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How to Do Fractions

Last Updated: November 28, 2022 Fact Checked

Understanding Fractions

Adding and subtracting, multiplying and dividing, expert q&a.

This article was co-authored by David Jia and by wikiHow staff writer, Hunter Rising . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 541,785 times.

Fractions represent how many parts of a whole you have, which makes them useful for taking measurements or calculating precise values. Fractions can be a difficult concept to learn since they have special terms and rules for using them in equations. Once you understand the parts of a fraction, practice doing addition and subtraction problems with them. When you know how to add and subtract fractions, you can move on to trying multiplication and division with fractions.

Step 1 Identify the numerator and denominator.

  • For example, in the fraction ½, the 1 is the numerator and 2 is the denominator.
  • You can also write fractions on a single line, like 4/5. The number on the left is always the numerator and the number on the right is the denominator.

Step 2 Know fractions are equal if you multiply the numerator and denominator by the same number.

  • For example, if you want to make an equivalent fraction to 3/5, you can multiply both numbers by 2 to make the fraction 6/10.
  • In a real-world example, if you have 2 equal slices of pizza and you cut one of them in half, the two halves are still the same amount as the other full slice.

Step 3 Simplify fractions by dividing the numerator and denominator by a common multiple.

  • For example, if you have the fraction 2/8, both the numerator and denominator are divisible by 2. Divide each number by 2 to get 2/8 = 1/4.

Step 4 Convert improper fractions to mixed numbers if the numerator is greater than the denominator.

  • For example, if you want to simplify 7/3, divide 7 by 3 to get the answer 2 with a remainder of 1. Your new mixed number will look like 2 ⅓.

Tip: If the numerator and denominator equal one another, then they can always be simplified to 1.

Step 5 Change mixed numbers into fractions when you need to use them in equations.

  • For example, if you want to convert 5 ¾ to an improper fraction, multiply 5 x 4 = 20. Add 20 to the numerator to get the fraction 23/4.

Step 1 Add or subtract just the numerators if the denominators are the same.

  • For example, if you wanted to solve 3/5 + 1/5, rewrite the equation as (3+1)/5 = 4/5.
  • If you want to solve 5/6 - 2/6, write it as (5-2)/6 = 3/6. Both the numerator and denominator are divisible by 3, so you can simplify the fraction to 1/2.
  • If you have mixed numbers, remember to change them to improper fractions first. For example, if you want to solve 2 ⅓ + 1 ⅓, change the mixed numbers so the problem reads 7/3 + 4/3. Rewrite the equation like (7 + 4)/3 = 11/3. Then convert it back to a mixed number, which would be 3 ⅔.

Warning: Never add or subtract the denominators. The denominators only represent how many parts make up a whole while the numerator represents how many parts you have.

Step 2 Find a common multiple for the denominators if they’re different.

  • For example, if you want to solve 1/6 + 2/4, list the multiples of 6 and 4.
  • Multiples of 6: 0, 6, 12, 18…
  • Multiples of 4: 0, 4, 8, 12, 16…
  • The least common multiple of 6 and 4 is 12.

Step 3 Make equivalent fractions so the denominators are the same.

  • In the example 1/6 + 2/4, multiply the numerator and denominator of 1/6 by 2 to get 2/12. Then multiply both numbers of 2/4 by 3 to equal 6/12.
  • Rewrite the equation as 2/12 + 6/12.

Step 4 Solve the equation as you normally would.

  • For example, rewrite 2/12 +6/12 as (2+6)/12 = 8/12.
  • Simplify your answer by dividing the numerator and denominator by 4 to get a final answer of ⅔.

Step 1 Multiply the numerators and denominators separately to find the product.

  • For example, if you want to solve 4/5 x 1/2, multiply the numerators for 4 x 1 = 4.
  • Then multiply the denominators for 5 x 2 = 10.
  • Write the new fraction 4/10 and simplify it by dividing the numerator and denominator by 2 to get the final answer of 2/5.
  • As another example, the problem 2 ½ x 3 ½ = 5/2 x 7/2 = (5 x 7)/(2 x 2) = 35/4 = 8 ¾.

Step 2 Flip the numerator and denominator for the second fraction in a division problem.

  • For example, the reciprocal of 3/8 is 8/3.
  • Convert a mixed number into an improper fraction before taking the reciprocal. For example, 2 ⅓ converts to 7/3 and the reciprocal is 3/7.

Step 3 Multiply the first fraction by the second fraction’s reciprocal to find the quotient.

  • For example, if your original problem was 3/8 ÷ 4/5, first find the reciprocal of 4/5, which is 5/4.
  • Rewrite your problem as multiplication with the reciprocal for 3/8 x 5/4.
  • Multiply the numerators for 3 x 5 = 15.
  • Multiply the denominators for 8 x 4 = 32.
  • Write the new fraction 15/32.

David Jia

  • Remember to never add or subtract denominators. Thanks Helpful 7 Not Helpful 0
  • Always simplify your answers to the lowest terms so they’re easy to read. Thanks Helpful 7 Not Helpful 0
  • Many calculators allow you to do fraction functions on them if you have trouble doing them on paper. Thanks Helpful 6 Not Helpful 1

how to solve problems using fraction

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Work Out a Fraction of an Amount

  • ↑ https://www.mathsisfun.com/definitions/fraction.html
  • ↑ https://www.mathsisfun.com/equivalent_fractions.html
  • ↑ https://www.mathsisfun.com/simplifying-fractions.html
  • ↑ https://youtu.be/nWZnyRTwBFM?t=15
  • ↑ https://www.calculatorsoup.com/calculators/math/mixed-number-to-improper-fraction.php
  • ↑ https://edu.gcfglobal.org/en/fractions/adding-and-subtracting-fractions/1/
  • ↑ https://www.bbc.co.uk/bitesize/topics/zhdwxnb/articles/z9n4k7h
  • ↑ https://www.mathsisfun.com/fractions_multiplication.html
  • ↑ https://www.chilimath.com/lessons/introductory-algebra/reciprocal-of-a-fraction/
  • ↑ https://openstax.org/books/prealgebra-2e/pages/4-2-multiply-and-divide-fractions

About This Article

David Jia

To understand a fraction, first identify the numerator and the denominator. The numerator is the number on top, which tells you how many parts of a whole the fraction represents. The denominator, which is on the bottom, represents the total number of possible parts making up the whole. For instance, the fraction ¾ describes 3 equal parts of a whole that has been divided up into 4 parts total. 3 is the numerator, while 4 is the denominator. Some fractions can be simplified, which means that you can divide the numerator and denominator by a common factor to create an equivalent fraction. For example, in the fraction 2/4, you can divide both the numerator and denominator by 2 to get the equivalent fraction ½. A fraction where the numerator is larger than the denominator is called an improper fraction. This kind of fraction represents a combination of a whole number and a fraction. You can convert improper fractions into mixed numbers by dividing the numerator by the denominator. The quotient in the division problem is the whole number, while the remainder is the numerator of the fraction. For instance, to turn 7/3 into a mixed number, divide 7 by 3 to get 2, with a remainder of 1. Write the fraction as the mixed number 2 1/3. You can add and subtract fractions like whole numbers, but only if they share the same denominator. For instance, 2/6 + 3/6 = 5/6. If the denominators are different, you’ll need to convert at least one of the fractions into an equivalent fraction so that the denominators match. Multiplying fractions is simple—just multiply the numerators by the numerators and the denominators by the denominators. To divide one fraction by another, flip the second fraction over to find its reciprocal, then multiply the two fractions together. Did this summary help you? Yes No

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Inverse operations

Solve equations with fractions

Here you will learn about how to solve equations with fractions, including solving equations with one or more operations. You will also learn about solving equations with fractions where the unknown is the denominator of a fraction.

Students will first learn how to solve equations with fractions in 7th grade as part of their work with expressions and equations and expand that knowledge in 8th grade.

What are equations with fractions?

Equations with fractions involve solving equations where the unknown variable is part of the numerator and/or denominator of a fraction.

The numerator (top number) in a fraction is divided by the denominator (bottom number).

To solve equations with fractions, you will use the “balancing method” to apply the inverse operation to both sides of the equation in order to work out the value of the unknown variable.

The inverse operation of addition is subtraction.

The inverse operation of subtraction is addition.

The inverse operation of multiplication is division.

The inverse operation of division is multiplication.

For example,

\begin{aligned} \cfrac{2x+3}{5} \, &= 7\\ \colorbox{#cec8ef}{$\times \, 5$} \; & \;\; \colorbox{#cec8ef}{$\times \, 5$} \\\\ 2x+3&=35 \\ \colorbox{#cec8ef}{$-\,3$} \; & \;\; \colorbox{#cec8ef}{$- \, 3$} \\\\ 2x & = 32 \\ \colorbox{#cec8ef}{$\div \, 2$} & \; \; \; \colorbox{#cec8ef}{$\div \, 2$}\\\\ x & = 16 \end{aligned}

What are equations with fractions?

Common Core State Standards

How does this relate to 7th grade and 8th grade math?

  • Grade 7: Expressions and Equations (7.EE.A.1) Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
  • Grade 8: Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • Grade 8: Expressions and Equations (8.EE.C.7b) Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

How to solve equations with fractions

In order to solve equations with fractions:

Identify the operations that are being applied to the unknown variable.

Apply the inverse operations, one at a time, to both sides of the equation.

Write the final answer, checking that it is correct.

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Solve equations with fractions examples

Example 1: equations with one operation.

Solve for x \text{: } \cfrac{x}{5}=4 .

The unknown is x.

Looking at the left hand side of the equation, the x is divided by 5.

\cfrac{x}{5}

2 Apply the inverse operations, one at a time, to both sides of the equation.

The inverse of “dividing by 5 ” is “multiplying by 5 ”.

You will multiply both sides of the equation by 5.

Solve equations with fractions example 1

3 Write the final answer, checking that it is correct.

The final answer is x=20.

You can check the answer by substituting the answer back into the original equation.

\cfrac{20}{5}=20\div5=4

Example 2: equations with one operation

Solve for x \text{: } \cfrac{x}{3}=8 .

Looking at the left hand side of the equation, the x is divided by 3.

\cfrac{x}{3}

The inverse of “dividing by 3 ” is “multiplying by 3 ”.

You will multiply both sides of the equation by 3.

Solve equations with fractions example 2

The final answer is x=24.

\cfrac{24}{3}=24\div3=8

Example 3: equations with two operations

Solve for x \text{: } \cfrac{x \, + \, 1}{2}=7 .

Looking at the left hand side of the equation, 1 is added to x and then divided by 2 (the denominator of the fraction).

\cfrac{x \, + \, 1}{2}

First, clear the fraction by multiplying both sides of the equation by 2.

Then, subtract 1 from both sides.

Solve equations with fractions example 3

The final answer is x=13.

\cfrac{13 \, +1 \, }{2}=\cfrac{14}{2}=14\div2=7

Example 4: equations with two operations

Solve for x \text{: } \cfrac{x}{4}-2=3 .

Looking at the left hand side of the equation, x is divided by 4 and then 2 is subtracted.

\cfrac{x}{4}-2

First, add 2 to both sides of the equation.

Then, multiply both sides of the equation by 4.

Solve equations with fractions example 4

\cfrac{20}{4}-2=20\div4-2=5-2=3

Example 5: equations with three operations

Solve for x \text{: } \cfrac{3x}{5}+1=7 .

Looking at the left hand side of the equation, x is multiplied by 3, then divided by 5 , and then 1 is added.

\cfrac{3x}{5}+1

First, subtract 1 from both sides of the equation.

Then, multiply both sides of the equation by 5.

Finally, divide both sides by 3.

Solve equations with fractions example 5

The final answer is x=10.

\cfrac{3 \, \times \, 10}{5}+1=\cfrac{30}{5}+1=6+1=7

Example 6: equations with three operations

Solve for x \text{: } \cfrac{2x-1}{7}=3 .

Looking at the left hand side of the equation, x is multiplied by 2, then 1 is subtracted, and the last operation is divided by 7 (the denominator).

\cfrac{2x-1}{7}

First, multiply both sides of the equation by 7.

Next, add 1 to both sides.

Solve equations with fractions example 6

The final answer is x=11.

\cfrac{2 \, \times \, 11-1}{7}=\cfrac{22-1}{7}=\cfrac{21}{7}=3

Example 7: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{24}{x}=6 .

Looking at the left hand side of the equation, x is the denominator. 24 is divided by x.

\cfrac{24}{x}

You need to multiply both sides of the equation by x.

Then, you can divide both sides by 6.

Solve equations with fractions example 7

The final answer is x=4.

\cfrac{24}{4}=24\div4=6

Example 8: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{18}{x}-6=3 .

Looking at the left hand side of the equation, x is the denominator. 18 is divided by x , and then 6 is subtracted.

\cfrac{18}{x}-6

First, add 6 to both sides of the equation.

Then, multiply both sides of the equation by x.

Finally, divide both sides by 9.

Solve equations with fractions example 8

The final answer is x=2.

\cfrac{18}{2}-6=9-6=3

Teaching tips for solving equations with fractions

  • When students first start working through practice problems and word problems, provide step-by-step instructions to assist them with solving linear equations.
  • Introduce solving equations with fractions with one-step problems, then two-step problems, before introducing multi-step problems.
  • Students will need lots of practice with solving linear equations. These standards provide the foundation for work with future linear equations in Algebra I and II.
  • Provide opportunities for students to explain their thinking through writing. Ensure that they are using key vocabulary, such as, absolute value, coefficient, equation, common factors, inequalities, simplify, etc.

Easy mistakes to make

  • The solution to an equation can be any type of number The unknowns do not have to be integers (whole numbers and their negative opposites). The solutions can be fractions or decimals. They can also be positive or negative numbers.
  • The unknown of an equation can be on either side of the equation The unknown, represented by a letter, is often on the left hand side of the equations; however, it doesn’t have to be. It could also be on the right hand side of an equation.

Solve equations with fractions image 2

  • Lowest common denominator (LCD) It is common to get confused between solving equations involving fractions and adding and subtracting fractions. When adding and subtracting, you need to work out the lowest/least common denominator (sometimes called the least common multiple or LCM). When you solve equations involving fractions, multiply both sides of the equation by the denominator of the fraction.

Related math equations lessons

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

Practice solve equations with fractions questions

1. Solve: \cfrac{x}{6}=3

GCSE Quiz False

You will multiply both sides of the equation by 6, because the inverse of “dividing by 6 ” is “multiplying by 6 ”.

Solve equations with fractions practice question 1

The final answer is x = 18.

\cfrac{18}{6}=18 \div 6=3

2. Solve: \cfrac{x \, + \, 4}{2}=7

Then subtract 4 from both sides.

Solve equations with fractions practice question 2

The final answer is x = 10.

\cfrac{10 \, + \, 4}{2}=\cfrac{14}{2}=14 \div 2=7

3. Solve: \cfrac{x}{8}-5=1

First, add 5 to both sides of the equation.

Then multiply both sides of the equation by 8.

Solve equations with fractions practice question 3

The final answer is x = 48.

\cfrac{48}{8}-5=48 \div 8-5=1

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

First, multiply both sides of the equation by 4.

Next, subtract 2 from both sides.

Solve equations with fractions practice question 4

The final answer is x = 2.

\cfrac{3 \, \times \, 2+2}{4}=\cfrac{6 \, + \, 2}{4}=\cfrac{8}{4}=8 \div 4=2

5. Solve: \cfrac{4x}{7}-2=6

Then multiply both sides of the equation by 7.

Finally, divide both sides by 4.

Solve equations with fractions practice question 5

The final answer is x = 14.

\cfrac{4 \, \times \, 14}{7}-2=\cfrac{56}{7}-2=56 \div 7-2=6

6. Solve: \cfrac{42}{x}=7

Then you divide both sides by 7.

Solve equations with fractions practice question 6

The final answer is x = 6.

\cfrac{42}{6}=42 \div 6=7

Solve equations with fractions FAQs

Yes, you still follow the order of operations when solving equations with fractions. You will start with any operations in the numerator and follow PEMDAS (parenthesis, exponents, multiply/divide, add/subtract), followed by any operations in the denominator. Then you will solve the rest of the equation as usual.

The next lessons are

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  • Coordinate plane
  • Number patterns
  • Algebraic expressions
  • Fractions operations

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Solving Problems that Include Fractions and Decimals

Introduction.

There are four important operations that you will encounter when solving problems in mathematics. The figures below indicate some of the actions in a problem that lead to different operations.

Addition and subtraction are related operations. Addition typically means to combine two or more numbers, and subtraction involves the difference , or removal, of one number from another.

Combining things, Accumulations, and Amounts of increase all pointing to Addition

Multiplication and division are also related operations. Both operations involve grouping and rates.

Combining several groups with the same size, Scaling a quantity, and Calculating Area all pointing at Multiplication

You have explored how to tell when to use which operation. Now, you will focus on identifying the operation from a word problem, and then use procedures to actually perform the operation and determine a solution to the problem.

Working with Signed Numbers

Signed numbers include integers and other rational numbers that have either a positive or a negative sign.

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Source: Badwater Elevation Sign, Complex01 and Elevation Benchmark, Jeff Kramer, Wikimedia Commons

Use the diagram below to review standard procedures for adding, subtracting, multiplying, and dividing integers.

A description of how to perform Subtraction, Addition, Division, and Multiplication

Adding and Subtracting Decimals

You have applied the rules of integers to solve word problems. Now, you will review ways to add and subtract decimals, and then use what you learn to solve problems relating to addition and subtraction of positive and negative decimals.

Click on the image below to open a base-ten model interactive in a new web browser tab or window. The interactive represents the two addends in an addition problem, or the minuend  and subtrahend in a subtraction problem. Use the manipulative to work through at least 3 problems.

  • Click on a block and drag it on top of its opposite block to remove zero pairs.
  • Click on a block and drag it to the next column to regroup.
  • Click “Next Problem” to move to the next problem when you are ready.

how to solve problems using fraction

Need additional help for addition?

Need additional help for subtraction?

Use what you noticed in the interactive to answer the following questions.

In the original problem, 4.3 – 1.5, when you dragged a ones rod into the tenths column, it split into 10 tenths. How does that relate to the regrouping that was recorded symbolically in the image shown below?

how to solve problems using fraction

In an addition problem, such as 6.4 + 4.8, when you regroup 10 tenths into 1 one and drag the ones rod into the ones place, how did that action appear in the regrouping that was recorded symbolically such as the regrouping shown in the image below?

how to solve problems using fraction

Pause and Reflect

1. Why is it important to line up the decimal point when adding or subtracting decimal numbers?

2. When regrouping 1 one and 3 tenths into 13 tenths, why do you cross out the original 3 in the tenths place and replace it with 13? 

Adding and Subtracting Fractions

You have used models and algorithms to add and subtract decimals, paying special attention to the regrouping that was necessary to perform the computations. Now, you will extend the idea of regrouping to models and procedures used to add and subtract fractions, including mixed numbers.

Consider the following problem.

apples

The example below shows how Marley used fraction strips to solve this problem.

Click the image below to view additional examples, including a video with a worked-out example for you to follow.

how to solve problems using fraction

1. How is regrouping when subtracting mixed numbers similar to regrouping when subtracting decimals?

2. When adding decimals, you regroup when the sum of the two digits in a place value that is greater than 10. When would you need to regroup as you add mixed numbers?

Multiplying and Dividing Decimals

Now that you’ve investigated addition and subtraction with decimals and fractions, let’s take a closer look at multiplication and division. You will start in this section with decimals, and then use a similar model to multiply and divide fractions and mixed numbers in the next section.

how to solve problems using fraction

  • Write an expression that you can use to determine the amount of oil that Rachel started with.
  • How would you represent 2.2 and 2.5 as improper fractions with denominators of 10?

The interactive below uses blocks to multiply decimals. When the blocks are combined, they will form a rectangle; the area of the rectangle is the product of the two decimals or the answer to Rachel’s problem.

how to solve problems using fraction

  • In the first activity, the first decimal is the length of the rectangle, and the second decimal is the width. Represent each decimal by dragging the appropriate blocks and moving them to the area for each decimal.
  • In the second activity, use the information from the decimals and drag the blocks to the open area to create a rectangle. You will use the green blocks to fill in the missing pieces of the rectangle.
  • Is the answer the same as what we found earlier in Anu's solution?
  • Adjust the numerators to create and represent two more multiplication problems. Record those problems on a piece of paper.

Based on what you saw in the interactive, why do you think that the product has the same number of digits to the right of the decimal as the total number of digits to the right of the decimal in the two factors ?

Multiplying and Dividing Fractions

In this section, you will look at models to represent multiplying and dividing fractions.

Multiplying Fractions

running shoes

Use the interactive below to represent the problem and graphically illustrate the product. Use the Numerator and Denominator sliders to create each fraction or mixed number. You may also need to use the Zoom in/out sliders to see the entire model.

how to solve problems using fraction

Need additional directions?

Use the interactive to answer the following questions:

  • What are the dimensions of the shaded rectangle in the solution? Check Your Answer
  • The solid lines represent the boundaries of a rectangle with an area of 1 square unit. The dashed lines represent the boundaries of a number of equal-sized regions within this area. What fraction of 1 does each smaller rectangle represent? Check Your Answer

how to solve problems using fraction

  • What mixed number does this rearranged figure represent? How does this compare with the product of 3 4 and 6 1 2 ? Check Your Answer

Dividing Fractions

cookies

To solve this problem, Barbara used a fraction strip generator, which gave her the following diagram.

cookies

  • Barbara knew this was a division problem, not a multiplication problem. How did she know that? Check Your Answer
  • Use the diagram to explain why the quotient of 6 1 2 ÷ 1 2 is 13. Check Your Answer

Use the same fraction strip generator that Barbara used to solve the problem below.

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Click the image below to open the fraction strip generator in a new web browser tab or window. Enter the key information from the problem, including the dividend and the divisor , and then use the results to answer the questions that follow.

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In the fraction diagrams, both 5 3 4 and 3 8 are marked off into eighths. Why do you think that is the case? Check Your Answer

To divide 5 3 4 by 3 8 , the number sentence beneath the diagrams shows multiplication of 5 3 4 by 8 3 , which is the reciprocal of 3 8 . Multiplying by 8 3 is the same as multiplying by 8 , and then dividing by 3 . Why do you need to multiply 5 3 4 by 8 , which is the numerator of the reciprocal? Check Your Answer

The next step in the number sentence divides the product of 5 3 4 and 8 by 3 (multiplies 5 3 4 by the fraction 8 3 ) . Why do you need to divide by 3 at this point? Check Your Answer

See the completed fraction diagram for Patrice's ornament problem.

Completed fraction diagram

1. How does the multiplication algorithm connect to the area model that you used in the first interactive?

2. How does the division algorithm connect to the fraction strip model that you used in the interactive?

You studied models that represent operations on rational numbers (fractions and decimals). You also connected those models to the standard algorithms for performing the operations.

The graphic below summarizes procedures to add, subtract, multiply, and divide decimals.

how to solve problems using fraction

The graphic below summarizes procedures to add, subtract, multiply, or divide fractions, including mixed numbers.

how to solve problems using fraction

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  • Add Fractions
  • Simplify Fractions

Fraction Word Problems (Difficult)

Here are some examples of more difficult fraction word problems. We will illustrate how block models (tape diagrams) can be used to help you to visualize the fraction word problems in terms of the information given and the data that needs to be found.

Related Pages Fraction Word Problems Singapore Math Lessons Fraction Problems Using Algebra Algebra Word Problems

Block modeling (also known as tape diagrams or bar models) are widely used in Singapore Math and the Common Core to help students visualize and understand math word problems.

Example: 2/9 of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?

Solution: Draw a diagram with 9 equal parts: 2 parts to represent the adults and 7 parts to represent the children.

5 units = 95 1 unit = 95 ÷ 5 = 19 7 units = 7 × 19 = 133

Answer: There are 133 children in the restaurant.

Example: Gary and Henry brought an equal amount of money for shopping. Gary spent $95 and Henry spent $350. After that Henry had 4/7 of what Gary had left. How much money did Gary have left after shopping?

350 – 95 = 255 3 units = 255 1 unit = 255 ÷ 3 = 85 7 units = 85 × 7 = 595

Answer: Gary has $595 after shopping.

Example: 1/9 of the shirts sold at Peter’s shop are striped. 5/8 of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?

Solution: Draw a diagram with 9 parts. One part represents striped shirts. Out of the remaining 8 parts: 5 parts represent the printed shirts and 3 parts represent plain colored shirts.

3 units = 81 1 unit = 81 ÷ 3 = 27 Printed shirts have 2 parts more than plain shirts. 2 units = 27 × 2 = 54

Answer: Peter’s shop has 54 more printed colored shirts than plain shirts.

Solve a problem involving fractions of fractions and fractions of remaining parts

Example: 1/4 of my trail mix recipe is raisins and the rest is nuts. 3/5 of the nuts are peanuts and the rest are almonds. What fraction of my trail mix is almonds?

How to solve fraction word problem that involves addition, subtraction and multiplication using a tape diagram or block model

Example: Jenny’s mom says she has an hour before it’s bedtime. Jenny spends 3/5 of the hour texting a friend and 3/8 of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time reading her book. How long did Jenny read?

How to solve a four step fraction word problem using tape diagrams?

Example: In an auditorium, 1/6 of the students are fifth graders, 1/3 are fourth graders, and 1/4 of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?

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Integration is an important tool in calculus that can give an antiderivative or represent area under a curve..

The indefinite integral of , denoted , is defined to be the antiderivative of . In other words, the derivative of is . Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example, , since the derivative of is . The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to .

Both types of integrals are tied together by the fundamental theorem of calculus. This states that if is continuous on and is its continuous indefinite integral, then . This means . Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can solve a broad range of integrals

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Easy Finger Math Tricks to Help Kids Solve Problems

While using your fingers isn't the fastest way to recall a multiplication fact while doing a problem, finger math tricks can help kids figure out how to answer the problem at hand — and as they work on their math, they will eventually learn all the facts by repetition.

Note that before your child can understand other finger tricks, they must be able to count by 2s, 5s, and 10s and multiply by 2s, 3s, and 4s.

Quick Finger Math Tricks for Threes and Fours

The tricks for multiplying by threes and fours are really a matter of counting out the answer on your fingers. As your children count out the answer repeatedly, they'll memorize it and then be able to move on to larger numbers.

Multiplying by Three

Did you realize that all of your fingers have three segments? Therefore, you can figure out anything from 3 x 1 to 3 x 10 by counting the segments on each finger. To start:

  • Hold up the number of fingers you're going to multiply by 3. For example, if the problem is 3 x 4 — hold up four fingers.
  • Count each segment on each finger you're holding up, and you should come up with 12 — which is the correct answer.

Multiplying by Four

Multiplying by four is the same as multiplying by two — twice. To start:

  • Hold up the number of fingers to correspond with the number you are multiplying by four. For example, if you are multiplying 4 x 6 — hold up six fingers.
  • Count each finger by two, moving from left to right. Then count each finger again, continuing to count by twos, until you've counted every finger twice.
Helpful Hack To keep track of the fingers you've counted twice, sometimes it's easier to put your finger down as you count the first time, and back up as you count the second time.

Finger Math Tricks for Multiplying by 6, 7, 8, and 9

While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little easier to work those problems out.

Multiplying 6, 7, 8, and 9 by Hand

To begin, assign each finger a number. For example, your thumbs represent 6, your index fingers each represent 7, etc. This will remain the same throughout the finger math hack.

Your left hand will represent the first number that you are multiplying and your right hand will represent the second number you are multiplying. In this example, we are multiplying 7 x 8. 

To Determine the Part of Your Answer:

  • On your left hand, put down the finger that represents the number you are multiplying as well as any fingers whose number value is less than this figure. In this example, you are multiplying 7 x 8, so the left hand will represent 7. You will drop your index finger (number 7) and your thumb (number 6).
  • Similarly, the right hand will represent eight, so you will drop down your middle finger (number 8), your index finger (number 7), and your thumb (number 6).
  • Now, just multiply the fingers that are still pointed upwards. In this case, you will have three fingers on your left hand and two on your right, so you will multiply 3 x 2 to get 6. This is the first part of your answer!

To Determine the Second Part of Your Answer:

  • Keeping your fingers in the same positions, count how many fingers are folded down. In the 7 x 8 example, you should have five fingers folded. 
  • You will count each of these in quantities of ten. So, 10, 20, 30, 40, 50.
  • 50 is your answer.

To Determine Your Final Answer:

  • Add your two numbers together. In this example, you would add 6 + 50, which gives you 56!

Another Finger Math Trick Just for Nine

There is a trick that works separately, just for multiplying by the number nine.

  • To start, hold up all ten fingers, with your palms facing you.
  • Assign each finger a number, starting with your left-hand thumb and ending with your right-hand thumb. The left-hand thumb will be one, the left-hand index finger will be two, and so on until you reach the number 10 for your right-hand thumb.
  • To tackle a problem, put down the corresponding finger of the number you're multiplying by nine. For example, if you are multiplying 9 x 8, you'd put down the eighth finger (which will be on your right hand).
  • Count all the fingers to the left of the finger you have folded down. This will give you 7. This is the first digit of your answer.
  • Count all the fingers to the right of the finger you have folded down. This will give you 2. This is the second digit of your answer.
  • Put the numbers together! Your answer is 72.

Finger Multiplication Tricks Can Make Math Easy and Fun

While the hope is that your kids will eventually memorize their multiplication charts , using some quick hand tricks for multiplication and letting them count things out on their fingers is not a bad way to learn. It keeps frustration at bay since the answer is always a fingertip away, and the repetition of having to figure it out will help cement those facts into their brains.

child counting fingers

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Mathematics LibreTexts

11.7: Dividing Fractions- Problems

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  • Page ID 83035

  • Michelle Manes
  • University of Hawaii

We’ve spent the last couple of chapters talking about dividing fractions: how to make sense of the operation, how to picture what’s going on, and how to do the computations. But all of this kind of begs the question: When would you ever want to divide fractions, anyway? How does that even come up?

It’s important that teachers are able to come up with situations and problems that model particular operations, which means you have to really understand what the operations mean and when they are used.

Think / Pair / Share

  • Use one of our methods (draw a picture, rectangles, common denominator, missing factor) to compute \(1 \frac{3}{4} \div \frac{1}{2}\).
  • Come up with a situation where you would want to compute \(1 \frac{3}{4} \div \frac{1}{2}\). (That is, write a word problem that would require you to do this computation to solve it.)

When to Multiply, When to Divide?

A common answer to

Come up with a situation where you would want to compute \(1 \frac{3}{4} \div \frac{1}{2}\).

Is something like this:

My recipe calls for \(1 \frac{3}{4} \div \frac{1}{2}\) cups of flour, but I only want to make half a recipe. How much flour should I use?

But that problem doesn’t ask you to divide fractions. It asks you to cut your recipe in half, which means dividing by 2 or multiplying by \(\frac{1}{2}\).

Why is it so hard to come up with division problems that use fractions? Maybe it’s because fractions are already the answer to a division problem, so you’re dividing and then dividing some more. Maybe it’s because they just make it look so complicated. In any case, it’s worth spending some time thinking about division problems that involve fractions and how to recognize and solve them.

One handy trick: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way.

Example \(\PageIndex{1}\):

Here are some division problems involving whole numbers:

  • I have 10 feet of ribbon. How many 2-inch pieces can I cut from it?
  • I have a fancy old clock that rings once every 15 minutes. How many times will it ring over the course of 2 hours (120 minutes)?
  • My fish tank needs 6 gallons of water, and my bucket holds 3 gallons. How many times will I need to fill my bucket in order to fill the tank?
  • A recipe calls for 6 cups of flour, and my largest scoop measures exactly 2 cups. How many times should I use it?
  • I ran 12 miles and went around the the same route 3 times. How long was the route?

Here are some very similar problems, rewritten to use fractions instead:

  • I have \(1 \frac{3}{4}\) feet of ribbon. How many 6-inch (that’s \(\frac{1}{2}\) a foot) pieces can I cut from it?
  • My watch alarm goes off every half hour, and I don’t know how to shut it off. How many times will it go off during the \(1 \frac{3}{4}\) hour movie?
  • My fish tank needs \(1 \frac{3}{4}\) gallons of water, and my bucket holds \(\frac{1}{2}\) gallon. How many times will I need to fill my bucket in order to fill the tank?
  • I want to measure \(1 \frac{3}{4}\) cups of flour for a recipe, but I only have a \(\frac{1}{2}\) cup measuring cup. How many times should I fill it?
  • I ran \(1 \frac{3}{4}\) miles before I twisted my ankle. I only finished half the race. How long was the race course?

For each one of the fraction division questions, we can understand why it’s a division problem:

  • I have \(1 \frac{3}{4}\) feet of ribbon. How many 6-inch (that’s \(\frac{1}{2}\) a foot) pieces can I cut from it? This means making equal groups of \(\frac{1}{2}\) foot each and asking how many groups. That’s quotative division.
  • My watch alarm goes off every half hour, and I don’t know how to shut it off. How many times will it go off during the \(1 \frac{3}{4}\) hour movie? Again, we’re making equal groups of \(\frac{1}{2}\) hour each, and asking how many groups. Quotative division.
  • My fish tank needs \(1 \frac{3}{4}\) gallons of water, and my bucket holds \(\frac{1}{2}\) gallon. How many times will I need to fill my bucket in order to fill the tank? Once again: we’re making equal groups of \(\frac{1}{2}\) gallon each, and asking how many groups (buckets).
  • I want to measure \(1 \frac{3}{4}\) cups of flour for a recipe, but I only have a \(\frac{1}{2}\) cup measuring cup. How many times should I fill it? This is making equal groups of \(\frac{1}{2}\) cup and asking how many groups.
  • I ran \(1 \frac{3}{4}\) miles before I twisted my ankle. I only finished half the race. How long was the race course? This one is a little different. This one is a little different. It’s the fraction version of partitive division.

Recall what partitive division asks: For \(20 \div 4\), we ask 20 is 4 groups of what size?

Partitive.png

So for \(1 \frac{3}{4} \div \frac{1}{2}\), we ask: \(1 \frac{3}{4}\) is half a group of what size?

partitivefrac-1-300x73.png

You try it.

  • First write five different division word problems that use whole numbers. (Try to write at least a couple each of partitive and quotative division problems.)
  • Then change the problems so that they are fraction division problems instead. You might need to rewrite the problem a bit so that it makes sense.
  • Solve your problems!

how to solve problems using fraction

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The research team from NVIDIA has introduced OpenMathInstruct-1, a novel dataset comprising 1.8 million problem-solution pairs to improve mathematical reasoning in LLMs. This dataset stands out due to its open license and the use of Mixtral, an open-source LLM, for data generation, allowing unrestricted use and fostering innovation in the field.

how to solve problems using fraction

OpenMathInstruct-1 was synthesized using a combination of brute-force scaling and novel prompting strategies with the Mixtral model. To synthesize solutions for GSM8K and MATH benchmarks, the research employed few-shot prompting, incorporating instructions, representative problems, their solutions in code-interpreter format, and a new question from the training set. If the base LLM generated a solution that led to the correct answer, it was included in the finetuning dataset. Solutions were sampled with constraints on token numbers and code blocks, using strategies like default, subject-specific, and masked text solution prompting, with the latter significantly increasing training set coverage by masking numbers in intermediate computations. Post-processing corrected syntactically noisy solutions. Data selection strategies included fair vs. naive downsampling and code-preferential selection, favoring code-based solutions. Models underwent training for four epochs, utilizing the AdamW optimizer, and were evaluated on benchmarks using greedy decoding and self-consistency/majority voting.

how to solve problems using fraction

Models finetuned on a mix of 512K downsampled GSM8K and MATH instances, totaling 1.2M, showcased competitive performance against gpt-distilled models across mathematical tasks. For example, when finetuned with OpenMathInstruct-1, the OpenMath-CodeLlama-70B model achieved competitive results, with 84.6% on GSM8K and 50.7% on MATH. Models notably outperformed MAmmoTH and MetaMath, with improvements sustained as model parameters increased. Enhanced by self-consistency decoding, their efficacy varied across tasks, subjects, and difficulty levels within the MATH dataset. Ablation studies highlighted the superiority of fair downsampling over naive approaches and the benefits of increasing dataset size. While code-preferential selection strategies improved greedy decoding, they had mixed effects on self-consistency decoding performance.

OpenMathInstruct-1 marks a significant advancement in the development of LLMs for mathematical reasoning. By offering a large-scale, openly licensed dataset, this work addresses the limitations of existing datasets and sets a new standard for collaborative and accessible research in the field. The success of the OpenMath-CodeLlama-70B model underscores the potential of open-source efforts to achieve breakthroughs in specialized domains like mathematical reasoning.

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how to solve problems using fraction

Nikhil is an intern consultant at Marktechpost. He is pursuing an integrated dual degree in Materials at the Indian Institute of Technology, Kharagpur. Nikhil is an AI/ML enthusiast who is always researching applications in fields like biomaterials and biomedical science. With a strong background in Material Science, he is exploring new advancements and creating opportunities to contribute.

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IMAGES

  1. 4 Ways to Solve Fraction Questions in Math

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  2. How to Solve Fraction Questions in Math: 10 Steps (with Pictures)

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  3. Fraction Division

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  4. lesson 6.9 problem solving fractions addition and subtraction

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  5. Problem solving with fractions

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  6. How To X Fractions

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VIDEO

  1. How to solve Fraction problem #भिन्न #maiths #ssc #uptet #upsc #Tricks

  2. Can You Solve This Fractional Equation ???

  3. How to solve the addition of fraction?

  4. How to Find the Fraction of a Fraction!

  5. fraction add solve in 2 sec

  6. ##How to solve Fraction with easy method ##Part 1😊😊

COMMENTS

  1. 3 Ways to Solve Fraction Questions in Math

    1 Add fractions with the same denominator by combining the numerators. To add fractions, they must have the same denominator. If they do, simply add the numerators together. [2] For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3. 2

  2. Problem Solving using Fractions (Definition, Types and Examples

    Solved Examples Frequently Asked Questions What are Fractions? Equal parts of a whole or a collection of things are represented by fractions. In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.

  3. 3 Ways to Do Fractions

    1 Identify the numerator and denominator. The top number of a fraction is known as the numerator and represents how many parts of the whole you have. The bottom number of the fraction is the denominator, which is the number of parts that would equal the whole. If the numerator is smaller than the denominator, then it is a proper fraction.

  4. Learn How to Solve Fraction Word Problems with Examples and Interactive

    Solving Word Problems by Adding and Subtracting Fractions and Mixed Numbers Learn How to Solve Fraction Word Problems with Examples and Interactive Exercises Example 1: Rachel rode her bike for one-fifth of a mile on Monday and two-fifths of a mile on Tuesday. How many miles did she ride altogether?

  5. Word Problems with Fractions

    Answer: Word problems with fractions: involving a fraction and a whole number Finally, we are going to look at an example of a word problem with a fraction and a whole number. Now we will have to convert all the information into a fraction with the same denominator (as we did in the example above) in order to calculate

  6. Equation with variables on both sides: fractions

    To solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal.

  7. 1.26: Solving Fractional Equations

    II. Multiple Fractions on Either Side of the Equation. Equations d) and e) in Example 24.1 fall into this category. We solve these equations here. We use the technique for combining rational expressions we learned in Chapter 23 to reduce our problem to a problem with a single fraction on each side of the equation. d) Solve \(\frac{3}{4}-\frac{1 ...

  8. 4.9: Solving Equations with Fractions

    Solution. To "undo" multiplying by 3/5, multiply both sides by the reciprocal 5/3 and simplify. 3 5x = 4 10 Original equation. 5 3(3 5x) = 5 3( 4 10) Multiply both sides by 5/3. (5 3 ⋅ 3 5)x = 20 30 On the left, use the associative property to regroup. On the right, multiply. 1x = 2 3 On the left, 5 3 ⋅ 3 5 = 1.

  9. Algebra: Fraction Problems (solutions, examples, videos)

    How to solve Fraction Word Problems using Algebra? Examples: (1) The denominator of a fraction is 5 more than the numerator. If 1 is subtracted from the numerator, the resulting fraction is 1/3. Find the original fraction. (2) If 3 is subtracted from the numerator of a fraction, the value of the resulting fraction is 1/2.

  10. 4.12: Solve Equations with Fractions (Part 1)

    Determine Whether a Fraction is a Solution of an Equation. As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation.In those sections, we found whole number and integer solutions to ...

  11. Solve Equations with Fractions

    How to solve equations with fractions In order to solve equations with fractions: Identify the operations that are being applied to the unknown variable. Apply the inverse operations, one at a time, to both sides of the equation. Write the final answer, checking that it is correct. [FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

  12. Fraction Word Problems

    For a complete lesson on fraction word problems, go to https://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every le...

  13. Fractions

    Type a math problem Solve Examples 124 − 79 124 × 89 124 ÷ 89 124 + 89 124 + 89 × 315 − 1026 81 + 2(79) ÷ 315 Quiz 124 − 79 124 ÷ 89 124 + 89 × 315 − 1026 Learn about fractions using our free math solver with step-by-step solutions.

  14. Solving Problems using Fractions and Mixed Numbers

    He has a master's degree in writing and literature. From understanding a recipe to deciding the winner of a competition, fractions have many uses in solving everyday problems. Learn how to use ...

  15. Solving Problems that Include Fractions and Decimals

    The graphic below summarizes procedures to add, subtract, multiply, or divide fractions, including mixed numbers. Given problem situations, the student will use addition, subtraction, multiplication, and division to solve problems involving positive and negative fractions and decimals.

  16. Fraction Word Problems: Examples

    Fraction Problems Using Algebra Lessons on solving fraction word problems using visual methods like bar models or tape diagrams. Share this page to Google Classroom Here are some examples and solutions of fraction word problems. The first example is a one-step word problem.

  17. Fractions: Solve fraction problems with the four operations

    Fractions: Solve fraction problems with the four operations In this lesson, we will revise calculating with fractions using the four operations and then apply this to word problems. This quiz includes images that don't have any alt text - please contact your teacher who should be able to help you with an audio description.

  18. Dividing fractions using fraction strips: An evidence ...

    Download Read: How to use this strategy Objective: Students will use fractions strips to show and solve the division of two fractions. Grade levels (with standards): 5 (Common Core 5.NF.B.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions)

  19. Solve

    Example: 2x-1=y,2y+3=x What can QuickMath do? QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose.

  20. Fraction Word Problems (Difficult)

    Fraction Word Problems - using block models (tape diagrams), Solve a problem involving fractions of fractions and fractions of remaining parts, how to solve a four step fraction word problem using tape diagrams, grade 5, grade 6, grade 7, with video lessons, examples and step-by-step solutions.

  21. How to Solve a Fractions Word Problem

    In this lesson, you will learn how to solve a multiplying fractions word problem by working through two multi-step problems.Tags: fractions word problem, mul...

  22. Mathway

    Free math problem solver answers your algebra homework questions with step-by-step explanations.

  23. Integral Calculator: Integrate with Wolfram|Alpha

    Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English. integrate x/(x-1) integrate x sin(x^2) integrate x sqrt(1-sqrt(x)) integrate x/(x+1)^3 from 0 to infinity

  24. Easy Finger Math Tricks to Help Kids Solve Problems

    Finger Math Tricks for Multiplying by 6, 7, 8, and 9 While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little ...

  25. 11.7: Dividing Fractions- Problems

    One handy trick: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way. Example 11.7.1 11.7. 1: Here are some division problems involving whole numbers: I have 10 feet of ribbon.

  26. CRASH MATH on Steam

    CRASH MATH You must solve math problems while trying to drive a car and avoid traps. This driving game has a lot of possibilities. The physics engine simulates every component of a vehicle in real-time, resulting in a funny behavior. While creating the excitement of solving math problems.

  27. NVIDIA AI Research Introduce OpenMathInstruct-1: A Math Instruction

    OpenMathInstruct-1 was synthesized using a combination of brute-force scaling and novel prompting strategies with the Mixtral model. To synthesize solutions for GSM8K and MATH benchmarks, the research employed few-shot prompting, incorporating instructions, representative problems, their solutions in code-interpreter format, and a new question from the training set.

  28. How to Use OpenPubkey to Solve Your Key Management Problems

    Second, to solve the problem of lost or stolen signing keys, OpenPubkey public keys and signing keys are ephemeral. That means the signing keys can be deleted and recreated at will. OpenPubkey generates a fresh public key and signing key for a user every time that user authenticates to their identity provider.