- Math Forum/Help
- Problem Solver
- College Math
Math Practice
Problems for 1st grade, problems for 2nd grade, problems for 3rd grade, problems for 4th grade, problems for 5th grade, problems for 6th grade, problems for 7th grade, problems for 8th grade, problems for 9-12 grade:, quadratic equations.
Views of a function Domain of a function Domain and range Range of a function Inverses of functions Shifting and reflecting functions Positive and negative parts of functions
Line graph intuition Slope of a line Slope intercept form Recognizing slope
Trygonometry
Probabilities, complex numbers.
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.
Get step-by-step solutions to your math problems
Try Math Solver
Get step-by-step explanations
Graph your math problems
Practice, practice, practice
Get math help in your language
- Prodigy Math
- Prodigy English
From our blog
- Is a Premium Membership Worth It?
- Promote a Growth Mindset
- Help Your Child Who's Struggling with Math
- Parent's Guide to Prodigy
- Math Curriculum Coverage
- English Curriculum Coverage
- Prodigy success stories
- Prodigy Teacher Dashboard Overview
- Help Students Learn at Home
- Remote Learning Engagement
- Teaching Strategies
- Parent Letter (English) PDF
- Game Portal
120 Math Word Problems To Challenge Students Grades 1 to 8
Engage and motivate your students with our adaptive, game-based learning platform!
- 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.
Step-by-Step Calculator
Solve problems from pre algebra to calculus step-by-step.
- Pre Algebra
- Pre Calculus
- Linear Algebra
- Trigonometry
- Conversions
Most Used Actions
Number line.
- x^{2}-x-6=0
- -x+3\gt 2x+1
- line\:(1,\:2),\:(3,\:1)
- prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
- \frac{d}{dx}(\frac{3x+9}{2-x})
- (\sin^2(\theta))'
- \lim _{x\to 0}(x\ln (x))
- \int e^x\cos (x)dx
- \int_{0}^{\pi}\sin(x)dx
- \sum_{n=0}^{\infty}\frac{3}{2^n}
Frequently Asked Questions (FAQ)
Is there a step by step calculator for math.
- Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
Is there a step by step calculator for physics?
- Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
How to solve math problems step-by-step?
- To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
Related Symbolab blog posts
We want your feedback.
Please add a message.
Message received. Thanks for the feedback.
Generating PDF...
Maths Problem-solving Examples With Solutions
What are some examples of problem-solving strategies used in mathematics?
Math can be studied independently if you can comprehend simple English and have access to the internet. You’ll discover that you are the only person who can teach yourself faster and better after putting everything in this manual into practice. After you implement everything in this guide, you’ll discover that there’s no one who can understand maths faster and better than you.
Just a small word of caution: even though I said anyone can do this, I’m absolutely certain not everyone will. Actually, it’s a little unsettling, particularly if you’re doing this for the first time.
As you work through a four-step problem-solving process, elementary math students using the “ Four-Step Problem Solving ” plan are encouraged to use sound reasoning and develop their mathematical language. The details, main idea, strategy, and how steps make up this problem-solving plan.
See also : How To Solve Maths Problems Quickly
Students may use “ graphic representations ” as they progress through each stage to arrange their thoughts, show how they think mathematically, and demonstrate how they plan to solve a problem.
Table of Contents
The student performs the roles of reader, thinker, and analyst in this step. The student begins by reading the article and scanning for any proper words (words with insight). If unusual names of people or places cause confusion, the student may substitute a familiar name and see if that is the problem. Now the puzzle is clear. Reread the issue, summarizing it, or putting the issue into a visual form may be helpful to the student.
The student should read the issue once more, word by word, slowly and attentively. By utilizing words, numbers, and phrases, you would recognize and document any details. The student searches for additional information or facts from the reading that is related to the answer. The student should continue to search for any hidden numbers that may be implied but not explicitly stated in this step.
For instance: The issue might concern “Frank and his three friends.” (In order to solve the problem, the student needs to understand that there are actually four people, even though “four” or “4” is not mentioned in the reading.)
The student picks a math strategy (or several strategies) to use in order to solve the problem and finds the solution using that strategy. The following are some potential tactics, according to the Texas Essential Knowledge and Skills (TEKS) curriculum.
- Use or create a visual
- Observe a pattern
- Construct a number of sentences.
- Use procedures (operations) like addition, subtraction, multiplication, and division
- Create or utilize a table
- Use or create a list.
- Tto solve a simpler issue
Students use words or phrases to explain how they came up with their solution in order to make sure that it is reasonable and that they fully comprehend the process. The following are some queries that students should ask themselves.
- What was the solution to the issue?
- How did I choose my approach?
- “What steps did I take?”
A student explanation of the chosen solution approach is required in this step. They need to justify their course of action and offer evidence that it is sound. From this step, students have the chance to show how they understood the mathematical ideas and vocabulary used in the problem they solved.
Mathematical Questions and Answers
1. In comparison to the morning, a salesman sold twice as many pears in the afternoon. If he sold 360 kilograms of pears that day, how many kilograms did he sell in the morning, and how many in the afternoon?
x is the number of kilograms he sold in the morning.
Then in the afternoon, he sold
2x kilograms. So, the total is
x + 2x = 3x
x+2x=3x. This must be equal to 360.
Therefore, the salesman sold 120 kg in the morning and
In the afternoon, he sold 2 X 120 = 240kg.
2. Mary, Peter, and Lucy were out picking chestnuts. Compared to Peter, Mary picked twice as many chestnuts. Peter picked up 2 kg less than Lucy did. The three of them collectively collected 26 kg of chestnuts. How many kilograms did each of them pick?
Let x be the amount Peter picked.
Then Mary picked 2x
Lucy picked x+2 respectively.
So x+2x+x+2=26.
Therefore, Peter, Mary, and Lucy picked 6, 12, and 8 kg, respectively.
3. An airplane that flies against the wind from A to B in 8 hours. The same aircraft takes seven hours to fly back from B to A in the same direction as the wind. Calculate the difference between the airplane’s speed in still air and the wind’s speed.
Let x = the speed of the airplane in still air, y = the speed of the wind, and D = the distance between A and B. Find the ratio x / y
Against the wind: D = 8(x – y), with the wind: D = 7(x + y).
8x – 8y = 7x + 7y, hence x / y = 15.
4. Find the area between two concentric circles defined by x 2 + y 2 -2x + 4y + 1 = 0 x 2 + y 2 -2x + 4y – 11 = 0
Rewrite equations of circles in standard form. Hence equation
x 2 + y 2 -2x + 4y + 1 = 0 may be written as
(x – 1) 2 + (y + 2) 2 = 4 = 2 2
and equation
x 2 + y 2 -2x + 4y – 11 = 0 as
(x – 1) 2 + (y + 2) 2 = 16 = 4 2
Knowing the radii, the area of the ring is π (4) 2 – π (2) 2 = 12
5. Find all values of parameter m (a real number) so that the equation 2x 2 – m x + m = 0 has no real solutions.
The given equation is a quadratic equation and has no solutions if its discriminant D is less than zero.
D = (-m) 2 – 4(2)(m) = m 2 – 8 m
We nos solve the inequality m 2 – 8 m < 0
The solution set of the above inequality is: (0 , 8)
Any value of m in the interval (0, 8) makes the discriminant D negative and, therefore, the equation has no real solutions.
6. The sum of an integer N and its reciprocal is equal to 78/15. What is the value of N?
This equation should be written in N.
N + 1/N = 78/15
Multiply all terms by N, obtain a quadratic equation, and solve to obtain N = 5.
7. M and N are integers, so that 4 m / 125 = 5 n / 64. Find values for m and n.
4 m / 125 = 5 n / 64
Cross multiply: 64 (4)m = 125 (5n)
Note that 64 = 4(3) and 125 = 5 (3)
The above equation may be written as: 4m + 3 = 5n + 3.
The only values of the exponents that make the two exponential expressions equal are: m + 3 = 0 and n + 3 = 0, which gives m = – 3 and n = – 3
The summary
If you believe that you are not a “math person,” you would require a different individual to teach you math in a classroom setting. Let me tell you this. With all the free resources available online, including lectures, syllabi, ebooks, and MOOCS, it is possible to self-study math with relative ease, just like you would in a college setting.
What’s great is that you can go at your own pace. No strict schedules, just self-commitment If you want to benefit, you must reconsider your perspective on this, though. You must bear in mind that the mental effort you spend mastering a math concept is the cost you must bear in exchange for future math skills becoming simpler or, more accurately, it’s the cost you incur to ensure that you don’t make learning challenging for yourself in the future.
Related posts:
- What Is Liberal Arts Education
- How To Apply For Alba Graduate Business School 2022
- 10 Best Colleges In Texas
- How To Read Without Sleeping – Effective Tips
- Gmail Password Recovery Tips Every Student Should Know
- Complete WAEC Past Questions And Answers For All Subjects
- Skip to main content
- Skip to primary sidebar
Additional menu
Khan Academy Blog
Free Math Worksheets — Over 100k free practice problems on Khan Academy
Looking for free math worksheets.
You’ve found something even better!
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.
What do Khan Academy’s interactive math worksheets look like?
Here’s an example:
What are teachers saying about Khan Academy’s interactive math worksheets?
“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”
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.
Is Khan Academy a company?
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?
Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.
Praxis Core Math
Unit 1: lesson 4.
- Algebraic properties | Lesson
- Algebraic properties | Worked example
- Solution procedures | Lesson
- Solution procedures | Worked example
- Equivalent expressions | Lesson
- Equivalent expressions | Worked example
- Creating expressions and equations | Lesson
- Creating expressions and equations | Worked example
Algebraic word problems | Lesson
- Algebraic word problems | Worked example
- Linear equations | Lesson
- Linear equations | Worked example
- Quadratic equations | Lesson
- Quadratic equations | Worked example
What are algebraic word problems?
What skills are needed.
- Translating sentences to equations
- Solving linear equations with one variable
- Evaluating algebraic expressions
- Solving problems using Venn diagrams
How do we solve algebraic word problems?
- Define a variable.
- Write an equation using the variable.
- Solve the equation.
- If the variable is not the answer to the word problem, use the variable to calculate the answer.
What's a Venn diagram?
- Your answer should be
- an integer, like 6 6 6 6
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text
- (Choice A) $ 4 \$4 $ 4 dollar sign, 4 A $ 4 \$4 $ 4 dollar sign, 4
- (Choice B) $ 5 \$5 $ 5 dollar sign, 5 B $ 5 \$5 $ 5 dollar sign, 5
- (Choice C) $ 9 \$9 $ 9 dollar sign, 9 C $ 9 \$9 $ 9 dollar sign, 9
- (Choice D) $ 14 \$14 $ 1 4 dollar sign, 14 D $ 14 \$14 $ 1 4 dollar sign, 14
- (Choice E) $ 20 \$20 $ 2 0 dollar sign, 20 E $ 20 \$20 $ 2 0 dollar sign, 20
- (Choice A) 10 10 1 0 10 A 10 10 1 0 10
- (Choice B) 12 12 1 2 12 B 12 12 1 2 12
- (Choice C) 24 24 2 4 24 C 24 24 2 4 24
- (Choice D) 30 30 3 0 30 D 30 30 3 0 30
- (Choice E) 32 32 3 2 32 E 32 32 3 2 32
- (Choice A) 4 4 4 4 A 4 4 4 4
- (Choice B) 10 10 1 0 10 B 10 10 1 0 10
- (Choice C) 14 14 1 4 14 C 14 14 1 4 14
- (Choice D) 18 18 1 8 18 D 18 18 1 8 18
- (Choice E) 22 22 2 2 22 E 22 22 2 2 22
Things to remember
Want to join the conversation.
- Upvote Button opens signup modal
- Downvote Button opens signup modal
- Flag Button opens signup modal
Problem Solving in Mathematics
- Math Tutorials
- Pre Algebra & Algebra
- Exponential Decay
- Worksheets By Grade
The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.
Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.
Use Established Procedures
Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.
Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.
Look for Clue Words
Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.
Common clue words for addition problems:
Common clue words for subtraction problems:
- How much more
Common clue words for multiplication problems:
Common clue words for division problems:
Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.
Read the Problem Carefully
This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:
- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?
Develop a Plan and Review Your Work
Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:
- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.
If it seems like you’ve solved the problem, ask yourself the following:
- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?
If you feel confident that the answer is “yes” to all questions, consider your problem solved.
Tips and Hints
Some key questions to consider as you approach the problem may be:
- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?
Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.
:max_bytes(150000):strip_icc():format(webp)/modern-office-shoot-531337705-5a0ae4fc13f1290037a9a205.jpg "sample of math problem solving with solution")
By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts.
- school Campus Bookshelves
- menu_book Bookshelves
- perm_media Learning Objects
- login Login
- how_to_reg Request Instructor Account
- hub Instructor Commons
- Download Page (PDF)
- Download Full Book (PDF)
- Periodic Table
- Physics Constants
- Scientific Calculator
- Reference & Cite
- Tools expand_more
- Readability
selected template will load here
This action is not available.
6.1.1: Practice Problems- Solution Concentration
- Last updated
- Save as PDF
- Page ID 217282
PROBLEM \(\PageIndex{1}\)
Explain what changes and what stays the same when 1.00 L of a solution of NaCl is diluted to 1.80 L.
The number of moles always stays the same in a dilution.
The concentration and the volumes change in a dilution.
PROBLEM \(\PageIndex{2}\)
What does it mean when we say that a 200-mL sample and a 400-mL sample of a solution of salt have the same molarity? In what ways are the two samples identical? In what ways are these two samples different?
The two samples contain the same proportion of moles of salt to liters of solution, but have different numbers of actual moles.
PROBLEM \(\PageIndex{3}\)
Determine the molarity for each of the following solutions:
- 0.444 mol of CoCl 2 in 0.654 L of solution
- 98.0 g of phosphoric acid, H 3 PO 4 , in 1.00 L of solution
- 0.2074 g of calcium hydroxide, Ca(OH) 2 , in 40.00 mL of solution
- 10.5 kg of Na 2 SO 4 ·10H 2 O in 18.60 L of solution
- 7.0 × 10 −3 mol of I 2 in 100.0 mL of solution
- 1.8 × 10 4 mg of HCl in 0.075 L of solution
PROBLEM \(\PageIndex{4}\)
Determine the molarity of each of the following solutions:
- 1.457 mol KCl in 1.500 L of solution
- 0.515 g of H 2 SO 4 in 1.00 L of solution
- 20.54 g of Al(NO 3 ) 3 in 1575 mL of solution
- 2.76 kg of CuSO 4 ·5H 2 O in 1.45 L of solution
- 0.005653 mol of Br 2 in 10.00 mL of solution
- 0.000889 g of glycine, C 2 H 5 NO 2 , in 1.05 mL of solution
5.25 × 10 -3 M
6.122 × 10 -2 M
1.13 × 10 -2 M
PROBLEM \(\PageIndex{5}\)
Calculate the number of moles and the mass of the solute in each of the following solutions:
(a) 2.00 L of 18.5 M H 2 SO 4 , concentrated sulfuric acid (b) 100.0 mL of 3.8 × 10 −5 M NaCN, the minimum lethal concentration of sodium cyanide in blood serum (c) 5.50 L of 13.3 M H 2 CO, the formaldehyde used to “fix” tissue samples (d) 325 mL of 1.8 × 10 −6 M FeSO 4 , the minimum concentration of iron sulfate detectable by taste in drinking water
37.0 mol H 2 SO 4
3.63 × 10 3 g H 2 SO 4
3.8 × 10 −6 mol NaCN
1.9 × 10 −4 g NaCN
73.2 mol H 2 CO
2.20 kg H 2 CO
5.9 × 10 −7 mol FeSO 4
8.9 × 10 −5 g FeSO 4
PROBLEM \(\PageIndex{6}\)
Calculate the molarity of each of the following solutions:
(a) 0.195 g of cholesterol, C 27 H 46 O, in 0.100 L of serum, the average concentration of cholesterol in human serum (b) 4.25 g of NH 3 in 0.500 L of solution, the concentration of NH 3 in household ammonia (c) 1.49 kg of isopropyl alcohol, C 3 H 7 OH, in 2.50 L of solution, the concentration of isopropyl alcohol in rubbing alcohol (d) 0.029 g of I 2 in 0.100 L of solution, the solubility of I 2 in water at 20 °C
5.04 × 10 −3 M
1.1 × 10 −3 M
PROBLEM \(\PageIndex{7}\)
There is about 1.0 g of calcium, as Ca 2+ , in 1.0 L of milk. What is the molarity of Ca 2+ in milk?
PROBLEM \(\PageIndex{8}\)
What volume of a 1.00- M Fe(NO 3 ) 3 solution can be diluted to prepare 1.00 L of a solution with a concentration of 0.250 M ?
PROBLEM \(\PageIndex{9}\)
If 0.1718 L of a 0.3556- M C 3 H 7 OH solution is diluted to a concentration of 0.1222 M , what is the volume of the resulting solution?
PROBLEM \(\PageIndex{10}\)
What volume of a 0.33- M C 12 H 22 O 11 solution can be diluted to prepare 25 mL of a solution with a concentration of 0.025 M ?
PROBLEM \(\PageIndex{11}\)
What is the concentration of the NaCl solution that results when 0.150 L of a 0.556- M solution is allowed to evaporate until the volume is reduced to 0.105 L?
PROBLEM \(\PageIndex{12}\)
What is the molarity of the diluted solution when each of the following solutions is diluted to the given final volume?
- 1.00 L of a 0.250- M solution of Fe(NO 3 ) 3 is diluted to a final volume of 2.00 L
- 0.5000 L of a 0.1222- M solution of C 3 H 7 OH is diluted to a final volume of 1.250 L
- 2.35 L of a 0.350- M solution of H 3 PO 4 is diluted to a final volume of 4.00 L
- 22.50 mL of a 0.025- M solution of C 12 H 22 O 11 is diluted to 100.0 mL
PROBLEM \(\PageIndex{13}\)
What is the final concentration of the solution produced when 225.5 mL of a 0.09988- M solution of Na 2 CO 3 is allowed to evaporate until the solution volume is reduced to 45.00 mL?
PROBLEM \(\PageIndex{14}\)
A 2.00-L bottle of a solution of concentrated HCl was purchased for the general chemistry laboratory. The solution contained 868.8 g of HCl. What is the molarity of the solution?
PROBLEM \(\PageIndex{15}\)
An experiment in a general chemistry laboratory calls for a 2.00- M solution of HCl. How many mL of 11.9 M HCl would be required to make 250 mL of 2.00 M HCl?
PROBLEM \(\PageIndex{16}\)
What volume of a 0.20- M K 2 SO 4 solution contains 57 g of K 2 SO 4 ?
PROBLEM \(\PageIndex{17}\)
The US Environmental Protection Agency (EPA) places limits on the quantities of toxic substances that may be discharged into the sewer system. Limits have been established for a variety of substances, including hexavalent chromium , which is limited to 0.50 mg/L. If an industry is discharging hexavalent chromium as potassium dichromate (K 2 Cr 2 O 7 ), what is the maximum permissible molarity of that substance?
4.8 × 10 −6 M
Contributors
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] ).
- Adelaide Clark, Oregon Institute of Technology
- Practice Problems
- Assignment Problems
- Show all Solutions/Steps/ etc.
- Hide all Solutions/Steps/ etc.
- Solving Equations and Inequalities Introduction
- Linear Equations
- Preliminaries
- Graphing and Functions
- Calculus II
- Calculus III
- Differential Equations
- Algebra & Trig Review
- Common Math Errors
- Complex Number Primer
- How To Study Math
- Cheat Sheets & Tables
- MathJax Help and Configuration
- Notes Downloads
- Complete Book
- Practice Problems Downloads
- Complete Book - Problems Only
- Complete Book - Solutions
- Assignment Problems Downloads
- Other Items
- Get URL's for Download Items
- Print Page in Current Form (Default)
- Show all Solutions/Steps and Print Page
- Hide all Solutions/Steps and Print Page
Section 2.1 : Solutions and Solution Sets
For each of the following determine if the given number is a solution to the given equation or inequality.
- Is \(x = 6\) a solution to \(2x - 5 = 3\left( {1 - x} \right) + 22\)? Solution
- Is \(t = 7\) a solution to \({t^2} + 3t - 10 = 4 + 8t\)? Solution
- Is \(t = - 3\) a solution to \({t^2} + 3t - 10 = 4 + 8t\)? Solution
- Is \(w = - 2\) a solution to \(\displaystyle \frac{{{w^2} + 8w + 12}}{{w + 2}} = 0\)? Solution
- Is \(z = 4\) a solution to \(6z - {z^2} \ge {z^2} + 3\)? Solution
- Is \(y = 0\) a solution to \(2\left( {y + 7} \right) - 1 Solution
- Is \(x = 1\) a solution to \({\left( {x + 1} \right)^2} > 3x + 1\)? Solution
Unlimited AI-generated practice problems and answers
With Wolfram Problem Generator, each question is generated instantly, just for you.
Get integrated Step-by-step solutions with a subscription to Wolfram|Alpha Pro. Pro subscribers can also create printable worksheets for study sessions and quizzes.
The most amazing part of Wolfram Problem Generator is something you can't even see.
Instead of pulling problems out of a database, Wolfram Problem Generator makes them on the fly, so you can have new practice problems and worksheets each time. Each practice session provides new challenges.
Practice for all ages
Wolfram Problem Generator offers beginner, intermediate, and advanced difficulty levels for a number of topics including algebra, calculus, statistics, number theory, and more.
Work with Step-by-step Solutions!
Only Wolfram Problem Generator directly integrates the popular and powerful Step-by-step Solutions from Wolfram|Alpha. You can use a single hint to get unstuck, or explore the entire math problem from beginning to end.
Math Problem Solving Strategies
In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess & Check Model and Find a Pattern Model.
Related Pages Solving Word Problems Using Block Models Heuristic Approach to Problem-Solving Algebra Lessons
Problem Solving Strategies
The strategies used in solving word problems:
- What do you know?
- What do you need to know?
- Draw a diagram/picture
Solution Strategies Label Variables Verbal Model or Logical Reasoning Algebraic Model - Translate Verbal Model to Algebraic Model Solve and Check.
Solving Word Problems
Step 1: Identify (What is being asked?) Step 2: Strategize Step 3: Write the equation(s) Step 4: Answer the question Step 5: Check
Problem Solving Strategy: Guess And Check
Using the guess and check problem solving strategy to help solve math word problems.
Example: Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills. If she gave the clerk 10 bills in all, how many of each bill did she use?
Problem Solving : Make A Table And Look For A Pattern
- Identify - What is the question?
- Plan - What strategy will I use to solve the problem?
- Solve - Carry out your plan.
- Verify - Does my answer make sense?
Example: Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2 more than the previous day. How much money did Marcus made in all after 5 days?
Find A Pattern Model (Intermediate)
In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.
Example: The figure shows a series of rectangles where each rectangle is bounded by 10 dots. a) How many dots are required for 7 rectangles? b) If the figure has 73 dots, how many rectangles would there be?
a) The number of dots required for 7 rectangles is 52.
b) If the figure has 73 dots, there would be 10 rectangles.
Example: Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7 layers of triangles.
The number of dots for 7 layers of triangles is 36.
Example: The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the following numbers belong? a) 25 b) 46 c) 269
Solution: The pattern is: The remainder when the number is divided by 6 determines the group. a) 25 ÷ 6 = 4 remainder 1 (Group I) b) 46 ÷ 6 = 7 remainder 4 (Group IV) c) 269 ÷ 6 = 44 remainder 5 (Group V)
Example: The following figures were formed using matchsticks.
a) Based on the above series of figures, complete the table below.
b) How many triangles are there if the figure in the series has 9 squares?
c) How many matchsticks would be used in the figure in the series with 11 squares?
b) The pattern is +2 for each additional square. 18 + 2 = 20 If the figure in the series has 9 squares, there would be 20 triangles.
c) The pattern is + 7 for each additional square 61 + (3 x 7) = 82 If the figure in the series has 11 squares, there would be 82 matchsticks.
Example: Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many handshakes were there?
Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.
The following video shows more examples of using problem solving strategies and models. Question 1: Approximate your average speed given some information Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row. Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?
The following are some other examples of problem solving strategies.
Explore it/Act it/Try it (EAT) Method (Basic) Explore it/Act it/Try it (EAT) Method (Intermediate) Explore it/Act it/Try it (EAT) Method (Advanced)
Finding A Pattern (Basic) Finding A Pattern (Intermediate) Finding A Pattern (Advanced)
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
Tips and Motivation to Master Problem Solving
We all want to develop problem-solving skills in math, programming, computer science, data structures and algorithms, and other fields. But as we move forward, we may face various challenges and barriers. Sometimes we handle these situations smartly, but sometime we may get stuck. In this blog, I will present solutions and motivation to tackle some of these scenarios in problem solving. Let’s begin step by step.
Never jump to solution quickly, practice patience and use your time wisely
Sometimes, we don’t know much about the problem and quickly suggest a solution. So habits of proposing or accepting such solutions can create a long-term problem in our ability to solve problems!
Such habits can develop blind spots in our problem-solving ability. We may start to rely on quick fixes rather than taking time to analyze the problem and develop comprehensive solutions. Over time, this can cause us to overlook important details or root causes of errors.
On the other hand, sometimes we lose patience early in the problem-solving. For example, after trying a problem for only 5 minutes, we may begin to think, “I will not be able to find a solution.” But staying with the problem longer is more effective. The fact is: Sometimes problems come with their own complexity, which requires patience, observation, and simplification.
If a 1-hour time limit is set for finding a solution, we need to develop a mindset to use that time wisely. As we move forward with an analytical focus, the probability of success will increase. We may get an idea for a solution at the end of 5 minutes, 15 minutes, 25 minutes, or some other time.
Problem solving is just like driving a Vehicle!
Sometimes, we are very close to getting the correct solution. But suddenly, we shift our focus to the wrong detail and deviate from the approach. Here are couple of ideas on how to handle such situations:
- Note every relevant detail (or step) and backtrack to the point of deviation. This approach is similar to steering a vehicle and returning to the correct track. By retracing your steps, you can identify where you went wrong and get back on track.
- Another idea is to learn from such failures while practicing problem-solving. By doing so, you can create a roadmap of similar situations and make smarter decisions in the future.
Solving a problem is much like driving a vehicle to a destination within a given time frame. If you have good driving experience, control over your driving tools, and an accurate map, handling any wrong deviation will be much easier.
Learn from failures and never give up!
Learning problem-solving is an art of continuous practice by using patience, experience, and knowledge. We should never lose faith in ourselves and keep trying until we find a solution in the given time!
Whenever a thought comes to our mind to give-up a problem, we should motivate ourselves, recollect all experiences and give one more best try! Such situations can be one of the defining moments. The fact is: mastering problem-solving is an iterative journey of learning from failures!
Follow quality over quantity!
Solving 100 problems with proper planning and in-depth understanding is much better than solving 1000 problems with no planning and poor understanding.
You may have observed some learners who focus on solving a lot of problems in a hurry to improve their problem-solving skills. Still, they don’t feel confident approaching new problems. Why does this happen? The idea is simple: During this process, they miss the reason behind underlying concept and develop a habit of memorization.
There is no limit to the number of problems you can solve, but you need to follow a proper strategy for each problem. By understanding the concepts behind approaches, you can avoid making the same mistakes repeatedly, identify common patterns, and apply appropriate solutions to different problems.
Always prepare your note to revise later
We all desire to keep problem-solving ideas in our memory for a long time, but sometimes we forget due to lack of application or other reasons. Later, when we need them, we have to start from scratch again.
To avoid such situations, one idea would be to write a simple note on paper whenever we explore a new idea for the first time. This requires patience and discipline, but it is worth in the long term.
When you look at these well-written notes later, it can provide a huge relief. It gives you an opportunity to build new ideas based on your previous work rather than starting from scratch. However, it is important to note that 80% of these notes may contain ideas about what not to do.
Trust yourself and never underestimate yourself!
Sometimes failures in problem-solving occur when we underestimate our abilities and experiences. The idea is simple: We should never ever underestimate ourselves!
As we know, confidence about our past experiences and knowledge have a significant impact on our ability to solve problems. But some programmers often underestimate themselves due to lack of confidence, fear of failure or the assumption that someone else knows better.
Underestimating ourselves can hinder our ability to think critically and creatively, make us hesitant to take risks, and impact our problem-solving efficiency. On the other hand, when we believe in our abilities and experiences, we can approach problems with a positive attitude, make bold decisions, and take the necessary steps to solve it. We are more likely to be open to new ideas and solutions, which can lead to more effective and innovative problem-solving.
Overcoming fear is a key step in achieving success
We mostly encounter two situations of fear in problem-solving:
- Just after looking at the problem
- In the process of designing the solution
The first type primarily occurs when the problem is totally new to us or when there is a little complex problem statement. One simple solution would be: Rather than overthinking, spend a little extra time in understanding the problem, extracting relevant details, and simplifying it. Of course, a good understanding of concepts works as a catalyst.
The second type is primarily due to the realization of an incorrect approach or inability to identify the next step. One simple solution would be: Gather all experiences and patterns of problem-solving, go back and forth, draw a simple picture of followed steps, and check: Are we moving in the right direction? Are we missing some details? What is the specific point where the error is happening? Do we need to switch to another approach?
Most important thing: Fear is just a current state of mind, nothing more! We should keep faith in our experiences, motivate ourselves and keep trying to get it done!
Think like a sport person to continue practice
The one basic rule of problem-solving: Keeping the distraction aside and developing a desire to get an efficient solution!
As we define the current form of a player in a sport, we can also define the current form of a person in problem-solving. So we should do a continuous practice like a sports person to achieve the best form! The best idea is: We should not bother about ups and downs. It’s a part of the journey, and focusing on the next ability level would be the best strategy!
Practice universal principles of problem solving
When presented with a problem — whether it is an algorithmic problem or a real-life problem, how do we solve it? Sharing here the five common strategies where different strategies have different action plans.
- Trial and Error: Continue trying different solutions until the problem is solved.
- Algorithmic problem solving: Step-by-step instruction used to achieve the desired outcome.
- Problem solving using heuristic: A mental shortcuts that are used to solve problems — a “rule of thumb” is a good example.
- Working backward: A helpful idea to begin solving the problem by focusing on the end result.
- Breaking problem into a series of smaller steps: We often use this method to complete a large research project or solve complex problem.
Track and monitor progress
Why is preparing a weekly execution note important? There could be several reasons, but the best idea is that it helps us track progress and formulate future strategies. Let’s understand from another perspective.
- How do you explain your progress in the last six days and six months?
- What is your plan for the coming six days and six months?
If you are not clear about the answers, then one thing is lacking — a strategy! We all have validated learnings at all stages of life, but our mind has one limitation — a habit of forgetting things as time passes.
Sometimes, we even make the same mistakes or solve the same problems again, which is an inefficient approach. So, the best long-term solution would be to write weekly execution notes and store the validated learnings. This is just like a dynamic programming approach of storing ideas in a cache or memory.
Practice focus and collaboration
Why are some people efficient in learning? For example, one person takes two months to learn something, another person takes two years, and there is a person who struggles to understand it in a lifetime. What are the differences in their approach? It is about two things — focus and collaboration.
- A person with good focus and good collaboration learns quickly.
- A person with good focus and poor collaboration learns at an average speed.
- A person with poor focus and poor collaboration learns slowly or struggles to learn it in a lifetime.
Focus is necessary for learning, but collaboration helps us understand other people’s experiences, avoid mistakes, and find solutions quickly.
Final conclusion and motivation!
- Understand and visualize the problem clearly.
- Use experiences, hypotheses, heuristics, and patterns.
- Do in-depth research about concepts.
- Try to break the problem into smaller problems or develop a step-by-step process to solve it efficiently.
- Think of solutions using multiple approaches!
- Take care of various edge cases and boundary conditions.
- Reach out to an expert or mentor if you get stuck.
- Note down critical steps that are working as a barrier towards the solution on a personal note.
- Refresh your perspective and try again later.
- Practice flexibility and avoid rigidity!
- Try to understand the reason; never memorize!
- Errors and mistakes are a part of continuous learning!
- Collaborate, exchange ideas, and keep ego aside!
Enjoy learning, Enjoy coding, enjoy problem solving :)
Share feedback
Don’t fill this out if you’re human:
More blogs to explore
Subscribe our newsletter.
Subscribe to get weekly content on data structure and algorithms, machine learning, system design and oops.
Follow us on:
© 2022 Code Algorithms Pvt. Ltd.
All rights reserved.
Venn Diagram Examples, Problems and Solutions
On this page:
- What is Venn diagram? Definition and meaning.
- Venn diagram formula with an explanation.
- Examples of 2 and 3 sets Venn diagrams: practice problems with solutions, questions, and answers.
- Simple 4 circles Venn diagram with word problems.
- Compare and contrast Venn diagram example.
Let’s define it:
A Venn Diagram is an illustration that shows logical relationships between two or more sets (grouping items). Venn diagram uses circles (both overlapping and nonoverlapping) or other shapes.
Commonly, Venn diagrams show how given items are similar and different.
Despite Venn diagram with 2 or 3 circles are the most common type, there are also many diagrams with a larger number of circles (5,6,7,8,10…). Theoretically, they can have unlimited circles.
Venn Diagram General Formula
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Don’t worry, there is no need to remember this formula, once you grasp the meaning. Let’s see the explanation with an example.
This is a very simple Venn diagram example that shows the relationship between two overlapping sets X, Y.
X – the number of items that belong to set A Y – the number of items that belong to set B Z – the number of items that belong to set A and B both
From the above Venn diagram, it is quite clear that
n(A) = x + z n(B) = y + z n(A ∩ B) = z n(A ∪ B) = x +y+ z.
Now, let’s move forward and think about Venn Diagrams with 3 circles.
Following the same logic, we can write the formula for 3 circles Venn diagram :
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
Venn Diagram Examples (Problems with Solutions)
As we already know how the Venn diagram works, we are going to give some practical examples (problems with solutions) from the real life.
2 Circle Venn Diagram Examples (word problems):
Suppose that in a town, 800 people are selected by random types of sampling methods . 280 go to work by car only, 220 go to work by bicycle only and 140 use both ways – sometimes go with a car and sometimes with a bicycle.
Here are some important questions we will find the answers:
- How many people go to work by car only?
- How many people go to work by bicycle only?
- How many people go by neither car nor bicycle?
- How many people use at least one of both transportation types?
- How many people use only one of car or bicycle?
The following Venn diagram represents the data above:
Now, we are going to answer our questions:
- Number of people who go to work by car only = 280
- Number of people who go to work by bicycle only = 220
- Number of people who go by neither car nor bicycle = 160
- Number of people who use at least one of both transportation types = n(only car) + n(only bicycle) + n(both car and bicycle) = 280 + 220 + 140 = 640
- Number of people who use only one of car or bicycle = 280 + 220 = 500
Note: The number of people who go by neither car nor bicycle (160) is illustrated outside of the circles. It is a common practice the number of items that belong to none of the studied sets, to be illustrated outside of the diagram circles.
We will deep further with a more complicated triple Venn diagram example.
3 Circle Venn Diagram Examples:
For the purposes of a marketing research , a survey of 1000 women is conducted in a town. The results show that 52 % liked watching comedies, 45% liked watching fantasy movies and 60% liked watching romantic movies. In addition, 25% liked watching comedy and fantasy both, 28% liked watching romantic and fantasy both and 30% liked watching comedy and romantic movies both. 6% liked watching none of these movie genres.
Here are our questions we should find the answer:
- How many women like watching all the three movie genres?
- Find the number of women who like watching only one of the three genres.
- Find the number of women who like watching at least two of the given genres.
Let’s represent the data above in a more digestible way using the Venn diagram formula elements:
- n(C) = percentage of women who like watching comedy = 52%
- n(F ) = percentage of women who like watching fantasy = 45%
- n(R) = percentage of women who like watching romantic movies= 60%
- n(C∩F) = 25%; n(F∩R) = 28%; n(C∩R) = 30%
- Since 6% like watching none of the given genres so, n (C ∪ F ∪ R) = 94%.
Now, we are going to apply the Venn diagram formula for 3 circles.
94% = 52% + 45% + 60% – 25% – 28% – 30% + n (C ∩ F ∩ R)
Solving this simple math equation, lead us to:
n (C ∩ F ∩ R) = 20%
It is a great time to make our Venn diagram related to the above situation (problem):
See, the Venn diagram makes our situation much more clear!
From the Venn diagram example, we can answer our questions with ease.
- The number of women who like watching all the three genres = 20% of 1000 = 200.
- Number of women who like watching only one of the three genres = (17% + 12% + 22%) of 1000 = 510
- The number of women who like watching at least two of the given genres = (number of women who like watching only two of the genres) +(number of women who like watching all the three genres) = (10 + 5 + 8 + 20)% i.e. 43% of 1000 = 430.
As we mentioned above 2 and 3 circle diagrams are much more common for problem-solving in many areas such as business, statistics, data science and etc. However, 4 circle Venn diagram also has its place.
4 Circles Venn Diagram Example:
A set of students were asked to tell which sports they played in school.
The options are: Football, Hockey, Basketball, and Netball.
Here is the list of the results:
The next step is to draw a Venn diagram to show the data sets we have.
It is very clear who plays which sports. As you see the diagram also include the student who does not play any sports (Dorothy) by putting her name outside of the 4 circles.
From the above Venn diagram examples, it is obvious that this graphical tool can help you a lot in representing a variety of data sets. Venn diagram also is among the most popular types of graphs for identifying similarities and differences .
Compare and Contrast Venn Diagram Example:
The following compare and contrast example of Venn diagram compares the features of birds and bats:
Tools for creating Venn diagrams
It is quite easy to create Venn diagrams, especially when you have the right tool. Nowadays, one of the most popular way to create them is with the help of paid or free graphing software tools such as:
You can use Microsoft products such as:
Some free mind mapping tools are also a good solution. Finally, you can simply use a sheet of paper or a whiteboard.
Conclusion:
A Venn diagram is a simple but powerful way to represent the relationships between datasets. It makes understanding math, different types of data analysis , set theory and business information easier and more fun for you.
Besides of using Venn diagram examples for problem-solving and comparing, you can use them to present passion, talent, feelings, funny moments and etc.
Be it data science or real-world situations, Venn diagrams are a great weapon in your hand to deal with almost any kind of information.
If you need more chart examples, our posts fishbone diagram examples and what does scatter plot show might be of help.
About The Author
Silvia Valcheva
Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.
One Response
Well explained I hope more on this one
Leave a Reply Cancel Reply
Currently you have JavaScript disabled. In order to post comments, please make sure JavaScript and Cookies are enabled, and reload the page. Click here for instructions on how to enable JavaScript in your browser.
This site uses Akismet to reduce spam. Learn how your comment data is processed .
Solving Equations
What is an equation.
An equation says that two things are equal. It will have an equals sign "=" like this:
That equations says:
what is on the left (x − 2) equals what is on the right (4)
So an equation is like a statement " this equals that "
What is a Solution?
A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .
Example: x − 2 = 4
When we put 6 in place of x we get:
which is true
So x = 6 is a solution.
How about other values for x ?
- For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
- For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .
In this case x = 6 is the only solution.
You might like to practice solving some animated equations .
More Than One Solution
There can be more than one solution.
Example: (x−3)(x−2) = 0
When x is 3 we get:
(3−3)(3−2) = 0 × 1 = 0
And when x is 2 we get:
(2−3)(2−2) = (−1) × 0 = 0
which is also true
So the solutions are:
x = 3 , or x = 2
When we gather all solutions together it is called a Solution Set
The above solution set is: {2, 3}
Solutions Everywhere!
Some equations are true for all allowed values and are then called Identities
Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities
Let's try θ = 30°:
sin(−30°) = −0.5 and
−sin(30°) = −0.5
So it is true for θ = 30°
Let's try θ = 90°:
sin(−90°) = −1 and
−sin(90°) = −1
So it is also true for θ = 90°
Is it true for all values of θ ? Try some values for yourself!
How to Solve an Equation
There is no "one perfect way" to solve all equations.
A Useful Goal
But we often get success when our goal is to end up with:
x = something
In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.
Example: Solve 3x−6 = 9
Now we have x = something ,
and a short calculation reveals that x = 5
Like a Puzzle
In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.
Here are some things we can do:
- Add or Subtract the same value from both sides
- Clear out any fractions by Multiplying every term by the bottom parts
- Divide every term by the same nonzero value
- Combine Like Terms
- Expanding (the opposite of factoring) may also help
- Recognizing a pattern, such as the difference of squares
- Sometimes we can apply a function to both sides (e.g. square both sides)
Example: Solve √(x/2) = 3
And the more "tricks" and techniques you learn the better you will get.
Special Equations
There are special ways of solving some types of equations. Learn how to ...
- solve Quadratic Equations
- solve Radical Equations
- solve Equations with Sine, Cosine and Tangent
Check Your Solutions
You should always check that your "solution" really is a solution.
How To Check
Take the solution(s) and put them in the original equation to see if they really work.
Example: solve for x:
2x x − 3 + 3 = 6 x − 3 (x≠3)
We have said x≠3 to avoid a division by zero.
Let's multiply through by (x − 3) :
2x + 3(x−3) = 6
Bring the 6 to the left:
2x + 3(x−3) − 6 = 0
Expand and solve:
2x + 3x − 9 − 6 = 0
5x − 15 = 0
5(x − 3) = 0
Which can be solved by having x=3
Let us check x=3 using the original question:
2 × 3 3 − 3 + 3 = 6 3 − 3
Hang On: 3 − 3 = 0 That means dividing by Zero!
And anyway, we said at the top that x≠3 , so ...
x = 3 does not actually work, and so:
There is No Solution!
That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!
This gives us a moral lesson:
"Solving" only gives us possible solutions, they need to be checked!
- Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
- Show all the steps , so it can be checked later (by you or someone else)
Microsoft Math Solver
Basic Math Examples
- Long Arithmetic
- Rational Numbers
- Operations with Fractions
- Ratios, Proportions, and Percents
- Measurement, Area, and Volume
- Factors, Fractions, and Exponents
- Unit Conversions
- Data Measurement and Statistics
- Points and Line Segments
- Terms ( Premium )
- DO NOT SELL MY INFO
- Mathway © 2023
Please ensure that your password is at least 8 characters and contains each of the following:
- a special character: @$#!%*?&
IMAGES
VIDEO
COMMENTS
Problems for 5th Grade. Multi-digit multiplication. Dividing completely. Writing expressions. Rounding whole numbers. Inequalities on a number line. Linear equation and inequality word problems. Linear equation word problems. Linear equation word problems.
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 Fractions of Measurements Fractions of Measurements 2 Adding Fractions Subtracting Fractions Adding Fractions: Fill in the Blank Multiplication: Fractions 1 Multiplication: Fractions 2
Practice Math Problems with Answers | Online Math Solver - Cymath Practice Problems Pre-Algebra Fractions GCF LCM Percentages Prime factorization Rationalizing Denominators Square and Cube Roots Algebra Completing the Square Exponents Factoring Imaginary Numbers Inequalities Logarithmic and Exponential Functions Partial Fractions
Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. ... Step-by-Step Examples. Basic Math. Long Arithmetic. Adding Using Long Addition. Long Subtraction. ... Solve for a Constant in a Given Solution. Solve the Bernoulli ...
Worked example: number of solutions to equations Number of solutions to equations Creating an equation with no solutions Creating an equation with infinitely many solutions Number of solutions to equations challenge Math > Algebra 1 > Solving equations & inequalities > Analyzing the number of solutions to linear equations
Free math problem solver answers your algebra homework questions with step-by-step explanations.
Get step-by-step solutions to your math problems Try Math Solver Type a math problem Solve Quadratic equation x2 − 4x − 5 = 0 Trigonometry 4sinθ cosθ = 2sinθ Linear equation y = 3x + 4 Arithmetic 699 ∗533 Matrix [ 2 5 3 4][ 2 −1 0 1 3 5] Simultaneous equation { 8x + 2y = 46 7x + 3y = 47 Differentiation dxd (x −5)(3x2 −2) Integration ∫ 01 xe−x2dx
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.
To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
Solution: Let x x is the number of kilograms he sold in the morning. Then in the afternoon, he sold Advertisement 2x 2x kilograms. So, the total is x + 2x = 3x x+2x=3x. This must be equal to 360. Advertisement 3x = 360 3x=360 x=3/360 x = 120 x=120 Therefore, the salesman sold 120 kg in the morning and In the afternoon, he sold 2 X 120 = 240kg. 2.
Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. Math Worksheets. Khan Academy. Math worksheets take forever to hunt down across the internet. Khan Academy is your one-stop-shop for practice from arithmetic to calculus. Math worksheets can vary in quality from ...
Solving algebraic word problems requires us to combine our ability to create equations and solve them. To solve an algebraic word problem: Define a variable. Write an equation using the variable. Solve the equation. If the variable is not the answer to the word problem, use the variable to calculate the answer.
The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation. Common clue words for addition problems: Sum Total In all Perimeter
Click here to see a video of the solution PROBLEM 6.1.1. 4 Determine the molarity of each of the following solutions: 1.457 mol KCl in 1.500 L of solution 0.515 g of H 2 SO 4 in 1.00 L of solution 20.54 g of Al (NO 3) 3 in 1575 mL of solution 2.76 kg of CuSO 4 ·5H 2 O in 1.45 L of solution 0.005653 mol of Br 2 in 10.00 mL of solution
Section 2.1 : Solutions and Solution Sets For each of the following determine if the given number is a solution to the given equation or inequality. Is x = 6 x = 6 a solution to 2x−5 = 3(1−x) +22 2 x − 5 = 3 ( 1 − x) + 22? Solution Is t = 7 t = 7 a solution to t2 +3t−10 = 4+8t t 2 + 3 t − 10 = 4 + 8 t? Solution
Solve a word problem: Rachel has 17 apples. She gives 9 to Sarah. How many apples does Rachel have now? Jack has 8 cats and 2 dogs. Jill has 7 cats and 4 dogs. How many dogs are there in all? if there are 40 cookies all together and A takes 10 and B takes 5 how many are left
Online practice problems for math, including arithmetic, algebra, calculus, linear algebra, number theory, and statistics. Get help from hints and Step-by-step solutions. ... Work with Step-by-step Solutions! Only Wolfram Problem Generator directly integrates the popular and powerful Step-by-step Solutions from Wolfram|Alpha. You can use a ...
Solving Word Problems Step 1: Identify (What is being asked?) Step 2: Strategize Step 3: Write the equation (s) Step 4: Answer the question Step 5: Check Show Video Lesson Problem Solving Strategy: Guess And Check Using the guess and check problem solving strategy to help solve math word problems. Example: Jamie spent $40 for an outfit.
Final conclusion and motivation! Understand and visualize the problem clearly. Use experiences, hypotheses, heuristics, and patterns. Do in-depth research about concepts. Try to break the problem into smaller problems or develop a step-by-step process to solve it efficiently. Think of solutions using multiple approaches!
The best way to explain how the Venn diagram works and what its formulas show is to give 2 or 3 circles Venn diagram examples and problems with solutions. Problem-solving using Venn diagram is a widely used approach in many areas such as statistics, data science, business, set theory, math, logic and etc.
What is a Solution? A Solution is a value we can put in place of a variable (such as x) that makes the equation true. Example: x − 2 = 4. ... You might like to practice solving some animated equations. More Than One Solution. There can be more than one solution. Example: (x−3)(x−2) = 0. When x is 3 we get:
Online math solver with free step by step solutions to algebra, calculus, and other math problems. ... Microsoft Math Solver. Solve Practice Download. Solve Practice. Topics ... Type a math problem. Type a math problem. Solve. Examples. Quadratic equation { x } ^ { 2 } - 4 x - 5 = 0 ...
Learn about algebra using our free math solver with step-by-step solutions.
Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.
quick equation solving #maths #shorts #equation #solving #solution