A ratio compares values .

A ratio says how much of one thing there is compared to another thing.

Ratios can be shown in different ways:

Use the ":" to separate the values:   3 : 1
Or we can use the word "to":   3 to 1
Or write it like a fraction:  

A ratio can be scaled up:

Try it Yourself

Using ratios.

The trick with ratios is to always multiply or divide the numbers by the same value .

is the same as ×2 ×2


Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk is 3 : 2

To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:

3 ×4 : 2 ×4 = 12 : 8

In other words, 12 cups of flour and 8 cups of milk .

The ratio is still the same, so the pancakes should be just as yummy.

"Part-to-Part" and "Part-to-Whole" Ratios

The examples so far have been "part-to-part" (comparing one part to another part).

But a ratio can also show a part compared to the whole lot .

Example: There are 5 pups, 2 are boys, and 3 are girls


The ratio of boys to girls is 2:3 or 2 / 3

The ratio of girls to boys is 3:2 or 3 / 2


The ratio of boys to all pups is 2:5 or 2 / 5

The ratio of girls to all pups is 3:5 or 3 / 5

Try It Yourself

We can use ratios to scale drawings up or down (by multiplying or dividing).

The height to width ratio of the Indian Flag is

So for every (inches, meters, whatever) of height
there should be of width.

If we made the flag 20 inches high, it should be 30 inches wide.

If we made the flag 40 cm high, it should be 60 cm wide (which is still in the ratio 2:3)

Example: To draw a horse at 1/10th normal size, multiply all sizes by 1/10th

This horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is

1500 : 2000

What is that ratio when we draw it at 1/10th normal size?

1500 : 2000   = 1500 : 2000

We can make any reduction/enlargement we want that way.

Allie measured her foot and it was 21cm long, and then she measured her Mother's foot, and it was 24cm long.

"I must have big feet, my foot is nearly as long as my Mom's!"

But then she thought to measure heights, and found she is 133cm tall, and her Mom is 152cm tall.

In a table this is:

  Allie Mom
Length of Foot: 21cm 24cm
Height: 133cm 152cm

The "foot-to-height" ratio in fraction style is:

Allie:   Mom:

We can simplify those fractions like this:

And we get this (please check that the calcs are correct):

"Oh!" she said, "the Ratios are the same".

"So my foot is only as big as it should be for my height, and is not really too big."

You can practice your ratio skills by Making Some Chocolate Crispies

Ratio Math Problems - Three Term Ratios

In these lessons, we will learn how to solve ratio word problems that involve three terms.

Related Pages Two-Term Ratio Word Problems More Ratio Word Problems Math Word Problems More Algebra Lessons

Ratio problems are word problems that use ratios to relate the different items in the question.

Ratio problems: Three-term Ratios

Example 1: A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a bag of the mixture contains 3 pounds of rice, how much corn does it contain?

Step 2: Solve the equation: Cross Multiply

2 × x = 3 × 5 2x = 15

Answer: The mixture contains 7.5 pounds of corn.

Example 2: Clothing store A sells T-shirts in only three colors: red, blue and green. The colors are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many T-shirts does it have altogether?

Solution: Step 1: Assign variables: Let x = red shirts y = green shirts

Step 2: Solve the equation: Cross Multiply both equations 3 × 20 = x × 4 60 = 4x x = 15

5 × 20 = y × 4 100 = 4y y = 25

The total number of shirts would be 15 + 25 + 20 = 60

Answer: There are 60 shirts.

How to solve Ratio Word Problems with three terms?

Example: A piece of string that is 63 inches long is cut into 3 parts such that the lengths of the parts of the string are in the ratio of 5 to 6 to 10. Find the lengths of the 3 parts.

How to solve Two Term and Three Term Ratio Problems?

A Ratio compares two things that have the same units A Part to Part Ratio compares one thing to another thing A Part to Total (whole) Ratio compares one thing to the total number

Example: In a class of 30 students, there are 18 girls and 12 boys. What is the ratio of boys to girls? What is the ratio of girls to boys? What is the ratio of girls to total?

We can have a three term ratio of red to blue to green marbles.

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Problem solving with fractions and ratios

I can use my knowledge of fractions and ratios to solve problems.

Lesson details

Key learning points.

  • Any fraction can be turned into a ratio.
  • Moving between fractions and ratios can make a problem easier.
  • There can be lots of information, it is important to think about what is required to solve the problem.

Common misconception

Always dividing the amount by the sum of the 'parts' of the ratio to get one 'part'.

Offer opportunities to match problems to bar models ensure the same numbers are used to highlight the differences.

Proportion - Proportionality means when variables are in proportion if they have a constant multiplicative relationship.

Ratio - A ratio shows the relative sizes of 2 or more values and allows you to compare a part with another part in a whole.

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Ratio Questions And Practice Problems: Differentiated Practice Questions Included

Beki Christian

Ratio questions appear throughout middle and high school, building on students’ knowledge year on year. Here we provide a range of ratio questions and practice problems of varying complexity to use with your own students in class or as inspiration for creating your own.

What is ratio?

Ratio is used to compare the size of different parts of a whole. For example, the total number of students in a class is 30. There are 10 girls and 20 boys. The ratio of girls:boys is 10:20 or 1:2. For every one girl there are two boys. 

Ratio Worksheet

Ratio Worksheet

Download this quiz to check your students’ understanding of ratios. Features 10 questions with answers to identify areas of strength and support!

Uses of ratio

You might see ratios written on maps to show the scale of the map or use ratios to determine the currency exchange rate if you are going on vacation to another country.

Ratio will be seen as a topic in its own right as well as appearing within other topics. An example of this might be the area of two shapes being in a given ratio or the angles of a shape being in a given ratio.

third space learning ratio lesson using visuals to support

Ratio in Middle School and High School

In middle school, ratio questions will involve writing and simplifying ratios, using equivalent ratios, dividing quantities into a given ratio and will begin to look at solving problems involving ratio. At high school, these skills are recapped and the focus will be more on ratio word problems which will require you to conduct problem solving using your knowledge of ratio.

Proportion and ratio

Ratio often appears alongside proportion and the two topics are related. Whereas ratio compares the size of different parts of a whole, proportion compares the size of one part with the whole. Given a ratio, we can find a proportion and vice versa.

Take the example of a box containing 7 counters; 3 red counters and 4 blue counters:

The ratio of red counters:blue counters is 3:4.

For every 3 red counters there are four blue counters.

The proportion of red counters is \frac{3}{7} and the proportion of blue counters is \frac{4}{7}.

3 out of every 7 counters are red and 4 out of every 7 counters are blue.

Direct proportion and inverse proportion

From 7th grade onwards, students learn about direct proportion and inverse proportion. When two things are directly proportional to each other, one can be written as a multiple of the other and therefore they increase at a fixed ratio.

How to solve a ratio problem

When looking at a ratio problem, the key pieces of information that you need are what the ratio is, whether you have been given the whole amount or a part of the whole and what you are trying to work out. 

If you have been given the whole amount you can follow these steps to answer the question:

  • Add together the parts of the ratio to find the total number of shares.
  • Divide the total amount by the total number of shares.
  • Multiply by the number of shares required.

If you have been been given a part of the whole you can follow these steps:

  • Identify which part you have been given and how many shares it is worth.
  • Use equivalent ratios to find the other parts.
  • Use the values you have to answer your problem.

How to solve a proportion problem

As we have seen, ratio and proportion are strongly linked. If we are asked to find what proportion something is of a total, we need to identify the amount in question and the total amount. We can then write this as a fraction:

Proportion problems can often be solved using scaling. To do this you can follow these steps:

  • Identify the values that you have been given which are proportional to each other.
  • Use division to find an equivalent relationship.
  • Use multiplication to find the required relationship.

Real life ratio problems and proportion problems

Ratio is all around us. Let’s look at some examples of where we may see ratio and proportion:

Cooking ratio question

When making yogurt, the ratio of starter yogurt to milk should be 1:9. I want to make 1,000 ml of yogurt. How much milk should I use?

Here we know the full amount – 1,000 ml.

The ratio is 1:9 and we want to find the amount of milk.

  • Total number of shares = 1 + 9 = 10
  • Value of each share: 1,000 ÷ 10 = 100
  • The milk is 9 shares so 9 × 100 = 900

I need to use 900ml of milk.

Maps ratio question

The scale on a map is 1:10,000. What distance would 3.5cm on the map represent in real life?

Here we know one part is 3.5. We can use equivalent ratios to find the other part.

The distance in real life would be 35,000cm or 350m.

Speed proportion question

I traveled 60 miles in 2 hours. Assuming my speed doesn’t change, how far will I travel in 3 hours?

This is a proportion question.

  • I traveled 60 miles in 2 hours.
  • Dividing by 2, I traveled 30 miles in one hour.
  • Multiplying by 3, I would travel 90 miles in 3 hours.

third space learning tuition slide on an exam style question

Middle School ratio questions

Ratio is introduced in middle school. Writing and simplifying ratios is explored and the idea of dividing quantities in a given ratio is introduced using proportion word problems such as the question below, before being linked with the mathematical notation of ratio.

Example Middle School worded question

Richard has a bag of 30 sweets. Richard shares the sweets with a friend. For every 3 sweets Richard eats, he gives his friend 2 sweets. How many sweets do they each eat?

Practice ratio questions for middle school

At this level, ratio questions ask you to write and simplify a ratio, to divide quantities into a given ratio and to solve problems using equivalent ratios. See below the example questions to support test prep.

Ratio questions for 6th grade

1. In Lucy’s class there are 12 boys and 18 girls. Write the ratio of girls to boys in its simplest form.

GCSE Quiz False

The question asks for the ratio of girls to boys, so girls must be first and boys second. It also asks for the answer in its simplest form.

2. The ratio of cups of flour to cups of water in a pizza dough recipe is 9:4. A pizza restaurant makes a large quantity of dough, using 36 cups of flour. How much water should they use?

The ratio of cups of flour to cups of water is 9:4. We have been given one part so we can work this out using equivalent ratios.

Ratio questions 7th grade

3. The ratio of men to women working in a company is 3:5. What proportion of the employees are women?

In this company, the ratio of men to women is 3:5 so for every 3 men there are 5 women.

This means that for every 8 employees, 5 of them are women.

Therefore \frac{5}{8} of the employees are women.

4. Mac traveled 30 miles in \frac{3}{4} of an hour. Assuming his speed doesn’t change, how far will Mac travel in 1 hour?

We have been given a part so we can work this out using equivalent ratios.

The ratio of miles to hours is 30: \frac{3}{4} .

To create an equivalent ratio, divide each side by the same number. Since we are solving to find how far Mac will travel in 1 hour, divide both sides by \frac{3}{4} .

30: \frac{3}{4}

30 \div \frac{3}{4}: \frac{3}{4} \div \frac{3}{4}

Mac will travel 40 miles in 1 hour.

Ratio questions 8th grade

While the Common Core State Standards does not explicitly include ratio and proportional relationships in the 8th grade, it may pop up on your own curriculums and offers a good opportunity to revisit and extend their knowledge of ratio and proportion before they enter high school.

5. The angles in a triangle are in the ratio 3:4:5. Work out the size of each angle.

30^{\circ} , 40^{\circ} and 50^{\circ}

22.5^{\circ},  30^{\circ} and 37.5^{\circ}

60^{\circ} , 60^{\circ} and 60^{\circ}

45^{\circ} , 60^{\circ} and 75^{\circ}

The angles in a triangle add up to 180 ^{\circ} . Therefore 180 ^{\circ} is the whole and we need to divide 180 ^{\circ} in the ratio 3:4:5.

The total number of shares is 3 + 4 + 5 = 12.

Each share is worth 180 ÷ 12 = 15 ^{\circ} .

3 shares is 3 x 15 = 45 ^{\circ} .

4 shares is 4 x 15 = 60 ^{\circ} .

5 shares is 5 x 15 = 75 ^{\circ} .

6. Paint Pro makes pink paint by mixing red paint and white paint in the ratio 3:4.

Colour Co makes pink paint by mixing red paint and white paint in the ratio 5:7. 

Which company uses a higher proportion of red paint in their mixture?

They are the same

It is impossible to tell

The proportion of red paint for Paint Pro is \frac{3}{7}

The proportion of red paint for Colour Co is \frac{5}{12}

We can compare fractions by putting them over a common denominator using equivalent fractions

\frac{3}{7} = \frac{36}{84} \hspace{3cm} \frac{5}{12}=\frac{35}{84}

\frac{3}{7} is a bigger fraction so Paint Pro uses a higher proportion of red paint. 

High school ratio questions

At high school, we apply the knowledge that we have of ratios to solve different problems. Ratio can be linked with many different topics, for example similar shapes and probability, as well as appearing as problems in their own right.

Ratio high school questions (low difficulty)

7. The students in Ellie’s class walk, cycle or drive to school in the ratio 2:1:4. If 8 students walk, how many students are there in Ellie’s class altogether?

We have been given one part so we can work this out using equivalent ratios.

The total number of students is 8 + 4 + 16 = 28

8. A bag contains counters. 40% of the counters are red and the rest are yellow.

Write down the ratio of red counters to yellow counters. Give your answer in the form 1:n.

If 40% of the counters are red, 60% must be yellow and therefore the ratio of red counters to yellow counters is 40:60. Dividing both sides by 40 to get one on the left gives us

Since the question has asked for the ratio in the form 1:n, it is fine to have decimals in the ratio.

9. Rosie and Jim share some sweets in the ratio 5:7. If Jim gets 12 sweets more than Rosie, work out the number of sweets that Rosie gets.

Jim receives 2 shares more than Rosie, so 2 shares is equal to 12.

Therefore 1 share is equal to 6. Rosie receives 5 shares: 5 × 6 = 30.

10. Rahim is saving for a new bike which will cost $480. Rahim earns $1,500 per month. Rahim spends his money on bills, food and extras in the ratio 8:3:4. Of the money he spends on extras, he spends 80% and puts 20% into his savings account.

How long will it take Rahim to save for his new bike?

Rahim’s earnings of $1,500 are divided in the ratio of 8:3:4.

The total number of shares is 8 + 3 + 4 = 15.

Each share is worth $ 1,500 ÷ 15 = £100 .

Rahim spends 4 shares on extras so 4 × $ 100 = $400 .

20% of $400 is $80.

The number of months it will take Rahim is $ 480 ÷ $ 80 = 6

Ratio GCSE exam questions higher

11. The ratio of milk chocolates to white chocolates in a box is 5:2. The ratio of milk chocolates to dark chocolates in the same box is 4:1.

If I choose one chocolate at random, what is the probability that that chocolate will be a milk chocolate?

To find the probability, we need to find the fraction of chocolates that are milk chocolates. We can look at this using equivalent ratios.

To make the ratios comparable, we need to make the number of shares of milk chocolate the same in both ratios. Since 20 is the LCM of 4 and 5 we will make them both into 20 parts.

We can now say that milk to white to dark is 20:8:5. The proportion of milk chocolates is \frac{20}{33} so the probability of choosing a milk chocolate is \frac{20}{33} .

12. In a school the ratio of girls to boys is 2:3. 

25% of the girls have school dinners.

30% of the boys have school dinners.

What  is the total percentage of students at the school who have school dinners?

In this question you are not given the number of students so it is best to think about it using percentages, starting with 100%.

100% in the ratio 2:3 is 40%:60% so 40% of the students are girls and 60% are boys.

25% of 40% is 10%.

30% of 60% is 18%.

The total percentage of students who have school dinners is 10 + 18 = 28%.

13. For the cuboid below, a:b = 3:1 and a:c = 1:2.

ratio question gcse higher

Find an expression for the volume of the cuboid in terms of a.

If a:b = 3:1 then b=\frac{1}{3}a

If a:c = 1:2 then c=2a.

GCSE Ratio question higher - answer - temp

Ratio high school questions (average difficulty)

14. Bill and Ben win some money in their local lottery. They share the money in the ratio 3:4. Ben decides to give $40 to his sister. The amount that Bill and Ben have is now in the ratio 6:7.

Calculate the total amount of money won by Bill and Ben.

Initially the ratio was 3:4 so Bill got $3a and Ben got $4a. Ben then gave away $40 so he had $(4a-40).

The new ratio is 3a:4a-40 and this is equal to the ratio 6:7.

Since 3a:4a-40 is equivalent to 6:7, 7 lots of 3a must be equal to 6 lots of 4a-40.

The initial amounts were 3a:4a. a is 80 so Bill received $240 and Ben received $320.

The total amount won was $560.

15. On a farm the ratio of pigs to goats is 4:1. The ratio of pigs to piglets is 1:6 and the ratio of goats to kids is 1:2.

What fraction of the animals on the farm are babies?

The easiest way to solve this is to think about fractions.

\\ \frac{4}{5} of the animals are pigs, \frac{1}{5} of the animals are goats.

\frac{1}{7} of the pigs are adult pigs, so  \frac{1}{7}   of  \frac{4}{5} is  \frac{1}{7} \times \frac{4}{5} = \frac{4}{35}

\frac{6}{7} of the pigs are piglets, so \frac{6}{7} of \frac{4}{5} is \frac{6}{7} \times \frac{4}{5} = \frac{24}{35}

\frac{1}{3}   of the goats are adult goats, so \frac{1}{3} of \frac{1}{5} is \frac{1}{3} \times \frac{1}{5} = \frac{1}{15}

\frac{2}{3}   of the goats are kids, so \frac{2}{3} of \frac{1}{5} is \frac{2}{3} \times \frac{1}{5} = \frac{2}{15}

The total fraction of baby animals is \frac{24}{35} + \frac{2}{15} = \frac{72}{105} +\frac{14}{105} = \frac{86}{105}

Looking for more middle school and high school ratio math questions?

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→ → Ratios

Find here an unlimited supply of worksheets with simple word problems involving ratios, meant for 6th-8th grade math. In , the problems ask for a specific ratio (such as, " "). In , the problems are the same but the ratios are supposed to be simplified.

contains varied word problems, similar to these:

Options include choosing the number of problems, the amount of workspace, font size, a border around each problem, and more. The worksheets can be generated as PDF or html files.

Each worksheet is randomly generated and thus unique. The and is placed on the second page of the file.

You can generate the worksheets — both are easy to print. To get the PDF worksheet, simply push the button titled " " or " ". To get the worksheet in html format, push the button " " or " ". This has the advantage that you can save the worksheet directly from your browser (choose File → Save) and then in Word or other word processing program.

Sometimes the generated worksheet is not exactly what you want. Just try again! To get a different worksheet using the same options:

What is the ratio given in the word problem? (grade 6)


What is the ratio given in the word problem? (with harder numbers; grade 6)


Solve ratio word problems (grade 7)


Solve ratio word problems
(more workspace; grade 7)


Use the generator to make customized ratio worksheets. Experiment with the options to see what their effect is.

(These determine the number of problems)

(only for levels 1 & 2):
      Range from
Page orientation:
    Font Size: 

Workspace: lines below each problem
Additional title & instructions  (HTML allowed)

Primary Grade Challenge Math cover

Primary Grade Challenge Math by Edward Zaccaro

A good book on problem solving with very varied word problems and strategies on how to solve problems. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Each chapter’s questions are broken down into four levels: easy, somewhat challenging, challenging, and very challenging.

solving problems with fractions and ratio

Visual maths worksheets, each maths worksheet is differentiated and visual.

Fraction, Percentage, and Ratio Problems Worksheet Suitable for Year 9, Year 10, and Year 11 Students

Fraction, Percentage and Ratio Problems worksheet

Total reviews: (0), fraction, percentage and ratio problems worksheet description.

This worksheet focuses on tackling problems that involve working with fractions, percentages, and ratios. These types of questions are very common in GCSE exams, and students studying either the foundation or higher tier curriculum will frequently encounter scenarios like these.

Learners will warm up by recapping how to find fractions and percentages of amounts, and how to share amounts into given ratios in Section A, by answering 9 quick questions.

Section B then consists of six problem-solving questions where the above skills are combined. Questions 5 and 6 are slightly different and require students to find a whole when given a fractional amount, and to find one part of a ratio when given another.

Related to Fraction, Percentage and Ratio Problems worksheet:

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Problem Solving with Ratio and Fractions

November 5, 2020.

Students are challenged to solve a range of problems involving arithmetic with fractions.

There are five problems that link to ratio, probability, mean averages and money.

Fractions  |  Foundation GCSE Maths | Higher GCSE Maths

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Grade 9 coordinate geometry problems.

How to solve geometrical problems involving the properties of shapes and straight line graphs.

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How do we solve ratio problems?


Definition: A comparison between quantities using division.
Examples : 3:2 , 3:2:88, 3 to 2, 3 to 2 to 88

A 2 to 5 ratio can be represented as 2:5

A ration between X and Y can be written

MEDIUM SAT PROBLEM #8 out of a 25 problem section

A bucket holds 4 quarts of popcorn. If 1/3 cup of corn kernels makes 2 quarts of popcorn, how many buckets can be filled with the popcorn made from 4 cups of kernels?

(Saturday/1.200) p. 524

C. 6



A recipe for making 10 loaves of bread requires 24 cups of flour and 4 tablespoons of baking powder. If the proportions in this recipe are to be used to make 3 loaves of bread, how many cups of flour will be needed? (Do not round your answer)

B) 7.2



The ratio of to to to to is 5 to 4 to 3 to 2 to 1. If = 60, what is the value of ?

(Saturday/5.2002 #16) p. 659


This should be fairly easy if you use the method we have practiced. Assuming you have the method down, the only reason to get this wrong is a careless arithmetic error.

For those of you making these errors, I hope you're starting to see why it's  a better use of your limited test taking time to double check your work on the medium questions than to rip your hair out and spend a lot of time trying to puzzle out the hardest questions which very few people can answer. (Listen try those last ones if you get there, make sure you think through these medium ones on which most people can more consistently score points.)

#19 out of 25

Five of the 12 members of a club are girls and the rest are boys. What is the ratio of boys to girls in the club?(Grid your ratio as a fraction)

Student Generated Response: Grid-in

( Sunday/5.1997 #19) p. 499

7/5 or 1.4


#17 out of a 25 problem section

The ratio of 1.5 to 32 is the same as the ratio of 0.15 to x. What is the value of x?

(Student generated response)
(#17 out of 25 Satuday/11.1996 SAT) p.383 *

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Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

Basic Math Solutions

Below are examples of basic math problems that can be solved.

  • Long Arithmetic
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  • Factors, Fractions, and Exponents
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Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

  • Variables, Expressions, and Integers
  • Simplifying and Evaluating Expressions
  • Solving Equations
  • Multi-Step Equations and Inequalities
  • Ratios, Proportions, and Percents
  • Linear Equations and Inequalities

Algebra Solutions

Below are examples of Algebra math problems that can be solved.

  • Algebra Concepts and Expressions
  • Points, Lines, and Line Segments
  • Simplifying Polynomials
  • Factoring Polynomials
  • Linear Equations
  • Absolute Value Expressions and Equations
  • Radical Expressions and Equations
  • Systems of Equations
  • Quadratic Equations
  • Inequalities
  • Complex Numbers and Vector Analysis
  • Logarithmic Expressions and Equations
  • Exponential Expressions and Equations
  • Conic Sections
  • Vector Spaces
  • 3d Coordinate System
  • Eigenvalues and Eigenvectors
  • Linear Transformations
  • Number Sets
  • Analytic Geometry

Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

  • Algebra Concepts and Expressions Review
  • Right Triangle Trigonometry
  • Radian Measure and Circular Functions
  • Graphing Trigonometric Functions
  • Simplifying Trigonometric Expressions
  • Verifying Trigonometric Identities
  • Solving Trigonometric Equations
  • Complex Numbers
  • Analytic Geometry in Polar Coordinates
  • Exponential and Logarithmic Functions
  • Vector Arithmetic

Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

  • Operations on Functions
  • Rational Expressions and Equations
  • Polynomial and Rational Functions
  • Analytic Trigonometry
  • Sequences and Series
  • Analytic Geometry in Rectangular Coordinates
  • Limits and an Introduction to Calculus

Calculus Solutions

Below are examples of Calculus math problems that can be solved.

  • Evaluating Limits
  • Derivatives
  • Applications of Differentiation
  • Applications of Integration
  • Techniques of Integration
  • Parametric Equations and Polar Coordinates
  • Differential Equations

Statistics Solutions

Below are examples of Statistics problems that can be solved.

  • Algebra Review
  • Average Descriptive Statistics
  • Dispersion Statistics
  • Probability
  • Probability Distributions
  • Frequency Distribution
  • Normal Distributions
  • t-Distributions
  • Hypothesis Testing
  • Estimation and Sample Size
  • Correlation and Regression

Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

  • Polynomials and Expressions
  • Equations and Inequalities
  • Linear Functions and Points
  • Systems of Linear Equations
  • Mathematics of Finance
  • Statistical Distributions

Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

  • Introduction to Matrices
  • Linear Independence and Combinations

Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

  • Unit Conversion
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  • Molecules and Compounds
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  • Behavior of Gases
  • Solutions and Concentrations

Physics Solutions

Below are examples of Physics math problems that can be solved.

  • Static Equilibrium
  • Dynamic Equilibrium
  • Kinematics Equations
  • Electricity
  • Thermodymanics

Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

  • Step By Step Graphing
  • Linear Equations and Functions
  • Polar Equations

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Math is tripping up community college students. Some schools are trying something new

ALBANY, Ore. – It’s 7:15 on a Monday morning in May at Linn-Benton Community College in northwestern Oregon. Math professor Michael Lopez, a tape measure on his belt, paces in front of the 14 students in his “math for welders” class. “I’m your OSHA inspector,” he says. “Three-sixteenths of an inch difference, you’re in violation. You’re going to get a fine.”

He has just given them a project they might have to do on the job: figuring out the rung spacing on a steel ladder that attaches to a wall. Thousands of dollars are at stake in such builds, and they’re complicated: Some clients want the fewest possible rungs to save money; others want a specific distance between steps. To pass inspection, rungs must be evenly spaced to within one-sixteenth of an inch.

Math is a giant hurdle for most community college students pursuing welding and other career and technical degrees. About a dozen years ago, Linn-Benton’s administrators looked at their data and found that many students in career and technical education, or CTE, were getting most of the way toward a degree but were stopped by a math course, said the college’s president, Lisa Avery. That’s not unusual: Up to 60% of students entering community college are unprepared for college-level work, and the subject they most often need help with is math .

The college asked the math department to design courses tailored to those students, starting with its welding, culinary arts and criminal justice programs. The first of those, math for welders, rolled out in 2013.

More than a decade later, welding department instructors say math for welders has had a huge effect on student performance. Since 2017, 93% of students taking it have passed, and 83% have achieved all the course’s learning goals, including the ability to use arithmetic, geometry, algebra and trigonometry to solve welding problems, school data shows. Two years ago, Linn-Benton asked Lopez to design a similar course for its automotive technology program; the college began to offer that course last fall.

Math for welders changed student Zane Azmane’s view of what he could do. “I absolutely hated math in high school. It didn’t apply to anything I needed at the moment,” said Azmane, 20, who failed several semesters of math early in high school but got a B in the Linn-Benton course last year. “We actually learned equations I’m going to use, like setting ladder rungs.”

Linn-Benton’s aim is to change how students pursuing technical degrees learn math by making it directly applicable to their technical specialties.

Some researchers say these small-scale efforts to teach math in context could transform how it’s taught more broadly.

Among the strategies to help college students who struggle with math, giving them contextual curriculums seems to have "the strongest theoretical base and perhaps the strongest empirical support,” according to a 2011 paper by Dolores Perin, now professor emerita at Columbia University Teachers College. (The Hechinger Report is an independent unit of Teachers College.)

Perin’s paper echoed the results of a 2006 study of math in high school CTE involving almost 3,000 students. Students in the study who were taught math through an applied approach performed significantly better on two of three standardized tests than those taught math in a more traditional way. (The applied math students also performed better on the third test, though the results were not statistically significant.)

There haven’t been systematic studies of math in CTE at the college level, according to James Stone, director of the National Research Center for Career and Technical Education at the Southern Regional Education Board, who ran the 2006 study.

Oregon appears to be one of the few places where this approach is spreading, if slowly.

Three hours south of Linn-Benton, Doug Gardner, an instructor in the Rogue Community College math department, had long struggled with a persistent question from students: “Why do we need to know this?”

“It became my life’s work to have an answer to that question,” said Gardner, now the department chair. 

Meanwhile, at the college, about a third of students taking Algebra I or a lower-level math course failed or withdrew. For many who stayed, the lack of math knowledge hurt their job prospects, preventing them from gaining necessary skills in fields like pipefitting.

So, in 2010, Gardner applied for and won a National Science Foundation grant to create two new applied algebra courses. Instead of abstract formulas, students would learn practical ones: how to calculate the volume of a wheelbarrow of gravel and the number of wheelbarrows needed to cover an area, or how much a beam of a certain size and type would bend under a certain load.

Since then, the pass rate in the applied algebra class has averaged 73%, while the rate for the traditional course has continued to hover around 59%, according to Gardner.

One day in May, math professor Kathleen Foster was teaching applied algebra in a sun-drenched classroom on Rogue’s campus. She launched into a lesson about the Pythagorean theorem and why it’s an essential tool for building home interiors and steel structures.

James Butler-Kyniston, 30, who is pursuing a degree as a machinist, said the exercises covered in Foster’s class are directly applicable to his future career. One exercise had students calculate how large a metal sheet you would need to manufacture a certain number of parts at a time. “Algebraic formulas apply to a lot of things, but since you don’t have any examples to tie them to, you end up thinking they’re useless,” he said.

In 2021, Oregon state legislators passed a law requiring all four-year colleges to accept an applied math community college course called Math in Society as satisfying the math requirement for a four-year degree. In that course, instead of studying theoretical algebra, students learn how to use probability and statistics to interpret the results in scientific papers and how political rules like apportionment and gerrymandering affect elections, said Kathy Smith, a math professor at Central Oregon Community College.

“If I had my way, this is how algebra would be taught to every student: the applied version,” Gardner said. “And then if a student says ‘This is great, but I want to go further,’ then you sign up for the theoretical version.”

But at individual schools, lack of money and time constrain the spread of applied math. Stone’s team works with high schools around the country to design contextual math courses for career and technical students. They tried to work with a few community colleges, but their CTE faculty, many of whom were part-timers on contract, didn’t have time to partner with their math departments to develop a new curriculum, a yearlong process, Stone said.

Linn-Benton was able to invest the time and money because its math department was big enough to take on the task, said Avery. Both Linn-Benton and Rogue may be outliers because they have math faculty with technical backgrounds: Lopez worked as a carpenter and sheriff’s deputy and served three tours as a machine gunner in Iraq, and Gardner was a construction contractor.

Back in Lopez’s class, students are done calculating where their ladder rungs should go and now must mark them on the wall.

As teams finish up, Lopez inspects their work. “That’s one-thirty-second shy. But I wouldn’t worry too much about it,” he tells one group. “OSHA’s not going to knock you down for that.”

Three teams pass, two fail – but this is the place to make mistakes, not out on the job, Lopez tells them.

“This stuff is hard,” said Keith Perkins, 40, who’s going for a welding degree and wants to get into the local pipe fitters union. “I hated math in school. Still hate it. But we use it every day.”

This story was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education.

More From Forbes

The most rigorous math program you've never heard of.

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Math-M-Addicts students eagerly dive into complex math problems during class.

In the building of the Speyer Legacy School in New York City, a revolutionary math program is quietly producing some of the city's most gifted young problem solvers and logical thinkers. Founded in 2005 by two former math prodigies, Math-M-Addicts has grown into an elite academy developing the skills and mindset that traditional schooling often lacks.

"We wanted to establish the most advanced math program in New York," explains Ruvim Breydo, co-founder of Math-M-Addicts. "The curriculum focuses not just on mathematical knowledge, but on developing a mastery of problem-solving through a proof-based approach aligned with prestigious competitions like the International Mathematical Olympiad."

From its inception, Math-M-Addicts took an unconventional path. What began as an attempt to attract only the highest caliber high school students soon expanded to offer multiple curriculum levels. "We realized we couldn't find enough kids at the most advanced levels," says Breydo. "So we decided to develop that talent from an earlier age."

The program's approach centers on rigor. At each of the 7 levels, the coursework comprises just a handful of fiendishly difficult proof-based math problems every week. "On average, we expect them to get about 50% of the solutions right," explains instructor Natalia Lukina. "The problems take hours and require grappling with sophisticated mathematical concepts."

But it's about more than just the content. Class sizes are small, with two instructors for every 15-20 students. One instructor leads the session, while the other teacher coordinates the presentation of the homework solutions by students. The teachers also provide customized feedback by meticulously reviewing each student's solutions. "I spend as much time analyzing their thought processes as I do teaching new material," admits instructor Bobby Lee.

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Lee and the Math-M-Addicts faculty embrace an unconventional pedagogy focused on developing logic, creativity, and a tenacious problem-solving mindset over procedures. "We don't dumb it down for them," says Breydo. "We use technical math language and allow students to struggle through the challenges because that's where real learning happens."

Impressive results of Math-M-addicts students in selective math competitions highlight their ... [+] preparation and dedication.

For the Math-M-Addicts team, finding the right teachers is as essential as shaping brilliant students. Prospective instructors go through a rigorous multi-stage vetting process. "We seek passionate mathematical problem solvers first," says program director Sonali Jasuja. "Teaching experience is great, but first and foremost, we need people who deeply understand and enjoy the reasoning behind mathematics."

Even exceptional instructors undergo extensive training by co-teaching for at least a year alongside veteran Math-M-Addicts faculty before taking the lead role. "Our approach is different from how most US teachers learned mathematics," explains instructor Tanya Gross, the director of Girls Adventures in Math (GAIM) competition. "We immerse them in our unique math culture, which focuses on the 'why' instead of the 'how,' empowering a paradigm shift."

That culture extends to the students as well. In addition to the tools and strategies imparted in class, Math-M-Addicts alumni speak of an unshakable confidence and camaraderie that comes from up to several thousands of hours grappling with mathematics at the highest levels alongside peers facing the same challenges.

As Math-M-Addicts ramps up efforts to expand access through online classes and global partnerships, the founders remain devoted to their core mission. "Math education should not obsess with speed and memorization of math concepts," argues Breydo. "This is not what mathematics is about. To unlock human potential, we must refocus on cognitive reasoning and problem-solving skills. We are seeking to raise young people unafraid to tackle any complex challenge they face"

Julia Brodsky

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