes to school. his class ate ¼ of them and ms. smith’s class ate ⅕ of them. how many cupcakes were eaten in all, 56. amy and laura each collected 15 seashells at the beach. amy gave 5 of her seashells to laura. how many seashells do amy and laura have now, 57. max bought 6 comic books. each book costs $3. he started with $25. how much money does max have left, 58. peter has a total of 128 trading cards. 39 of them are from baseball and 56 are from football. the rest are from basketball. how many basketball trading cards does peter have, 59. linda and sue bought 14 bags of candies. each bag contains 10 candies. after a party, there are only 35 candies left. how many candies did they have at the start, 60. bella has a collection of 200 stickers. she gives 8 stickers to each of her friends and then has 80 stickers left. how many friends did she give stickers to, 61. richard has 48 pieces of chewing gum. he gives an equal amount to each of his 4 friends. how many pieces of gum does each friend receive, 62. molly has 9 packs of paper. each pack has 200 sheets. she used some for a project and now has 1050 sheets left. how many packs of paper did she use for her project, 63. david has 72 toy cars. 16 are blue, 30 are red, and the rest are yellow. how many yellow cars does david have, 64. helen baked 150 cupcakes for a bake sale. during the sale, 67 cupcakes were sold and the rest were given away. how many cupcakes were given away, 65. jacob is reading a 600 page book. he read 60 pages each day for the first 4 days. after that, he read 40 pages each day for 8 days. how many pages does he still have to read, 66. fred and george each bought 5 packets of trading cards. each packet contains 15 cards.  if they gave 12 cards to their friend ron, how many cards do they have now, 67. olivia has 450 flowers in her garden. she wants to arrange them in 9 equal bouquets. how many flowers will be in each bouquet, 68. abigail and natalie have $85 to buy presents for their friends. they spent $30 on books and the rest on art supplies. how much did they spend on art supplies, 69. paul has $200 saved. he wants to buy a toy that costs $15. if he buys the toy, how much money will he have left, 70. mr. turner teaches a class of 30 students. one-third of them play basketball and the rest play soccer. how many students play soccer, 71. ava saw some birds in the park. she counted 48 legs in total. how many birds did she see, 72. ethan picked 21 strawberries. his brother, aaron picked twice as many strawberries as ethan. how many strawberries did aaron pick, 73. mandy bought 80 marbles. she bought four times as many marbles as tim. how many marbles did tim buy, 74. sarah planted 12 rows of roses with 5 roses in each row. how many roses did she plant, 75. a dozen balloons cost $6. how much do 9 dozen balloons cost.

Problem Solving Strategies for Grade 3

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Nonlinear system of equations

Here you will learn about a nonlinear system of equations, including what they are and how to solve them.

Students will first learn about a nonlinear system of equations as part of algebra in high school.

What is a nonlinear system of equations?

A nonlinear system of equations is two or more equations that form a system, where at least one equation is nonlinear.

One example of a nonlinear equation is a quadratic equation , which has variables that are raised to powers of 2, for example, x^2 and y^2.

There are other types of nonlinear equations, but this page will cover simple systems that involve a linear and a quadratic equation.

One key difference of a nonlinear system of equations compared to linear system of equations is that you can expect multiple answers . This is because of the way the graphs of linear and quadratic functions can intersect.

On the graph below, the straight line of the linear equation crosses the curved parabola of the quadratic equation at two points of intersection .

This means the system of equations has two valid answers .

This is an example of a nonlinear system of equations:

y=x^{2}-4 (quadratic)

y=x+2 (linear)

These points of intersection are the solutions to the system of equations. They are ordered pairs that satisfy all equations in the system.

Here, the solutions are x=3, \, y=5 and x=- \, 2, \, y=0. This can also be written as a coordinate pair (3, \, 5) and (- \, 2, \, 0).

What is a Nonlinear System of Equations?

Common Core State Standards

How does this relate to high school math?

• Algebra – Reasoning with Equations and Inequalities (HS.A.REI.C.7) Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=- \, 3x and the circle x^{2}+y^{2}=3.

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

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

How to solve nonlinear system of equations

To solve a set of system of equations you need to:

Use substitution to eliminate one of the variables.

Solve for the value(s) of one variable.

Use substitution to find the value(s) of the remaining variable.

Clearly state the final answer(s).

Check your answer by substituting both values into either of the original equations.

Nonlinear system of equations examples

Example 1: equation of a straight line and a quadratic (factor).

Use the fact that y=x+3 to substitute the value of y into the second equation.

Therefore x+3=x^{2}+5x-2.

Another way to think about this is that as both equations are equal to y, they must therefore be equal to one another.

2 Solve for the value(s) of one variable.

Since the equation is now quadratic, rearrange so that the equation is equal to 0.

Then solve:

x^{2}+4x-5=0 can be factored:

x+5=0 so x=- \, 5

x-1=0 so x=1

The two values of x are x=- \, 5 and x=1.

3 Use substitution to find the value(s) of the remaining variable.

The two values of x can be substituted into one of the original equations to find the two possible values of y.

Remember you can use either equation, so why not pick the easiest!

As y=x+3, when x=- \, 5, \, y=- \, 5+3=- \, 2.

As y=x+3, when x=1, \, y=1+3=4.

4 Clearly state the final answer(s).

x=- \, 5, \, y=- \, 2 and x=1, \, y=4.

5 Check your answer by substituting both values into either of the original equations.

Let’s use the quadratic equation to check the answers.

As y=x^{2}+5x-2, when x=- \, 5,

As y=x^{2}+5x-2, when x=1,

Both are valid so the answers are correct.

Let’s look at the graph of the system.

When graphed, these two equations intersect at (– \, 5, \, – \, 2) and (1, \, 4). So the solutions to the system of equations are:

Example 2: equation of a straight line and a quadratic (factor)

Use the fact that y=3x-12 to substitute the value of y into the second equation.

\begin{aligned}y&=3x-12 \\\\ y&=x^{2}-2x-6 \end{aligned}

Therefore 3x-12=x^{2}-2x-6.

x^{2}-5x+6=0 can be factored:

(x-3)(x-2)=0

x-3=0 so x=3

x-2=0 so x=2

The two values of x are x=3 and x=2.

As y=3x-12, when x=3, \, y=3(3)-12=- \, 3.

As y=3x-12, when x=2, \, y=3(2)-12=- \, 6.

x=2, \, y=- \, 6 and x=3, \, y=- \, 3.

As y=x^{2}-2x-6, when x=3,

\begin{aligned}y&=(3)^{2}-2(3)-6 \\\\ &=9-6-6 \\\\ &=- \, 3 \end{aligned}

As y=x^{2}-2x-6, when x=2,

\begin{aligned}y&=(2)^{2}-2(2)-6 \\\\ &=4-4-6 \\\\ &=- \, 6 \end{aligned}

Both are valid so the answer is correct.

When graphed, these two equations intersect at (2, \, – \, 6) and (3, \, – \, 3). So the solutions to the system of equations are:

Example 3: equation of a straight line and a circle (factor)

Use y=x+7 to substitute the value of y into the first equation:

(x+7)^{2}+x^{2}=29

x^{2}+7x+10=0 can be factored:

(x+5)(x+2)=0

x+2=0 so x=- \, 2

The two values of x are x=- \, 5 and x=- \, 2.

As x+7=y, when x=- \, 5, \, y=- \, 5+7=2.

As x+7=y, when x=- \, 2, \, y=- \, 2+7=5.

x=- \, 5, \, y=2 and x=- \, 2, \, y=5.

Let’s use the equation of the circle to check the answers.

Substitute x=- \, 5, \, y=2 into y^2+x^2=29\text{:}

2^{2}+(- \, 5)^{2}=4+25=29.

Substitute x=- \, 2, \, y=5 into y^2+x^2=29\text{:}

5^{2}+(- \, 2)^{2}=25+4=29.

Let’s look at the graph of the system:

When graphed these two equations intersect at (– \, 2, \, 5) and (- \, 5, \, 2), so the solutions to the system of equations are:

Notice that the graph of y^2+x^2=29 is a circle!

Example 4: equation of a straight line and a circle (quadratic formula)

Use x=y+2 to substitute the value of x into the first equation:

y^2+(y+2)^2=14

This equation does not factor, so use the quadratic formula to determine the value(s) of x\text{:}

y^{2}+2y-5=0 with a=1, \, b=2, \, c=- \, 5

\begin{aligned}y&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, 2\pm\sqrt{2^{2}-4(1)(- \, 5)}}{2(1)} \\\\ &=\cfrac{- \, 2\pm\sqrt{4+20}}{2} \\\\ &=\cfrac{- \, 2\pm2\sqrt{6}}{2} \end{aligned}

y=- \, 1+\sqrt{6} or y=- \, 1-\sqrt{6}

The two values of y can be substituted into one of the original equations to find the two possible values of x.

As x=y+2, when y=-1+\sqrt{6}\text{:}

x=- \, 1+\sqrt{6}+2=1+\sqrt{6}

As x=y+2, when y=- \, 1-\sqrt{6}\text{:}

x=- \, 1-\sqrt{6}+2=1-\sqrt{6}

Substitute x=1+\sqrt{6},~y=- \, 1+\sqrt{6} into y^{2}+x^{2}=14\text{:}

\begin{aligned}&(- \, 1+\sqrt{6})^{2}+(1+\sqrt{6})^{2} \\\\ &=(- \, 1+\sqrt{6})(-1+\sqrt{6})+(1+\sqrt{6})(1+\sqrt{6}) \\\\ &=(1-\sqrt{6}-\sqrt{6}+6)+(1+\sqrt{6}+\sqrt{6}+6) \\\\ &=7-2\sqrt{6}+7+2\sqrt{6} \\\\ &=14. \end{aligned}

Substitute x=1-\sqrt{6},~y=- \, 1-\sqrt{6} into y^{2}+x^{2}=14\text{:}

\begin{aligned}&(- \, 1-\sqrt{6})^{2}+(1-\sqrt{6})^{2} \\\\ &=(- \, 1-\sqrt{6})(-1-\sqrt{6})+(1-\sqrt{6})(1-\sqrt{6}) \\\\ &=(1+\sqrt{6}+\sqrt{6}+6)+(1-\sqrt{6}-\sqrt{6}+6) \\\\ &=7+2\sqrt{6}+7-2\sqrt{6} \\\\ &=14.\end{aligned}

When graphed these two equations intersect at (1+\sqrt{6}, \, - \, 1+\sqrt{6}) and (1-\sqrt{6}, \, - \, 1-\sqrt{6}), so the solutions to the system of equations are:

x=1+\sqrt{6}, \, y=- \, 1+\sqrt{6} and x=1-\sqrt{6}, \, y=- \,1-\sqrt{6}

Notice that the graph of y^2+x^2=14 is a circle!

Example 5: equation of a straight line and a hyperbola (quadratic formula/factor)

Solve the system of equations:

First, make y the subject of the formula in the second equation.

\begin{aligned}3x+4y&=7 \\\\ 4y&=7-3x \\\\ y&=\cfrac{7-3x}{4} \end{aligned}

Then, use the fact that y=\cfrac{7-3x}{4} to substitute the value of y into the first equation:

2x^{2}-8\left(\cfrac{7-3x}{4}\right)^{2}=18.

\begin{aligned}2x^{2}-8\left(\cfrac{7-3x}{4}\right)^{2}&=18 \\\\ 2x^{2}-8\left(\cfrac{7-3x}{4}\right)\left(\cfrac{7-3x}{4}\right)&=18 \\\\ 2x^{2}-8\left(\cfrac{49-21x-21x+9x^{2}}{16}\right)&=18 \\\\ 2x^{2}-8\left(\cfrac{49-42x+9x^{2}}{16}\right)&=18 \\\\ 2x^{2}- \cfrac{49-42x+9x^{2}}{2}&=18 \\\\ 4x^{2}-\left(49-42x+9x^{2}\right)&=36 \\\\ 4x^{2}-49+42x-9x^{2}&=36 \\\\ - \, 49+42x-5x^{2}&=36 \\\\ 42x-5x^{2}&=85 \\\\ - \, 5x^{2}&=- \, 42x+85 \\\\ 0&=5x^{2}-42x+85 \end{aligned}

This equation appears harder to factor, so use the quadratic formula:

5x^{2}-42x+85=0 where a=5, \, b =- \, 42, \, c= 85

\begin{aligned}x&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, (- \, 42)\pm\sqrt{(- \, 42)^{2}-4(5)(85)}}{2(5)} \\\\ &=\cfrac{42\pm\sqrt{1764-1700}}{10} \\\\ &=\cfrac{42\pm\sqrt{64}}{10} \\\\ &=\cfrac{42\pm{8}}{10} \end{aligned}

x=3.4 or x=5

Notice that the solutions can be decimals as well as integers.

Since there are two values of x, substitute both values into one of the original equations and find the two possible values of y.

As 3x+4y=7, when x=3.4,

\begin{aligned}3(3.4)+4y&=7 \\\\ 10.2+4y&=7 \\\\ 4y&=- \, 3.2 \\\\ y&=- \, 0.8 \end{aligned}

As 3x+4y=7, when x=5,

\begin{aligned}3(5)+4y&=7 \\\\ 15+4y&=7 \\\\ 4y&=- \, 8 \\\\ y&=- \, 2 \end{aligned}

x=3.4, \, y=- \, 0.8 and x=5, \, y=- \, 2.

As 2x^{2}-8y^{2}=18 where x=3.4, \, y=- \, 0.8,

2(3.4)^{2}-8(- \, 0.8)^{2}=23.12-5.12=18

As 2x^{2}-8y^{2}=18 where x=5, \, y=- \, 2,

2(5)^{2}-8(- \, 2)^{2}=50-32=18

Let’s look at part of the graph of the system.

When graphed, these two equations intersect at (3.4, \, – \,0.8) and (5, \, – \, 2), so the solutions to the system of equations are:

Example 6: equation of a straight line and an ellipse (quadratic formula)

\begin{aligned}7x+y&=28 \\\\ y&=28-7x \end{aligned}

Then, use the fact that y=28-7x to substitute the value of y into the first equation:

x^{2}+8\left(28-7x\right)^{2}=20.

\begin{aligned}x^{2}+8(28-7x)^{2}&=20 \\\\ x^{2}+8(28-7x)(28-7x)&=20 \\\\ x^{2}+8(784-196x-196x+49x^{2})&=20 \\\\ x^{2}+8(784-392x+49x^{2})&=20 \\\\ x^{2}+6272-3136x+392x^{2}&=20 \\\\ 393x^{2}-3136x+6272&=20 \\\\ 393x^{2}-3136x+6252=0 \end{aligned}

This equation would be very hard to factor, so use the quadratic formula:

393x^{2}-3136x+6252=0 where a=393, \, b=- \, 3136, \, c=6252

\begin{aligned}x&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, (- \, 3136)\pm\sqrt{(- \, 3136)^{2}-4(393)(6252)}}{2(393)} \\\\ &=\cfrac{3136\pm\sqrt{9834496-9828144}}{786} \\\\ &=\cfrac{3136\pm\sqrt{6352}}{786} \\\\ &=\cfrac{3136\pm4\sqrt{397}}{786} \end{aligned}

x=4.091220656… or x=3.88842311…

As 7x+y=28, when x=4.091 \, (3dp),

\begin{aligned}7(4.091)+y&=28. \\\\ 28.637+y&=28 \\\\ y&=- \, 0.637 \end{aligned}

As 7x+y=28, when x=3.888 \, (3dp),

\begin{aligned}7(3.888)+y&=28. \\\\ 27.216+y&=28 \\\\ y&=0.784 \end{aligned}

x=4.091, \, y=- \, 0.637 and x=3.888, \, y=0.784.

As x^{2}+8y^{2}=20 where x=4.091, \, y=-0.637,

(4.091)^{2}+8(- \, 0.637)^{2}=16.736281+3.246152=20 (nearest integer)

As x^{2}+8y^{2}=20 where x=3.888, \, y=0.784,

(3.888)^{2}+8(0.784)^{2}=15.116544+4.917248=20 (nearest integer)

When graphed, these two equations intersect at (4.091, \, - \, 0.637) and (3.888, \, 0.784), so the solutions to the system of equations are:

Notice that the graph of x^{2}+8y^{2}=20 is an ellipse!

Teaching tips for nonlinear system of equations

• Explore systems of linear equations before beginning this topic. It is also a good idea to review skills like changing the subject, operating with square roots and expanding quadratic equations, before asking students to solve systems of nonlinear equations.
• Show students a variety of strategies, such as the substitution method or the elimination method, but never require students to solve in a certain way (unless necessary on an assessment). Letting students decide how to solve helps them understand the methods in their own time and allows them to develop analytical skills, such as thinking about which way to solve would be the most efficient.
• For students to truly be proficient at solving nonlinear systems, they need to see equations of all kinds, not just equations in standard form.

Easy mistakes to make

• Forgetting that squaring a negative number results in a positive The product of two negative numbers is always positive. It is also important to use parentheses when solving with a calculator to prevent this mistake. See how a calculator solves differently for each type of equation. For example, \begin{aligned}- \, 9^2&=- \, 81 \\\\ \left(- \, 9^2\right)&=81 \\\\ - \, \left(9^2\right)&=- \, 81 \end{aligned}
• Missing solutions It is easy to forget that a nonlinear system of equations can have two pairs of solutions. Use substitution to find all of the possible solutions.
• Substitute \textbf{y} or \textbf{x} incorrectly A term from the second equation should be isolated, before being used to substitute. Additionally, if the term isolated is y (as is common), then the expression it is equal to should be used as y in the first equation. Not doing this correctly will lead to mistakes. For example, The following system x^{2}-y^{2}=10 and y=4x+3 should be substituted in this way: x^{2}-(4x+3)^{2}=10
• Not subtracting the entire term within the parentheses If a term is being subtracted, all coefficients and constants in the term need to be subtracted. For example, 4x^2-\left(49-42x+9x^2\right)=4x^2-49+42x-9x^2
• Not simplifying algebraic fractions When using algebraic fractions to remove the denominator make sure each term is carefully multiplied. It is much easier to make a mistake when operations involve algebraic fractions. For example, \begin{aligned}&2x^{2}-\cfrac{49-42x+9x^{2}}{2} \\\\ &4x^{2}-(49-42x+9x^{2}) \end{aligned}

Related systems of equations lessons

• Systems of equations
• Intersecting lines

Practice nonlinear system of equations questions

1. Solve the nonlinear system of equations:

x=- \, 1, \, y=- \, 2 and x=- \, 3, \, y=- \, 2

x=1, \, y=2 and x=3, \, y=2

x=1, \, y=2 and x=- \, 3, \, y=- \, 2

x=1, \, y=- \, 2 and x=3, \, y=- \, 2

Use the fact that y=x+1 to substitute the value of y into the second equation: x+1=x^{2}+3x-2.

x^{2}+2x-3=0 can be factored.

x+3=0 so x=- \, 3

The two values of x can be substituted into one of the original equations and find the two possible values of y.

When x=- \, 3, \, \, y=- \, 3+1=- \, 2

When x=1, \, y=1+1=2

The two solutions are x=- \, 3, \, y=- \, 2 and x=1, \, y=2.

Note: Graph of the system for reference:

2. Solve the nonlinear system of equations:

x=- \, 3, \, y=0 and x=0, \, y=- \, 3

Use the fact that y=x+3 to substitute the value of y into the second equation: x+3=x^{2}+7x+12.

x^{2}+6x+9=0 can be factored.

x+3=0 so x=- \, 3 only.

The value of x can be substituted into one of the original equations and find the value of y. Note, in this case there is only one solution.

When x=- \, 3, \, y=- \, 3+3=0

The solution is x=- \, 3, \, y=0.

3. Solve the nonlinear system of equations:

x=5, \, y=9 and x=1, \, y=3

x=- \, 5, \, y=9 and x=1, \, y=3

x=- \, 5, \, y=9 and x=- \, 1, \, y=- \, 3

x=- \, 5, \, y=9 and x=1, \, y=- \, 3

Use the fact that y=4-x to substitute the value of y into the second equation: 4-x=x^{2}+3x-1.

x^{2}+4x-5=0 can be factored.

When x=- \, 5, \, – \, 5+y=4 so y=9

When x=1, \, 1+y=4 so y=3

The two solutions are x=- \, 5, \, y=9 and x=1, \, y=3.

4. Solve the nonlinear system of equations:

Use the fact that y=2+5x to substitute the value of y into the second equation: 2+5x=9x^2+11x+3.

This equations appears harder to factor, so use the quadratic formula:

9x^{2}+6x+1=0 where a=9, \, b=6, \, c=1

x=- \, \cfrac{1}{3} only

When x=- \, \cfrac{1}{3},

The one solution is x=- \, \cfrac{1}{3},~y=- \, \cfrac{11}{3}.

5. Solve the nonlinear system of equations:

x=1+2\sqrt{3},~y=3+2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3}

x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=1-2\sqrt{3},~y=3-2\sqrt{3}

x=1+2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=3-2\sqrt{3}

x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3}

Use the fact that   y= x-2 to substitute the value of y into the second equation: 2x^{2}-x(x-2)=11.

x^{2}+2x-11=0 with a=1, \, b=2, \, c=- \, 11

x=- \, 1+2\sqrt{3} or x=- \, 1-2\sqrt{3}

When x=- \, 1+2\sqrt{3},

y=- \, 1+2\sqrt{3}-2 so y=- \, 3+2\sqrt{3}

When x=- \, 1-2\sqrt{3},

y=- \, 1-2\sqrt{3}-2 so y=- \, 3-2\sqrt{3}

The two solutions are x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3} and x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3}.

6. Solve the nonlinear system of equations:

x=0.5,~y=- \, 0.75 and x=- \, 2,~y=- \, 2

x=- \, 0.5,~y=- \, 0.75 and x=2,~y=- \, 2

x=0.5,~y=0.75 and x=2,~y=2

x=0.5,~y=- \, 0.75 and x=2,~y=- \, 2

First, make y the subject of the formula in the first equation.

Use the fact that   \cfrac{x-2}{2} to substitute the value of y into the second equation: \cfrac{x-2}{2}=x^{2}+2x-2.

2x^{2}+3x-2=0 with a=2, \, b=3, \, c=- \, 2

x=\cfrac{- \, 3+5}{4}=\cfrac{1}{2} or x=\cfrac{- \, 3-5}{4}=- \, 2

When x=\cfrac{1}{2},

When x=- \, 2,

The two solutions are x=\cfrac{1}{2},~y=- \, \cfrac{3}{4} and x=- \, 2,~y=- \, 2.

Nonlinear systems of equations FAQs

Functions that create straight line graphs.

Yes, any equation can be part of a nonlinear system of equation.

A mathematical expression that has variables, coefficients and exponents.

The next lessons are

• Number patterns
• Functions in algebra
• Laws of exponents

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