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## How to Solve Fraction Questions in Math

Last Updated: February 24, 2023 Fact Checked

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Sophia Latorre . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 7 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 1,170,677 times.

Fraction questions can look tricky at first, but they become easier with practice and know-how. Start by learning the terminology and fundamentals, then pratice adding, subtracting, multiplying, and dividing fractions. [1] X Research source Once you understand what fractions are and how to manipulate them, you'll be breezing through fraction problems in no time.

## Doing Calculations with Fractions

- 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.

- For instance, to solve 6/8 - 2/8, all you do is take away 2 from 6. The answer is 4/8, which can be reduced to 1/2.

- For example, if you need to add 1/2 and 2/3, start by determining a common multiple. In this case, the common multiple is 6 since both 2 and 3 can be converted to 6. To turn 1/2 into a fraction with a denominator of 6, multiply both the numerator and denominator by 3: 1 x 3 = 3 and 2 x 3 = 6, so the new fraction is 3/6. To turn 2/3 into a fraction with a denominator of 6, multiply both the numerator and denominator by 2: 2 x 2 = 4 and 3 x 2 = 6, so the new fraction is 4/6. Now, you can add the numerators: 3/6 + 4/6 = 7/6. Since this is an improper fraction, you can convert it to the mixed number 1 1/6.
- On the other hand, say you're working on the problem 7/10 - 1/5. The common multiple in this case is 10, since 1/5 can be converted into a fraction with a denominator of 10 by multiplying it by 2: 1 x 2 = 2 and 5 x 2 = 10, so the new fraction is 2/10. You don't need to convert the other fraction at all. Just subtract 2 from 7, which is 5. The answer is 5/10, which can also be reduced to 1/2.

- For instance, to multiply 2/3 and 7/8, find the new numerator by multiplying 2 by 7, which is 14. Then, multiply 3 by 8, which is 24. Therefore, the answer is 14/24, which can be reduced to 7/12 by dividing both the numerator and denominator by 2.

- For example, to solve 1/2 ÷ 1/6, flip 1/6 upside down so it becomes 6/1. Then just multiply 1 x 6 to find the numerator (which is 6) and 2 x 1 to find the denominator (which is 2). So, the answer is 6/2 which is equal to 3.

## Practicing the Basics

- For instance, in 3/5, 3 is the numerator so there are 3 parts and 5 is the denominator so there are 5 total parts. In 7/8, 7 is the numerator and 8 is the denominator.

- If you need to turn 7 into a fraction, for instance, write it as 7/1.

- For example, if you have the fraction 15/45, the greatest common factor is 15, since both 15 and 45 can be divided by 15. Divide 15 by 15, which is 1, so that's your new numerator. Divide 45 by 15, which is 3, so that's your new denominator. This means that 15/45 can be reduced to 1/3.

- Say you have the mixed number 1 2/3. Stary by multiplying 3 by 1, which is 3. Add 3 to 2, the existing numerator. The new numerator is 5, so the mixed fraction is 5/3.

Tip: Typically, you'll need to convert mixed numbers to improper fractions if you're multiplying or dividing them.

- Say that you have the improper fraction 17/4. Set up the problem as 17 ÷ 4. The number 4 goes into 17 a total of 4 times, so the whole number is 4. Then, multiply 4 by 4, which is equal to 16. Subtract 16 from 17, which is equal to 1, so that's the remainder. This means that 17/4 is the same as 4 1/4.

## Fraction Calculator, Practice Problems, and Answers

## Community Q&A

- Take the time to carefully read through the problem at least twice so you can be sure you know what it's asking you to do. Thanks Helpful 2 Not Helpful 1
- Check with your teacher to find out if you need to convert improper fractions into mixed numbers and/or reduce fractions to their lowest terms to get full marks. Thanks Helpful 2 Not Helpful 0
- To take the reciprocal of a whole number, just put a 1 over it. For example, 5 becomes 1/5. Thanks Helpful 1 Not Helpful 0

## You Might Also Like

- ↑ https://www.sparknotes.com/math/prealgebra/fractions/terms/
- ↑ https://www.bbc.co.uk/bitesize/articles/z9n4k7h
- ↑ https://www.mathsisfun.com/fractions_multiplication.html
- ↑ https://www.mathsisfun.com/fractions_division.html
- ↑ https://medium.com/i-math/the-no-nonsense-straightforward-da76a4849ec
- ↑ https://www.youtube.com/watch?v=PcEwj5_v75g
- ↑ https://sciencing.com/solve-math-problems-fractions-7964895.html

## About This Article

To solve a fraction multiplication question in math, line up the 2 fractions next to each other. Multiply the top of the left fraction by the top of the right fraction and write that answer on top, then multiply the bottom of each fraction and write that answer on the bottom. Simplify the new fraction as much as possible. To divide fractions, flip one of the fractions upside-down and multiply them the same way. If you need to add or subtract fractions, keep reading! Did this summary help you? Yes No

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## Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

## Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

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

## What can QuickMath do?

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

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

## Math Topics

More solvers.

- Add Fractions
- Simplify Fractions

Home / United States / Math Classes / 5th Grade Math / Problem Solving using Fractions

## Problem Solving using Fractions

Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. Here we will learn to solve some real-life problems using fractions. ...Read More Read Less

## Table of Contents

## What are Fractions?

Types of fractions.

- Fractions with like and unlike denominators
- Operations on fractions
- Fractions can be multiplied by using
- Let’s take a look at a few examples

## Solved Examples

- Frequently Asked Questions

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.

For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction \(\frac{1}{2}\) .

Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\) .

Proper fractions

A fraction in which the numerator is less than the denominator value is called a proper fraction.

For example , \(\frac{3}{4}\) , \(\frac{5}{7}\) , \(\frac{3}{8}\) are proper fractions.

Improper fractions

A fraction with the numerator higher than or equal to the denominator is called an improper fraction .

Eg \(\frac{9}{4}\) , \(\frac{8}{8}\) , \(\frac{9}{4}\) are examples of improper fractions.

Mixed fractions

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.

We express improper fractions as mixed numbers.

For example , 5\(\frac{1}{3}\) , 1\(\frac{4}{9}\) , 13\(\frac{7}{8}\) are mixed fractions.

Unit fraction

A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction .

## Fractions with Like and Unlike Denominators

Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.

For example,

\(\frac{1}{4}\) and \(\frac{3}{4}\) are like fractions as they both have the same denominator, that is, 4.

\(\frac{1}{3}\) and \(\frac{1}{4}\) are unlike fractions as they both have a different denominator.

## Operations on Fractions

We can perform addition, subtraction, multiplication and division operations on fractions.

Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.

There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.

## Fractions can be Multiplied by Using:

Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor.

## Let’s Take a Look at a Few Examples

Addition and subtraction using common denominator

( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )

We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.

\(\frac{1}{6} ~+ ~\frac{2}{5}\)

= \(\frac{5~+~12}{30}\)

= \(\frac{17}{30}\)

( \(\frac{5}{2}~-~\frac{1}{6}\) )

= \(\frac{12~-~5}{30}\)

= \(\frac{7}{30}\)

Examples of Multiplication and Division

Multiplication:

(\(\frac{1}{6}~\times~\frac{2}{5}\))

= (\(\frac{1~\times~2}{6~\times~5}\)) [Multiplying numerator of fractions and multiplying denominator of fractions]

= \(\frac{2}{30}\)

(\(\frac{2}{5}~÷~\frac{1}{6}\))

= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\)) [Multiplying dividend with the reciprocal of divisor]

= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))

= \(\frac{12}{5}\)

Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)

Let’s add \(\frac{7}{8}\) and \(\frac{2}{3}\) using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.

\(\frac{7}{8}\) + \(\frac{2}{3}\)

= \(\frac{21~+~16}{24}\)

= \(\frac{37}{24}\)

Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)

Solution:

Let’s subtract \(\frac{12}{17}\) from \(\frac{11}{13}\) using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.

\(\frac{11}{13}\) – \(\frac{12}{17}\)

= \(\frac{187~-~156}{221}\)

= \(\frac{31}{221}\)

Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)

Multiply the numerators and multiply the denominators of the 2 fractions.

\(\frac{15}{13}~\times~\frac{18}{17}\)

= \(\frac{15~~\times~18}{13~~\times~~17}\)

= \(\frac{270}{221}\)

Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)

Divide by multiplying the dividend with the reciprocal of the divisor.

\(\frac{25}{33}~\div~\frac{41}{45}\)

= \(\frac{25}{33}~\times~\frac{41}{45}\) [Multiply with reciprocal of the divisor \(\frac{41}{45}\) , that is, \(\frac{45}{41}\) ]

= \(\frac{25~\times~45}{33~\times~41}\)

= \(\frac{1125}{1353}\)

Example 5:

Sam was left with \(\frac{7}{8}\) slices of chocolate cake and \(\frac{3}{7}\) slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared \(\frac{10}{11}\) slices from the total number he had with his parents. What is the number of slices he has remaining?

To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,

= \(\frac{7}{8}\) + \(\frac{3}{7}\)

= \(\frac{49~+~24}{56}\)

= \(\frac{73}{56}\)

To find out the remaining number of slices Sam has \(\frac{10}{11}\) slices need to be deducted from the total number,

= \(\frac{73}{56}~-~\frac{10}{11}\)

= \(\frac{803~-~560}{616}\)

= \(\frac{243}{616}\)

Hence, after sharing the cake with his friends, Sam has \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had \(\frac{243}{616}\) slices of cake left with him.

Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?

First \(\frac{15}{8}\) l needs to be converted to milliliters.

\(\frac{15}{8}\)l into milliliters = \(\frac{15}{8}\) x 1000 = 1875 ml

To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.

The number of oranges required for 1875 m l of juice = \(\frac{1875}{25}\) ml = 75 oranges

To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has

= \(\frac{1875}{200}~=~9\frac{3}{8}\) cups

We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups, \(\frac{3}{8}\) th of a cup cannot be sold alone.

Money made on selling 9 cups = 9 x 64 = 576 cents

Hence she makes 576 cents from her juice stand.

## What is a mixed fraction?

A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.

## How will you add fractions with unlike denominators?

When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions.

## Check out our other courses

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## Google can now solve trickier math problems for you with these new features

Math is a challenging subject because it requires an understanding of how to perform the operation to reach an answer, which makes it more difficult to Google an equation to find the answer difficult -- until now.

Google added new updates to Search and Lens that make it easier for users to get assistance when solving math problems. All users have to do now is type the equation or integral into the Search bar, or take a picture with Lens to get a step-by-step explanation or solution.

Also: Chrome on iOS unveils a much-anticipated feature. Here's how to access it

To test out the experience for yourself, on desktop, you can type in an equation or type the term "Math Solver" on Google Search where you will be prompted to enter a math problem or select from the examples to see how it works. The math solver experience will be coming to mobile soon.

Lens can also be leveraged by users to take a photo of geometry triangle problems, solving the challenge of trying to put primarily visual problems into words.

Advancements in Google's large language models also give Search the capability to solve word problems.

All you have to do is type the problem into Search, where you will be met with steps that tell you how to solve the problem by identifying the known and unknown values and providing correct formulas.

Also: The AI I want to see in the world: 5 ways it could manage my Gmail inbox for me

Lastly, Google is also making it easier to explore STEM-related concepts on Search by including 3D models and interactive diagrams for almost 1,000 biology, chemistry, physics, astronomy, and related topics, according to Google.

For example, if you Google "mitochondrion" you will have the opportunity to click on and learn from an interactive diagram that provides an overview, as well as specific details about the individual parts.

## Google will soon roll out a fix for an issue that can render devices inoperable

## How Google's AI Bard helped me fix a Gmail technical problem

## 5 ways Google can help you score deals this holiday season

## I need to know how to solve a math problem 45.2÷0.2

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what is the place value of the 8 in 281,480,100 ?

## The place value of 8 is ten million and the second place value of 8 is ten thousand.

I hope it helps.

9 ten thousands divided by ten

The answer is 9,000 or 9 thousand.

## Further explanation

9 ten thousand are a unit form. The standard form is 90,000.

Explanations and an example of questions about the four types of number forms can be found at https://brainly.com/question/4725342.

The key problem is working out the value of 90,000 divide by 10. This is a division operation.

[tex]\boxed{90,000 \div 10 = ?}[/tex]

Both numbers are divided by 10.

[tex]\boxed{90,000 \div 10 = 9,000 \div 1}[/tex]

The result is [tex]\boxed{90,000 \div 10 = 9,000}[/tex]

Division operations can be written as fractions, like this.

[tex]\boxed{\frac{90,000}{10} = ?}[/tex]

The numerator and denominator are both divided by 10, or 'cross out a zero from both'.

[tex]\boxed{ = \frac{9,000}{1}}[/tex]

- - - - - - -

If x is divided by y equals z, then x is equal to y multiplied by z.

[tex]\boxed{If \ x \div y = z \ or \ \frac{x}{y} = z, then \ x= y \cdot z}[/tex]

Let's prove it!

[tex]\boxed{If \ 90,000 \div 10 = 9,000 \ or \ \frac{90,000}{10} = 9,000, then \ 90,000 = 10 \cdot 9,000}[/tex]

This is proven.

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Keywords: 9 ten thousand divided by ten, division operation, unit form, standard form, fractions, cross out a zero, the numerator and denominator

The standard form of nine ten thousand is 90,000.

Here, our task is to divide 90,000 by 10.

If we divide 90,000 by 10 then the answer is 9,000.

=90,000/10=9,000.

Further details:

Related terms:

Standard form:

Standard form is a technique of writing down very large or very small numbers simply.

The number being divided into parts is termed as dividend, the number that is doing the dividing is termed as divisor, and the outcome is termed as quotient. “In division, the dividend is divided by the divisor to get a quotient.”

Answer details:

Subject: Mathematics

Level: Middle school

• Standard form

Learn more to evaluate:

https://brainly.com/question/4997492

https://brainly.com/question/4010464

https://brainly.com/question/1754173

What is 100,203 in expanded form

when I double my number, add four, and then subtract my number and subtract 3, I get my number plus 1

The number x is equal to all real number.

It is required to find the required number.

## What is real number?

Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers.

Let the no.= x

Double the number=2x

Add 4 = 2x+4

Subtract same number=2x+4-x

Subtract 3=2x+4-x-3

I get my number plus 1=x+1

So equation we get

2x+4-x-3=x+1

Therefore, the number x is equal to all real number.

Learn more about real number here:

https://brainly.com/question/545116

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The first 6 is in the thousands place and the second 6 is in the hundreds place.

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- Choose the point-slope form of the equation below that represents the line that passes through the point (1, 6) and has a slope of 3. A.y 6 = 3x 3 B.y 6 = 3(x + 1) C.y = 3x + 3 D.3x + y = 3
- The structure of DNA resembles a twisted ladder. Which structural components form the rungs of the ladder?
- What is the main function of veins in a leaf?PhotosynthesisTranspirationTransport of substances
- Simplify: 3(x 5) 8(y + 2) + (3)(6 4b) A. 3x 8y 12b 15 B. 3x 8y + 12b + 15 C. 3x 8y + 12b + 20 D. 3x 8y + 12b 19
- can the square root of 130 be simplified

## Easy Finger Math Tricks to Help Kids Solve Problems

Posted: November 9, 2023 | Last updated: November 9, 2023

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.

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## After 90 Years, Mathematicians Finally Solved the Most Notorious Ramsey Problem

“Everyone’s tried to figure it out,” says the scientist who actually did.

- For 90 years, mathematicians have been stumped by a problem that attempts to explain how many people would need to be invited to a party so at least 4 people always know one another.
- To finally crack the case, researchers from UC San Diego delved into the world of finite geometry and found a pseudograph to estimate an answer to the question .
- The researchers hope that this approach could help estimate other mathematical functions that are still left unsolved.

Okay ... let’s back up a little.

Named after mathematician Frank Ramsey, who first proved his eponymous theorem in the 1920s, Ramsey problems seek to find order in disorder. Ramsey’s original theory says that if a graph is large enough—in mathematics , a graph is a collection of points and lines between those points—you can find order within all that chaos. Chiefly, the theorem states that a set of points will have no lines between them, or a set of points with all possible lines between them (aka cliques). This is expressed as r(s, t ), where s means “points with lines” and t means “points without lines.”

## More From Popular Mechanics

While this is enough to make a non-mathematician dizzy, the concept is usually described as “the theorem on friends and strangers,” and is often allegorically explained as the idea that a party of six people will inevitably produce three people who know each other and three people who don’t. This is expressed as r(3,3), with the answer being six. (While I’m sure you have five close friends you can invite to a party, we’re talking about points and lines on a graph here—not actually people.)

“It’s a fact of nature , an absolute truth,” Jacques Verstraete, whose groundbreaking finding is currently under review with the journal Annals of Mathematics , says in a press release . “It doesn’t matter what the situation is or which six people you pick—you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

Expanding this idea, mathematicians Paul Erdös and George Szekeres discovered that r(4,4) equals 18 in 1935, and the solution to r(5,5) is still unknown. However, Verstraete and UC San Diego researcher Sam Mattheus have found an estimated solution for r(4, t ), where t means that “points without lines” is variable.

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations ,” Verstraete said in the press statement. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

Verstraete first saw the problem stated in a book about Erdös written by two UC San Diego researchers. These problems are usually solved using random graphs, but this Ramsey problem required more out-of-the-box thinking.

Using a concept known as pseudorandom graphs, Verstaraete discovered a solution for the r(3, t ) in 2019 . But the technique hit a brick wall when applied to the notoriously tricky r(4, t ) problem, so Verstraete ventured into other areas of mathematics—specifically, finite geometry —to figure out the answer.

“Many people have thought about r(4, t )—it’s been an open problem for over 90 years ,” Verstraete said in a press statement. “But it wasn’t something that was at the forefront of my research. Everybody knows it’s hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Drawing on Mattheus’s expertise in the field, the researchers discovered a pseudograph in finite geometry called an O’Nan configuration that worked for their problem. After a year of fine-tuning the math, Verstraete found that the answer was very close to a cubic function of t . This means that if you want a party where there will always be four people who know one another then, you need to invite t 3 people who don’t know each other. Reminder: this is just a close estimate, not an exact answer.

Now that it has produced a close estimate, Verstraete hopes that the technique could help open up new mathematical frontiers that are still waiting to be explored.

Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough.

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