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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

• Trigonometry

Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
• Measurement, data, and geometry
• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
• Negative numbers: multiplication and division
• Fractions, decimals, & percentages
• Rates & proportional relationships
• Expressions, equations, & inequalities
• Numbers and operations
• Solving equations with one unknown
• Linear equations and functions
• Systems of equations
• Geometric transformations
• Data and modeling
• Volume and surface area
• Pythagorean theorem
• Transformations, congruence, and similarity
• Arithmetic properties
• Factors and multiples
• Reading and interpreting data
• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
• Quadratic functions & equations
• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
• Confidence intervals
• Significance tests (hypothesis testing)
• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
• Thinking about multivariable functions
• Derivatives of multivariable functions
• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

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What are teachers saying about Khan Academy’s interactive math worksheets?

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Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

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Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

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

What can QuickMath do?

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

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

Math Topics

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Unit 2: Solving equations & inequalities

About this unit.

There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions.

Linear equations with variables on both sides

• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
• Intro to equations with variables on both sides (Opens a modal)
• Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
• Equation with variables on both sides: fractions (Opens a modal)
• Equation with the variable in the denominator (Opens a modal)
• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

Linear equations with parentheses

• Equations with parentheses (Opens a modal)
• Reasoning with linear equations (Opens a modal)
• Multi-step equations review (Opens a modal)
• Equations with parentheses Get 3 of 4 questions to level up!
• Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
• Reasoning with linear equations Get 3 of 4 questions to level up!

Analyzing the number of solutions to linear equations

• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Linear equations with unknown coefficients

• Linear equations with unknown coefficients (Opens a modal)
• Why is algebra important to learn? (Opens a modal)
• Linear equations with unknown coefficients Get 3 of 4 questions to level up!

Multi-step inequalities

• Inequalities with variables on both sides (Opens a modal)
• Inequalities with variables on both sides (with parentheses) (Opens a modal)
• Multi-step inequalities (Opens a modal)
• Using inequalities to solve problems (Opens a modal)
• Multi-step linear inequalities Get 3 of 4 questions to level up!
• Using inequalities to solve problems Get 3 of 4 questions to level up!

Compound inequalities

• Compound inequalities: OR (Opens a modal)
• Compound inequalities: AND (Opens a modal)
• A compound inequality with no solution (Opens a modal)
• Double inequalities (Opens a modal)
• Compound inequalities examples (Opens a modal)
• Compound inequalities review (Opens a modal)
• Solving equations & inequalities: FAQ (Opens a modal)
• Compound inequalities Get 3 of 4 questions to level up!

MathPapa Practice

MathPapa Practice has practice problems to help you learn algebra.

Basic Arithmetic

Subtraction, multiplication, basic arithmetic review, multi-digit arithmetic, addition (2-digit), subtraction (2-digit), multiplication (2-digit by 1-digit), division (2-digit answer), multiplication (2-digit by 2-digit), multi-digit division, negative numbers, addition: negative numbers, subtraction: negative numbers, multiplication: negative numbers, division: negative numbers, order of operations, order of operations 1, basic equations, equations: fill in the blank 1, equations: fill in the blank 2, equations: fill in the blank 3 (order of operations), fractions of measurements, fractions of measurements 2, adding fractions, subtracting fractions, adding fractions: fill in the blank, multiplication: fractions 1, multiplication: fractions 2, division: fractions 1, division: fractions 2, division: fractions 3, addition (decimals), subtraction (decimals), multiplication 2 (example problem: 3.5*8), multiplication 3 (example problem: 0.3*80), division (decimals), division (decimals 2), percentages, percentages 1, percentages 2, chain reaction, balance arithmetic, number balance, basic balance 1, basic balance 2, basic balance 3, basic balance 4, basic balance 5, basic algebra, basic algebra 1, basic algebra 2, basic algebra 3, basic algebra 4, basic algebra 5, algebra: basic fractions 1, algebra: basic fractions 2, algebra: basic fractions 3, algebra: basic fractions 4, algebra: basic fractions 5.

Math Practice

Geogebra math practice.

Math Practice is a tool for mastering algebraic notation. It supports students in their step-by-step math work, let's them explore different solution paths, and helps build confidence, fluency, and understanding.

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Immerse yourself in the world of algebraic problems to fine-tune your mathematical abilities and elevate your skillset

Linear equations

Order of Operations

Algebraic Expressions

Polynomials

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Use interactive notation to build comfort and fluency with algebraic transformations

Get adaptive hints and in-the-moment feedback to explore different solution paths

Get comprehensive help and guidance focused on improving understanding of basic concepts

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If you require any guidance on how to use GeoGebra Math Practice, explore the articles in our help center.

120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

• Teaching Tools
• Subtraction
• Multiplication
• Mixed operations
• Ordering and number sense
• Comparing and sequencing
• Physical measurement
• Ratios and percentages
• Probability and data relationships

You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

Algebra Worksheets

Free worksheets with answer keys.

Enjoy these free printable sheets . Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Distance Formula
• Equation of Circle
• Factor Trinomials Worksheet
• Domain and Range
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line from Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials Worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Mulitplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Scientific Notation
• Operations with Scientific Notation

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Math Word Problems

Welcome to the math word problems worksheets page at Math-Drills.com! On this page, you will find Math word and story problems worksheets with single- and multi-step solutions on a variety of math topics including addition, multiplication, subtraction, division and other math topics. It is usually a good idea to ensure students already have a strategy or two in place to complete the math operations involved in a particular question. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8.

There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills.com problem-solving strategy:

• Question : Understand what the question is asking. What operation or operations do you need to use to solve this question? Ask for help to understand the question if you can't do it on your own.
• Estimate : Use an estimation strategy, so you can check your answer for reasonableness in the evaluate step. Try underestimating and overestimating, so you know what range the answer is supposed to be in. Be flexible in rounding numbers if it will make your estimate easier.
• Strategize : Choose a strategy to solve the problem. Will you use mental math, manipulatives, or pencil and paper? Use a strategy that works for you. Save the calculator until the evaluate stage.
• Calculate : Use your strategy to solve the problem.
• Evaluate : Compare your answer to your estimate. If you under and overestimated, is the answer in the correct range. If you rounded up or down, does the answer make sense (e.g. is it a little less or a little more than the estimate). Also check with a calculator.

Most Popular Math Word Problems this Week

Arithmetic Word Problems

• Addition Word Problems One-Step Addition Word Problems Using Single-Digit Numbers One-Step Addition Word Problems Using Two-Digit Numbers
• Subtraction Word Problems Subtraction Facts Word Problems With Differences from 5 to 12
• Multiplication Word Problems One-Step Multiplication Word Problems up to 10 × 10
• Division Word Problems Division Facts Word Problems with Quotients from 5 to 12
• Multi-Step Word Problems Easy Multi-Step Word Problems

Copyright © 2005-2024 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

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Short problems for Starters, Homework and Assessment

Number - Short Problems

A collection of short Stage 3 and 4 problems on number.

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A collection of short Stage 3 and 4 problems on algebra.

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From her bedroom window a girl drops a water-filled balloon to the ground 6.0 m...

A 730-N man stands in the middle of a frozen pond of radius 5.0 m....

A 5.00 kg package slides 1.50 m down a long ramp that is inclined at12.0∘below...

Ropes 3m and 5m in length are fastened to a holiday decoration that is suspended...

A skier of mass 70 kg is pulled up a slope by a motor driven...

A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid,...

A sled with rider having a combined mass of 120 kg travels over the perfectly...

A 7.00- kg bowling ball moves at 3.00 m/s. How fast must a 2.45- g...

Two point chargesq1=+2.40nC andq2=−6.50nC are 0.100 m apart. Point A is midway between them and...

A block of mass m slides on a horizontal frictionless table with an initial speed...

A space traveler weights 540 N on earth. what will the traveler weigh on another...

A block of mass m=2.20 kg slides down a 30 degree incline which is 3.60...

A weatherman carried an aneroid barometer from the groundfloor to his office atop a tower....

If a negative charge is initially at rest in an electric field, will it move...

A coin with a diameter of 2.40cm is dropped on edge on to a horizontal...

An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction...

An 80.0-kg skydiver jumps out of a balloon at an altitude of1000 m and opens...

A 0.145 kg baseball pitched at 39.0 m/s is hit on a horizontal line drive...

A 1000 kg safe is 2.0 m above a heavy-duty spring when the rope holding...

A 500 g ball swings in a vertical circle at the end of a1.5-m-long string....

A rifle with a weight of 30 N fires a 5.0 g bullet with a...

The tires of a car make 65 revolutions as the car reduces its speed uniformly...

A 2.0- kg piece of wood slides on the surface. The curved sides are perfectly...

A 292 kg motorcycle is accelerating up along a ramp that is inclined 30.0° above...

A projectile is shot from the edge of a cliff 125 m above ground level...

A lunch tray is being held in one hand, as the drawing illustrates. The mass...

The initial velocity of a car, vi, is 45 km/h in the positivex direction. The...

An Alaskan rescue plane drops a package of emergency rations to a stranded party of...

Raindrops make an angle theta with the vertical when viewed through a moving train window....

A 0.50 kg ball that is tied to the end of a 1.1 m light...

If the coefficient of static friction between your coffeecup and the horizontal dashboard of your...

A car is initially going 50 ft/sec brakes at a constant rate (constant negative acceleration),...

A swimmer is capable of swimming 0.45m/s in still water (a) If sheaim her body...

A block is hung by a string from inside the roof of avan. When the...

A race driver has made a pit stop to refuel. Afterrefueling, he leaves the pit...

A relief airplane is delivering a food package to a group of people stranded on...

The eye of a hurricane passes over Grand Bahama Island. It is moving in a...

An extreme skier, starting from rest, coasts down a mountainthat makes an angle25.0∘with the horizontal....

Four point charges form a square with sides of length d, as shown in the...

In a scene in an action movie, a stuntman jumps from the top of one...

The spring in the figure (a) is compressed by length delta x . It launches...

An airplane propeller is 2.08 m in length (from tip to tip) and has a...

A helicopter carrying dr. evil takes off with a constant upward acceleration of5.0ms2. Secret agent...

A 15.0 kg block is dragged over a rough, horizontal surface by a70.0 N force...

A box is sliding with a speed of 4.50 m/s on a horizontal surface when,...

3.19 Win the Prize. In a carnival booth, you can win a stuffed giraffe if...

A car is stopped at a traffic light. It then travels along a straight road...

a. When the displacement of a mass on a spring is12A, what fraction of the...

At a certain location, wind is blowing steadily at 10 m/s. Determine the mechanical energy...

A jet plane lands with a speed of 100 m/s and can accelerate at a...

In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints...

An antelope moving with constant acceleration covers the distance between two points 70.0 m apart...

A bicycle with 0.80-m-diameter tires is coasting on a level road at 5.6 m/s. A...

The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of...

A proton with an initial speed of 800,000 m/s is brought to rest by an...

The volume of a cube is increasing at the rate of 1200 cm supmin at...

An airplane starting from airport A flies 300 km east, then 350 km at 30...

To prove: In the following figure, triangles ABC and ADC are congruent. Given: Figure is...

Conduct a formal proof to prove that the diagonals of an isosceles trapezoid are congruent....

The distance between the centers of two circles C1 and C2 is equal to 10...

Segment BC is Tangent to Circle A at Point B. What is the length of...

Find an equation for the surface obtained by rotating the parabola y=x2 about the y-axis.

Find the area of the parallelogram with vertices A(-3, 0), B(-1 , 3), C(5, 2),...

If the atomic radius of lead is 0.175 nm, find the volume of its unit...

At one point in a pipeline the water’s speed is 3.00 m/s and the gauge...

Find the volume of the solid in the first octant bounded by the coordinate planes,...

A paper cup has the shape of a cone with height 10 cm and radius...

A light wave has a 670 nm wavelength in air. Its wavelength in a transparent...

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50...

Find the equation of the sphere centered at (-9, 3, 9) with radius 5. Give...

Determine whether the congruence is true or false. 5≡8 mod 3

Find all whole number solutions of the congruence equation. (2x+1)≡5 mod 4

Determine whether the congruence is true or false. 100≡20 mod 8

I want example of an undefined term and a defined term in geometry and explaining...

Two fair dice are rolled. Let X equal the product of the 2dice. Compute P{X=i}...

Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The...

Based on the Normal model N(100, 16) describing IQ scores, what percent of peoples

The probability density function of the net weight in pounds of a packaged chemical herbicide...

Let X represent the difference between the number of heads and the number of tails...

An urn contains 3 red and 7 black balls. Players A and B withdraw balls...

80% A poll is given, showing are in favor of a new building project. 8...

The probability that the San Jose Sharks will win any given game is 0.3694 based...

Find the value of P(X=7) if X is a binomial random variable with n=8 and...

Find the value of P(X=8) if X is a binomial random variable with n=12 and...

On a 8 question multiple-choice test, where each question has 2 answers, what would be...

If you toss a fair coin 11 times, what is the probability of getting all...

A coffee connoisseur claims that he can distinguish between a cup of instant coffee and...

Two firms V and W consider bidding on a road-building job, which may or may...

Two cards are drawn without replacement from an ordinary deck, find the probability that the...

In August 2012, tropical storm Isaac formed in the Caribbean and was headed for the...

A local bank reviewed its credit card policy with the intention of recalling some of...

The accompanying table gives information on the type of coffee selected by someone purchasing a...

A batch of 500 containers for frozen orange juice contains 5 that are defective. Two...

The probability that an automobile being filled with gasoline also needs an oil change is...

Let the random variable X follow a normal distribution with μ=80 and σ2=100. a. Find...

A card is drawn randomly from a standard 52-card deck. Find the probability of the...

The next number in the series 38, 36, 30, 28, 22 is ?

What is the coefficient of x8y9 in the expansion of (3x+2y)17?

A boat on the ocean is 4 mi from the nearest point on a straight...

How many different ways can you make change for a quarter? (Different arrangements of the...

Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue, and...

Approximately 80,000 marriages took place in the state of New York last year. Estimate the...

The probability that a student passes the Probability and Statistics exam is 0.7. (i)Find the...

Customers at a gas station pay with a credit card (A), debit card (B), or...

It is conjectured that an impurity exists in 30% of all drinking wells in a...

Assume that the duration of human pregnancies can be described by a Normal model with...

According to a renowned expert, heavy smokers make up 70% of lung cancer patients. If...

Two cards are drawn successively and without replacement from an ordinary deck of playing cards...

Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L),...

A bag contains 6 red, 4 blue and 8 green marbles. How many marbles of...

A normal distribution has a mean of 50 and a standard deviation of 4. Please...

Seven women and nine men are on the faculty in the mathematics department at a...

An automatic machine in a manufacturing process is operating properly if the lengths of an...

Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings)...

Among 157 African-American men, the mean systolic blood pressure was 146 mm Hg with a...

A TIRE MANUFACTURER WANTS TO DETERMINE THE INNER DIAMETER OF A CERTAIN GRADE OF TIRE....

Differentiate the three measures of central tendency: ungrouped data.

Find the mean of the following data: 12,10,15,10,16,12,10,15,15,13

A wallet containing four P100 bills, two P200 bills, three P500 bills, and one P1,000...

The number of hours per week that the television is turned on is determined for...

Data was collected for 259 randomly selected 10 minute intervals. For each ten-minute interval, the...

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in...

A normal distribution has a mean of 80 and a standard deviation of 14. Determine...

True or false: a. All normal distributions are symmetrical b. All normal distributions have a...

Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or...

Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8408 1374 1872 8879...

The velocity function (in meters per second) is given for a particle moving along a...

Find the area of the parallelogram with vertices A(-3,0) , B(-1,6) , C(8,5) and D(6,-1)

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4),...

The integral represents the volume of a solid. Describe the solid. π∫01(y4−y8)dy a) The integral...

Two components of a minicomputer have the following joint pdf for their useful lifetimes X...

Use the table of values of f(x,y) to estimate the values of fx(3,2), fx(3,2.2), and...

Calculate net price factor and net price. Dollars list price −435.20$ Trade discount rate −26%,15%,5%.

Represent the line segment from P to Q by a vector-valued function and by a...

(x2+2xy−4y2)dx−(x2−8xy−4y2)dy=0

If f is continuous and integral 0 to 9 f(x)dx=4, find integral 0 to 3...

Find the parametric equation of the line through a parallel to ba=[3−4],b=[−78]

Find the velocity and position vectors of a particle that has the given acceleration and...

If we know that the f is continuous and integral 0 to 4f(x)dx=10, compute the...

Integration of (y⋅tan⁡xy)

For the matrix A below, find a nonzero vector in the null space of A...

Find a nonzero vector orthogonal to the plane through the points P, Q, and R....

Suppose that the augmented matrix for a system of linear equations has been reduced by...

Find two unit vectors orthogonal to both (3 , 2, 1) and (- 1, 1,...

What is the area of the parallelogram whose vertices are listed? (0,0), (5,2), (6,4), (11,6)

Using T defined by T(x)=Ax, find a vector x whose image under T is b,...

Use the definition of Ax to write the matrix equation as a vector equation, or...

We need to find the volume of the parallelepiped with only one vertex at the...

List five vectors in Span {v1,v2}. For each vector, show the weights on v1 and...

(1) find the projection of u onto v and (2) find the vector component of...

Find the area of the parallelogram determined by the given vectors u and v. u...

(a) Find the point at which the given lines intersect. r = 2,...

(a) find the transition matrix from B toB′,(b) find the transition matrix fromB′to B,(c) verify...

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If...

Given the following vector X, find anon zero square marix A such that AX=0; You...

Construct a matrix whose column space contains (1, 1, 5) and (0, 3.1) and whose...

At what point on the paraboloid y=x2+z2 is the tangent plane parallel to the plane...

Label the following statements as being true or false. (a) If V is a vector...

Find the Euclidean distance between u and v and the cosine of the angle between...

Write an equation of the line that passes through (3, 1) and (0, 10)

There are 100 two-bedroom apartments in the apartment building Lynbrook West.. The montly profit (in...

State and prove the linearity property of the Laplace transform by using the definition of...

The analysis of shafts for a compressor is summarized by conformance to specifications. Suppose that...

The Munchies Cereal Company combines a number of components to create a cereal. Oats and...

Movement of a Pendulum A pendulum swings through an angle of 20∘ each second. If...

If sin⁡x+sin⁡y=aandcos⁡x+cos⁡y=b then find tan⁡(x−y2)

Find the values of x such that the angle between the vectors (2, 1, -1),...

Find the dimensions of the isosceles triangle of largest area that can be inscribed in...

Suppose that you are headed toward a plateau 50 meters high. If the angle of...

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport....

Find an equation of the plane. The plane through the points (2, 1, 2), (3,...

Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following...

two small spheres spaced 20.0cm apart have equal charges. How many extra electrons must be...

The base of a pyramid covers an area of 13.0 acres (1 acre =43,560 ft2)...

Find out these functions' domain and range. To find the domain in each scenario, identify...

Your bank account pays an interest rate of 8 percent. You are considering buying a...

Whether f is a function from Z to R ifa)f(n)=±n.b)f(n)=n2+1.c)f(n)=1n2−4.

The probability density function of X, the lifetime of a certain type of electronic device...

A sandbag is released by a balloon that is rising vertically at a speed of...

A proton is located in a uniform electric field of2.75×103NCFind:a) the magnitude of the electric...

A rectangular plot of farmland are finite on one facet by a watercourse and on...

A solenoid is designed to produce a magnetic field of 0.0270 T at its center....

I want to find the volume of the solid enclosed by the paraboloidz=2+x2+(y−2)2and the planesz=1,x=−1y=0,andy=4

Let W be the subspace spanned by the u’s, and write y as the sum...

Can u find the point on the planex+2y+3z=13that is closest to the point (1,1,1). You...

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force...

A force of 250 Newtons is applied to a hydraulic jack piston that is 0.01...

Three identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface...

A credit card contains 16 digits between 0 and 9. However, only 100 million numbers...

Every real number is also a complex number? True of false?

Let F be a fixed 3x2 matrix, and let H be the set of all...

Find a vector a with representation given by the directed line segment AB. Draw AB...

Find A such that the given set is Col A. {[2s+3tr+s−2t4r+s3r−s−t]:r,s,t real}

Find the vector that has the same direction as (6, 2, -3) but is four...

For the matrices (a) find k such that Nul A is a subspace of Rk,...

How many subsets with an odd number of elements does a set with 10 elements...

In how many ways can a set of five letters be selected from the English...

Suppose that f(x) = x/8 for 3 < x < 5. Determine the following probabilities:...

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to...

Find two vectors parallel to v of the given length. v=PQ→ with P(1,7,1) and Q(0,2,5);...

A dog in an open field runs 12.0 m east and then 28.0 m in...

Can two events with nonzero probabilities be both independent and mutually exclusive? Explain your reasoning.

Use the Intermediate Value Theorem to show that there is a root of the given...

In a fuel economy study, each of 3 race cars is tested using 5 different...

A company has 34 salespeople. A board member at the company asks for a list...

A dresser drawer contains one pair of socks with each of the following colors: blue,...

A restaurant offers a $12 dinner special with seven appetizer options, 12 choices for an...

A professor writes 40 discrete mathematics true/false questions. Of the statements in these questions, 17...

Suppose E(X)=5 and E[X(X–1)]=27.5, find ∈(x2) and the variance.

A Major League baseball diamond has four bases forming a square whose sides measure 90...

Express f(x)=4x3+6x2+7x+2 in term of Legendre Polynomials.

Find a basis for the space of 2×2 diagonal matrices. Basis ={[],[]}

Which of the following expressions are meaningful? Which are meaningless? Explain. a) (a⋅b)⋅c (a⋅b)⋅c has...

Vectors V1 and V2 are different vectors with lengths V1 and V2 respectively. Find the...

Find an equation for the plane containing the two (parallel) lines v1=(0,1,−2)+t(2,3,−1) and v2=(2,−1,0)+t(2,3,−1).

Find, correct to the nearest degree, the three angles of the triangle with the given...

Find the vector, not with determinants, but by using properties of cross products. (i+j)×(i−j)

Find the curve’s unit tangent vector. Also, find the length of the indicated portion of...

Construct a 4×3 matrix with rank 1

Find x such that the matrix is equal to its inverse.A=[7x−8−7]

Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3...

Write in words how to read each of the following out loud.a.{x∈R′∣0<x<1}b.{x∈R∣x≤0orx⇒1}c.{n∈Z∣nisafactorof6}d.{n∈Z⋅∣nisafactorof6}

Pets Plus and Pet Planet are having a sale on the same aquarium. At Pets...

Find the average value of F(x, y, z) over the given region. F(x,y,z)=x2+9 over the...

Find the trace of the plane in the given coordinate plane. 3x−9y+4z=5,yz

Determine the level of measurement of the variable. Favorite color Choose the correct level of...

How wide is the chasm between what men and women earn in the workplace? According...

Write an algebraic expression for: 6 more than a number c.

Please, can u convert 3.16 (6 repeating) to a fraction.

Evaluate the expression. P(8, 3)

In a poker hand consisting of 5 cards, find the probability of holding 3 aces.

Give an expression that generates all angles coterminal with each angle. Let n represent any...

An ideal Otto cycle has a compression ratio of 10.5, takes in air at 90...

A piece of wire 10 m long is cut into two pieces. One piece is...

Put the following equation of a line into slope intercept form, simplifying all fractions 3x+3y=24

Find the point on the hyperbola xy = 8 that is closest to the point...

Water is pumped from a lower reservoir to a higher reservoir by a pump that...

A piston–cylinder device initially contains 0.07m3 of nitrogen gas at 130 kPa and 180∘. The...

Write an algebraic expression for each word phrase. 4 more than p

A club has 25 members. a) How many ways are there to choose four members...

For each of the sets below, determine whether {2} is an element of that set....

Which expression has both 8 and n as factors?

If repetitions are not permitted (a) how many 3 digit number can be formed from...

To determine the sum of all multiples of 3 between 1 and 1000

On average, there are 3 accidents per month at one intersection. We need to find...

One number is 2 more than 3 times another. Their sum is 22. Find the...

The PMF for a flash drive with X (GB) of memory that was purchased is...

An airplane needs to reach a velocity of 203.0 km/h to takeoff. On a 2000...

A racquetball strikes a wall with a speed of 30 m/s and rebounds with a...

Assuming that the random variable x has a cumulative distribution function,F(x)={0,x<00.25x,0≤x<51,5≤xDetermine the following:a)p(x<2.8)b)p(x>1.5)c)p(x<−z)d)p(x>b)

At t = 0 a grinding wheel has an angular velocity of 24.0 rad/s. It...

How many 3/4's are in 1?

You’re driving down the highway late one night at 20 m/s when a deer steps...

Table salt contains 39.33 g of sodium per 100 g of salt. The U.S. Food...

The constant-pressure heat capacity of a sample of a perfect gas was found to vary...

Coffee is draining from a conical filter into a cylindrical coffepot at the rate of...

Cart is driven by a large propeller or fan, which can accelerate or decelerate the...

A vending machine dispenses coffee into an eight-ounce cup. The amounts of coffee dispensed into...

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 m/s encounters a...

The gage pressure in a liquid at a depth of 3 m is read to...

Consider a cylindrical specimen of a steel alloy 8.5 mm (0.33 in.) in diameter and...

Calculate the total kinetic energy, in Btu, of an object with a mass of 10...

A 0.500-kg mass on a spring has velocity as a function of time given by...

An Australian emu is running due north in a straight line at a speed of...

Another pitfall cited is expecting to improve the overall performance of a computer by improving...

You throw a glob of putty straight up toward the ceiling, which is 3.60 m...

A 0.150-kg frame, when suspended from a coil spring, stretches the spring 0.070 m. A...

A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips....

A rock climber stands on top of a 50-m-high cliff overhanging a pool of water....

A tank whose bottom is a mirror is filled with water to a depth of...

Two sites are being considered for wind power generation. In the first site, the wind...

0.250 kilogram of water at75.0∘Care contained in a tiny, inert beaker. How much ice, at...

Two boats start together and race across a 60-km-wide lake and back. Boat A goes...

A roller coaster moves 200 ft horizontally and the rises 135 ft at an angle...

A tow truck drags a stalled car along a road. The chain makes an angle...

Consider the curve created by2x2+3y2–4xy=36(a) Show thatdydx=2y−2x3y−2x(b) Calculate the slope of the line perpendicular to...

The current entering the positive terminal of a device is i(t)=6e−2t mA and the voltage...

The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m/s....

Calculate the total potential energy, in Btu, of an object that is 20 ft below...

A chemist in an imaginary universe, where electrons have a different charge than they do...

When jumping, a flea reaches a takeoff speed of 1.0 m/s over a distance of...

Determine the energy required to accelerate a 1300-kg car from 10 to 60 km/h on...

The deepest point in the ocean is 11 km below sea level, deeper than MT....

A golfer imparts a speed of 30.3 m/s to a ball, and it travels the...

Calculate the frequency of each of the following wavelengths of electromagnetic radiation. A) 632.8 nm...

Prove that there is a positive integer that equals the sum of the positive integers...

A hurricane wind blows across a 6.00 m×15.0 m flat roof at a speed of...

If an electron and a proton are expelled at the same time,2.0×10−10mapart (a typical atomic...

The speed of sound in air at 20 C is 344 m/s. (a) What is...

Which of the following functions f has a removable discontinuity at a? If the discontinuity...

A uniform steel bar swings from a pivot at one end with a period of...

A wind farm generator uses a two-bladed propellermounted on a pylon at a height of...

A copper calorimeter can with mass 0.100 kg contains 0.160 kgof water and 0.018 kg...

Jones figures that the total number of thousands of miles that a used auto can...

Assign a binary code in some orderly manner to the 52 playingcards. Use the minimum...

A copper pot with mass 0.500 kg contains 0.170 kg of water ata temperature of...

Ea for a certain biological reaction is 50 kJ/mol, by what factor ( how many...

When a person stands on tiptoe (a strenuous position), the position of the foot is...

A solution was prepared by dissolving 1210 mg of K3Fe(CN)6 (329.2 g/mol) in sufficient waterto...

A 58-kg skier is going down a slope oriented 35 degree abovethe horizontal. The area...

The mechanics at lincoln automotive are reboring a 6-in deepcylinder to fit a new piston....

A 0.48 kg piece of wood floats in water but is found to sinkin alcohol...

A 50-g ice cube at 0oC is heated until 45-g hasbecome water at 100oC and...

A solution containing 6.23 ppm of KMnO4 had a transmittance of 0.195 in a 1.00-cm...

A black body at 7500K consists of an opening of diameter 0.0500mm, looking into an...

A new absolute temperature scale is proposed. On thisscale the ice point of water is...

A 65.0 mm focal length converging lens is 78.0 mm away from a sharp image....

A crate of fruit with mass 35.0 kg and specific heat capacity 3650 J/Kg ....

A freezer has a thermal efficiency of 2.40. Thefreezer is to convert 1.80 kg of...

A horizontal force of 210N is exerted on a 2.0 kg discus as it rotates...

Lead has a specific heat of 0.030 cal/gC. In an insulated container, 300 grams of...

A parachutist relies on air resistance mainly on her parachute to decrease her downward velocity....

The distance between a carbon atom (m=12 u) and an oxygen atom (m + 16...

A car heading north collides at an intersection with a truckheading east. If they lock...

Water stands at a depth H in a large, open tank whose sidewalls are vertical....

The heaviest invertebrate is the giant squid, which is estimated to have a weight of...

Which of the following is a correct comment? */ Comments */ ** Comment ** /*...

The concentrated sulfuric acid we use in the laboratory is 98% H2SO4 by mass. Calculate...

Consider the reaction N2(g)+3H2(g)→2NH3(g) suppose that a particular moment during the reaction molecular hydrogen on...

use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F...

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25 Fun Maths Problems For KS2 And KS3 (From Easy To Very Hard!)

Fun maths problems are one of the things mathematicians love about the subject; they provide an opportunity to apply mathematical knowledge, logic and problem solving skills all at once.  In this article, we’ve compiled 25 fun maths problems, each covering various topics and question types. They’re aimed at students in KS2 & KS3. We’ve categorised them as:

Maths word problems

Maths puzzles, fraction problems, multiplication and division problems, geometry problems, problem solving questions, maths puzzles are everywhere, how should teachers use these maths problems.

Teachers could make use of these maths problem solving questions in a number of ways, such as:

• embed into a relevant maths topic’s teaching.
• settling tasks at the beginning of lessons.
• break up or extend a maths worksheet.
• keep students thinking mathematically after the main lesson has finished.

Some are based on real life or historical maths problems, and some include ‘bonus’ maths questions to help to extend the problem solving fun! As you read through these problems, think about how you could adjust them to be relevant to your students or to practise different skills. 

These maths problems can also be used as introductory puzzles for maths games such as those introduced at the following links:

• KS2 maths games
• KS3 maths games

Need more support teaching reasoning, problem solving and planning for depth ? Read here for free CPD for you and your team of teachers.

1. Home on time – easy

Type: Time, Number, Addition

A cinema screening starts at 14:35. The movie lasts for 2 hours, 32 minutes after 23 minutes of adverts. It took 20 minutes to get to the cinema. What time should you tell your family that you’ll be home?

Answer: 17:50

2. A nugget of truth – mixed

Type: Times Tables, Multiplication, Multiples, Factors, Problem Solving 

Chicken nuggets come in boxes of 6, 9 or 20, so you can’t order 7 chicken nuggets. How many other impossible quantities can you find (not including fractions or decimals)?

Answer: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, or 43

There is actually a theorem which can be used to prove that every integer quantity greater than 43 can be ordered.

3. A pet problem – mixed

Type: Number, Problem Solving, Forming and Solving Equations, Simultaneous Equations, Algebra

Eight of my pets aren’t dogs, five aren’t rabbits, and seven aren’t cats. How many pets do I have?

Answer: 10 pets (5 rabbits, 3 cats, 2 dogs)

4. The price of things – mixed

Type: lateral thinking problem

A mouse costs £10, a bee costs £15, and a spider costs £20. How much does a duck cost? Answer: £5 (£2.50 per leg)

Looking for more word problems, solutions and explanations? Read our article on word problems for primary school.

25 Fun Maths Problems - Printable

Download a printable version of these fun maths problems together with answers and mark scheme.

5. A dicey maths challenge – easy

Type: Place value, number, addition, problem solving

Roll three dice to generate three place value digits. What’s the biggest number you can make out of these digits? What’s the smallest number you can make?

Add these two numbers together. What do you get?

Answer: In most cases, 1,089.

Bonus: Who got a different result? Why?

6. PIN problem solving – mixed

Type: Logic, problem solving, reasoning

I’ve forgotten my PIN. Six incorrect attempts locks my account: I’ve used five! Two digits are displayed after each unsuccessful attempt: “2, 0” means 2 digits from that guess are in the PIN, but 0 are in the right place.

What should my sixth attempt be?

Answer: 6347

7. So many birds – mixed

Type: Triangular Numbers, Sequences, Number, Problem Solving

On the first day of Christmas my true love gave me one gift. On the second day they gave me another pair of gifts plus a copy of what they gave me on day one. On day 3, they gave me three new gifts, plus another copy of everything they’d already given me. If they keep this up, how many gifts will I have after twelve days?

Answer: 364

Bonus: This could be calculated as 1 + (1 + 2) + (1 + 2 + 3) + … but is there an easier way? What percentage of my gifts do I receive on each day?

8. I 8 sum maths questions – mixed

Type: Number, Place Value, Addition, Problem Solving, Reasoning

Using only addition and the digit 8, can you make 1,000? You can put 8s together to make 88, for example.

Answer: 888 + 88 + 8 + 8 + 8 = 1,000 Bonus: Which other digits allow you to get 1,000 in this way?

9. Quizzical – easy

Type: Fractions, Adding Fractions, Equivalent Fractions, Fractions to Percentages

4 friends entered a maths quiz. One answered \frac{1}{5} of the maths questions, one answered \frac{1}{10} , one answered \frac{1}{4} , and the other answered \frac{4}{25} . What percentage of the questions did they answer altogether?

Answer: 71%

10. Ancient problem solving – mixed

Type: Fractions, Reasoning, Problem Solving

Ancient Egyptians only used unit fractions (like \frac{1}{2} , \frac{1}{3} or \frac{1}{4} ). For \frac{2}{3} , they’d write \frac{1}{3} + \frac{1}{3} . How might they write \frac{5}{8} ?

Answer: \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} is correct. So is \frac{1}{2} + \frac{1}{8} .

Bonus: Which solution is better? Why? Can you find any more? What if subtractions are allowed?

Learn more about unit fractions here

11. everybody wants a pizza the action – hard.

An infinite number of mathematicians buy pizza. The first wants \frac{1}{2} pizza. The second wants \frac{1}{4} pizza. The third & fourth want \frac{1}{8} and \frac{1}{16} each, and so on. How many pizzas should they order?

Answer: 1 Each successive mathematician wants a slice that is exactly half of what is left:

12. Shade it black – hard

Type: Fractions, Reasoning, Problem Solving What fraction of this image is shaded black?

Answer: \frac{1}{3}

Look at the L-shaped part made up of two white and one black squares: \frac{1}{3} of this part is shaded. Zoom in on the top-right quarter of the image, which looks exactly the same as the whole image, and use the same reasoning to find what fraction of its L-shaped portion is shaded. Imagine zooming in to do the same thing again and again…

13. Giving is receiving – easy

Type: Number, Reasoning, Problem Solving

5 people give each other a present. How many presents are given altogether?

14. Sharing is caring – mixed

I have 20 sweets. If I share them equally with my friends, there are 2 left over. If one more person joins us, there are 6 sweets left. How many friends am I with?

Answer: 6 people altogether (so 5 friends!)

15. Times tables secrets – mixed

Type: Area, 2D Shape, Rectangles

Here are 77 letters:

BYHRCGNGNEOEAAHGHGCURPUTSTSASHHSBOBOREOPEEMEMEELATPEPEFADPHLTLTUT IEEOHOHLENRYTITIIAGBMTNTNFCGEIIGIG

How many different rectangular grids could you arrange all 77 letters into?

Answer: Four: 1⨉77, 77⨉1, 11⨉7 & 7⨉11. If the letters are arranged into one of these, a message appears, reading down each column starting from the top left.

Bonus: Can you find any more integers with the same number of factors as 77? What do you notice about these factors (think about prime numbers)? Can you use this system to hide your own messages?

16. Laugh it up – hard

Type: Multiples, Lowest Common Multiple, Times Tables, Division, Time

One friend jumps every \frac{1}{3} of a minute. Another jumps every 31 seconds. When will they jump together? Answer: After 620 seconds

17. Pictures of matchstick triangles – easy

Type: 2D Shapes, Equilateral Triangles, Problem Solving, Reasoning

Look at the matchsticks arranged below. How many equilateral triangles are there?

Answer: 13 (9 small, 3 medium, 1 large)

Bonus: What if the biggest triangle only had two matchsticks on each side? What if it had four?

18. Dissecting squares – mixed

Type: Reasoning, Problem Solving

What’s the smallest number of straight lines you could draw on this grid such that each square has a line going through it?

19. Make it right – mixed

Type: Pythagoras’ theorem

This triangle does not agree with Pythagoras’ theorem. 

Adding, subtracting, multiplying or dividing each of the side lengths by the same integer can fix it. What is the integer?

Answer: 3 

8 – 3 = 5

The new side lengths are 3, 4 and 5 and  32 + 42 = 52.

20. A most regular maths question – hard

Type: Polygons, 2D Shapes, tessellation, reasoning, problem-solving, patterns

What is the regular polygon with the largest number of sides that will self-tessellate?

Answer: Hexagon.

Regular polygons tessellate if one interior angle is a factor of 360°. The interior angle of a hexagon is 120°. This is the largest factor less than 180°.

21. Pleased to meet you – easy

Type: Number Problem, Reasoning, Problem-Solving

5 people meet; each shakes everyone else’s hand once. How many handshakes take place?

Person A shakes 4 people’s hands. Person B has already shaken Person A’s hand, so only needs to shake 3 more, and so on.

Bonus: How many handshakes would there be if you did this with your class?

22. All relative – easy

Type: Number, Reasoning, Problem-Solving

When I was twelve my brother was half my age. I’m 40 now, so how old is he?

23. It’s about time – mixed

Type: Time, Reasoning, Problem-Solving

When is “8 + 10 = 6” true?

Answer: When you’re telling the time (8am + 10 hours = 6pm)

24. More than a match – mixed

Type: Reasoning, Problem-Solving, Roman Numerals, Numerical Notation

Here are three matches:

How can you add two more matches, but get eight? Answer: Put the extra two matches in a V shape to make 8 in Roman Numerals:

25. Leonhard’s graph – hard

Type: Reasoning, Problem-Solving, Logic

Leonhard’s town has seven bridges as shown below. Can you find a route around the town that crosses every bridge exactly once?

Answer: No!

This is a classic real life historical maths problem solved by mathematician Leonhard Euler (rhymes with “boiler”). The city was Konigsberg in Prussia (now Kaliningrad, Russia). Not being able to find a solution is different to proving that there aren’t any! Euler managed to do this in 1736, practically inventing graph theory in the process.

Many of these 25 maths problems are rooted in real life, from everyday occurrences to historical events. Others are just questions that might arise if you say “what if…?”. The point is that although there are many lists of such problem solving maths questions that you can make use of, with a little bit of experience and inspiration you could create your own on almost any topic – and so could your students. 

For a kick-starter on creating your own maths problems, read our article on KS3 maths problem solving .

Looking for additional support and resources at KS3? You are welcome to download any of the secondary maths resources from Third Space Learning’s resource library for free. There is a section devoted to GCSE maths revision with plenty of maths worksheets and GCSE maths questions . There are also maths tests for KS3, including a Year 7 maths test , a Year 8 maths test and a Year 9 maths test For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach.

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Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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AI Reveals New Possibilities in Matrix Multiplication

November 23, 2022

Matrix multiplication is not unlike solving an unthinkably large Rubik’s Cube.

Mahmet Emin Güzel for Quanta Magazine

Introduction

Mathematicians love a good puzzle. Even something as abstract as multiplying matrices (two-dimensional tables of numbers) can feel like a game when you try to find the most efficient way to do it. It’s a little like trying to solve a Rubik’s Cube in as few moves as possible — challenging, but alluring. Except that for a Rubik’s Cube, the number of possible moves at each step is 18; for matrix multiplication, even in relatively simple cases, every step can present more than 10 12 options.

Over the past 50 years, researchers have approached this problem in many ways, all based on computer searches aided by human intuition. Last month, a team at the artificial intelligence company DeepMind showed how to tackle the problem from a new direction, reporting in a paper in Nature that they’d successfully trained a neural network to discover new fast algorithms for matrix multiplication. It was as if the AI had found an unknown strategy for solving a monstrously complex Rubik’s Cube.

“It’s a very neat result,” said Josh Alman , a computer scientist at Columbia University. But he and other matrix multiplication specialists also emphasized that such AI assistance will complement rather than replace existing methods — at least in the near term. “It’s like a proof of concept for something that could become a breakthrough,” Alman said. The result will simply help researchers on their quest.

As if to prove the point, three days after the Nature paper came out, a pair of Austrian researchers illustrated how new and old methods might complement each other. They used a conventional computer-aided search to further improve one of the algorithms that the neural network had discovered.

The results suggest that, like the process of solving a Rubik’s Cube, the path to better algorithms will be full of twists and turns.

Video : DeepMind researchers trained an AI system called AlphaTensor to find new, faster algorithms for matrix multiplication. AlphaTensor quickly rediscovered — and surpassed, for some cases — the reigning algorithm discovered by German mathematician Volker Strassen in 1969.

Christopher Webb Young/ Quanta Magazine ; Emily Zhang for Quanta Magazine

Multiplying Matrices

Matrix multiplication is one of the most fundamental and ubiquitous operations in all of mathematics. To multiply a pair of n -by- n matrices, each with n 2 elements, you multiply and add these elements together in particular combinations to generate the product, a third n -by- n matrix. The standard recipe for multiplying two n -by- n matrices requires n 3 multiplication operations, so a 2-by-2 matrix, for example, requires eight multiplications.

For larger matrices, with thousands of rows and columns, this process quickly becomes cumbersome. But in 1969, the mathematician Volker Strassen discovered a procedure for multiplying a pair of 2-by-2 matrices using seven rather than eight multiplication steps, at the cost of introducing more addition steps.

Strassen’s algorithm is needlessly convoluted if you just want to multiply a pair of 2-by-2 matrices. But he realized it would also work for bigger matrices. That’s because the elements of a matrix can themselves be matrices. For example, a matrix with 20,000 rows and 20,000 columns can be reimagined as a 2-by-2 matrix whose four elements are each 10,000-by-10,000 matrices. Each of these matrices can then be subdivided into four 5,000-by-5,000 blocks, and so on. Strassen could apply his method to multiply 2-by-2 matrices at each level of this nested hierarchy. As the matrix size increases, the savings from fewer multiplications grows.

Strassen’s discovery led to a search for efficient algorithms for matrix multiplication, which has since inspired two distinct subfields. One focuses on a question of principle: If you imagine multiplying two n -by- n matrices and let n run toward infinity, how does the number of multiplication steps in the fastest possible algorithm scale with n ? The current record for the best scaling, n 2.3728596 , belongs to Alman and Virginia Vassilevska Williams , a computer scientist at the Massachusetts Institute of Technology. (A recent unpublished preprint reported a tiny improvement using a new technique.) But these algorithms are of purely theoretical interest, winning out over methods like Strassen’s only for absurdly large matrices.

The second subfield thinks on a smaller scale. Soon after Strassen’s work, the Israeli American computer scientist Shmuel Winograd showed that Strassen had reached a theoretical limit: It’s not possible to multiply 2-by-2 matrices with fewer than seven multiplication steps. But for all other matrix sizes, the minimum number of required multiplications remains an open question. And fast algorithms for small matrices could have an outsize impact, since repeated iterations of such an algorithm might beat Strassen’s algorithm when reasonably sized matrices are being multiplied.

Unfortunately, the sheer number of possibilities is huge. Even for 3-by-3 matrices, “the number of possible algorithms exceeds the number of atoms in the universe,” said Alhussein Fawzi , a DeepMind researcher and one of the leaders of the new work.

Faced with this dizzying menu of options, researchers have made progress by transforming matrix multiplication into what seems like a totally different math problem — one that is easier for computers to handle. It’s possible to represent the abstract task of multiplying two matrices as a specific kind of mathematical object: a three-dimensional array of numbers called a tensor. Researchers can then break this tensor up into a sum of elementary components, called “rank-1” tensors; each of these will represent a different step in the corresponding matrix multiplication algorithm. That means that finding an efficient multiplication algorithm amounts to minimizing the number of terms in a tensor decomposition — the fewer the terms, the fewer the steps involved.

In this way, researchers have discovered new algorithms that multiply n -by- n matrices using fewer than the standard n 3 multiplication steps for many small matrix sizes. But algorithms that outperform not only the standard but also Strassen’s algorithm for small matrices have remained out of reach — until now.

The DeepMind team approached the problem by turning tensor decomposition into a single-player game. They started with a deep learning algorithm descended from AlphaGo — another DeepMind AI that in 2016 learned to play the board game Go well enough to beat the top human players.

All deep learning algorithms are built around neural networks: webs of artificial neurons sorted into layers, with connections that can vary in strength representing how much each neuron influences those in the next layer. The strength of these connections is tweaked over many iterations of a training procedure, during which the neural network learns to transform each input it receives into an output that helps the algorithm accomplish its overall goal.

In DeepMind’s new algorithm, dubbed AlphaTensor, the inputs represent steps along the way to a valid matrix multiplication scheme. The first input to the neural network is the original matrix multiplication tensor, and its output is the rank-1 tensor that AlphaTensor has chosen for its first move. The algorithm subtracts this rank-1 tensor from the initial input, yielding an updated tensor that is fed back into the network as a new input. The process repeats until eventually every element in the starting tensor has been reduced to zero, meaning there are no more rank-1 tensors to take out.

At that point, the neural network has discovered a valid tensor decomposition, since it’s mathematically guaranteed that the sum of all the rank-1 tensors is exactly equal to the starting tensor. And the steps it took to get there can be translated back into steps of the corresponding matrix multiplication algorithm.

Here’s the game: AlphaTensor repeatedly decomposes a tensor to a set of rank-1 components. Each time, AlphaTensor gets rewarded if it finds a way to reduce the number of steps. But shortcuts to victory are not at all intuitive, just as you sometimes must scramble a perfectly ordered face on a Rubik’s Cube before you can solve the whole thing.

The team now had an algorithm that could, theoretically, solve their problem. They just had to train it first.

Like all neural networks, AlphaTensor needs a lot of data to train on, but tensor decomposition is a notoriously hard problem. There were few examples of efficient decompositions that the researchers could feed the network. Instead, they helped the algorithm get started by training it on the much easier inverse problem: adding up a bunch of randomly generated rank-1 tensors.

“They’re using the easy problem to produce more data for the hard problem,” said Michael Littman , a computer scientist at Brown University. Combining this backward training procedure with reinforcement learning, wherein AlphaTensor generated its own training data as it blundered around looking for efficient decompositions, worked much better than either training method on its own.

The DeepMind team trained AlphaTensor to decompose tensors representing the multiplication of matrices up to 12-by-12. It sought fast algorithms for multiplying matrices of ordinary real numbers and also algorithms specific to a more constrained setting known as modulo 2 arithmetic. (This is math based on only two numbers, so matrix elements can only be 0 or 1, and 1 + 1 = 0.) Researchers often start with this more restricted but still vast space, in hopes that algorithms discovered here can be adapted to work on matrices of real numbers.

After training, AlphaTensor rediscovered Strassen’s algorithm within minutes. It then discovered up to thousands of new fast algorithms for each matrix size. These were different from the best-known algorithms but had the same number of multiplication steps.

In a few cases, AlphaTensor even beat existing records. Its most surprising discoveries happened in modulo 2 arithmetic, where it found a new algorithm for multiplying 4-by-4 matrices in 47 multiplication steps, an improvement over the 49 steps required for two iterations of Strassen’s algorithm. It also beat the best-known algorithm for 5-by-5 modulo 2 matrices, reducing the number of required multiplications from the previous record of 98 to 96. (But this new record still lags behind the 91 steps that would be required to beat Strassen’s algorithm using 5-by-5 matrices.)

The new high-profile result created a lot of excitement, with some researchers heaping praise on the AI-based improvement on the status quo. But not everyone in the matrix multiplication community was so impressed. “I felt like it was a little overhyped,” said Vassilevska Williams. “It’s another tool. It’s not like, ‘Oh, the computers beat the humans,’ you know?”

Researchers also emphasized that immediate applications of the record-breaking 4-by-4 algorithm would be limited: Not only is it valid only in modulo 2 arithmetic, but in real life there are important considerations besides speed.

Fawzi agreed that really, this is just the beginning. “There is a lot of room for improvement and research, and that’s a good thing,” he said.

A Final Twist

AlphaTensor’s greatest strength relative to well-established computer search methods is also its greatest weakness: It’s not constrained by human intuition about what good algorithms look like, so it can’t explain its choices. That makes it difficult for researchers to learn from its achievements.

But this may not be as big a drawback as it seems. A few days after the AlphaTensor result, the mathematician Manuel Kauers and his graduate student Jakob Moosbauer , both of Johannes Kepler University Linz in Austria, reported another step forward.

Manuel Kauers tweaked the DeepMind approach to generate further improvements.

Jakob Moosbauer

When the DeepMind paper came out, Kauers and Moosbauer were in the process of searching for new multiplication algorithms using a conventional computer-aided search. But unlike most such searches, which start afresh with a new guiding principle, their method works by repeatedly tweaking an existing algorithm in hopes of squeezing more multiplication savings out of it. Taking AlphaTensor’s algorithm for 5-by-5 modulo 2 matrices as a starting point, they were surprised to find that their method reduced the number of multiplication steps from 96 to 95 after just a few seconds of computation.

AlphaTensor also helped them make another improvement indirectly. Previously, Kauers and Moosbauer hadn’t bothered to explore the space of 4-by-4 matrices, believing that it would not be possible to beat two iterations of Strassen’s algorithm. The AlphaTensor result prompted them to reconsider, and after a week of computation time starting from scratch, their method turned up another 47-step algorithm unrelated to the one AlphaTensor had discovered. “If somebody had told us that there is something to discover for 4-by-4, we could have done that earlier,” said Kauers. “But OK, well, that’s how it works.”

Littman expects more such surprises, likening the situation to the first time a runner finished a mile in under four minutes, a feat that had widely been considered impossible. “People were like, ‘Oh, wait, we can do this,’ and now lots of people can do it,” he said.

Looking forward, Fawzi hopes to generalize AlphaTensor to tackle a broader range of mathematical and computational tasks, just as its ancestor AlphaGo eventually branched out into other games.

Kauers sees this as the true litmus test for the application of machine learning to discovering new algorithms. He points out that the quest for fast matrix multiplication algorithms is a combinatorial problem to which computer searches, with or without human assistance, are well suited. But not all mathematical problems are so easy to pin down. If machine learning can discover a qualitatively new algorithmic idea, he said, “this would then be a game changer.”

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