problem solving of fractions examples

Fractions Questions and Problems with Solutions

Questions and problems with solutions on fractions are presented. Detailed solutions to the examples are also included. In order to master the concepts and skills of fractions, you need a thorough understanding (NOT memorizing) of the rules and properties and lot of practice and patience. I hope the examples, questions, problems in the links below will help you.

• Fractions and Mixed Numbers , define fractions and mixed numbers, and introduce important vocabulary.
• Evaluate Fractions of Quantities .
• Fractions Rules including questions with Solutions
• Properties of Fractions
• Complex Fractions with Variables
• Equivalent Fractions examples and questions with solutions.
• Reduce Fractions examples and questions with solutions.
• Reduce Fractions Calculator
• Simplify Fractions , examples and questions including solutions.
• Factor Fractions , examples with questions including solutions.
• Adding Fractions. Add fractions with same denominator or different denominator. Several examples with detailed solutions and exercises.
• Multiply Fractions. Multiply a fraction by another fraction or a number by a fraction. Examples with solutions and exercises.
• Divide Fractions. Divide a fraction by a fraction, a fraction by a number of a number by a fraction. Several examples with solutions and exercises with answers.

Fractions Per Grade

• Fractions and Mixed Numbers , questions and problems with solutions for grade 7
• Fractions and Mixed Numbers , questions and problems with solutions for grade 6
• Fractions , questions for grade 5 and their solutions
• Fractions , questions for grade 4 their answers.

Fractions Calculators

• Fractions Calculator including reducing, adding and multiplying fractions.
• Add Mixed Numbers Calculator

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. 

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Word Problems: Fractions

Word problem is a or two sentences that require students to apply their math skills and whatever they have learned to a real-life scenario. Some of the word problems of number fractions are:

• Addition and subtraction of fractions
• Dividing fractions
• Multiplication of fractions
• Simplification of fractions

Word Problems of Number Fractions – Examples 1

Kevin has 9 chocolates. He wants to share them with 5 friends. How much will each get?

Step 1: Divide 9 chocolates by 5+his=6: Step 2: Then, simplify the product if it is possible.

\(9÷6=\frac{9}{1}÷\frac{6}{1}=\frac{9}{1}×\frac{1}{6}=\frac{9×1}{1×6}=\frac{9}{6}\)

\(\frac{9}{6}=\frac{3}{2}\)

Word Problems of Number Fractions – Examples 2

Alice needs to order grilled chicken for 15 students. Each student should get \(\frac{1}{5}\) of a grilled chicken. How many grilled chickens should Alice order?

Step 1: Multiply 15 grilled chickens by \(\frac{1}{5}\). Step 2: Then, simplify the product if it is possible.

\(15×\frac{1}{5}=\frac{15}{1}×\frac{1}{5}=\frac{15}{5}\frac{3}{1}=3\)

by: Effortless Math Team about 1 year ago (category: Articles )

Effortless Math Team

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Word Problems on Fraction

In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

1.  4/7 of a number is 84. Find the number. Solution: According to the problem, 4/7 of a number = 84 Number = 84 × 7/4 [Here we need to multiply 84 by the reciprocal of 4/7]

= 21 × 7 = 147 Therefore, the number is 147.

2.  Rachel took \(\frac{1}{2}\) hour to paint a table and \(\frac{1}{3}\) hour to paint a chair. How much time did she take in all?

3. If 3\(\frac{1}{2}\) m of wire is cut from a piece of 10 m long wire, how much of wire is left?

Total length of the wire = 10 m

Fraction of the wire cut out = 3\(\frac{1}{2}\) m = \(\frac{7}{2}\) m

Length of the wire left = 10 m – 3\(\frac{1}{2}\) m

            = [\(\frac{10}{1}\) - \(\frac{7}{2}\)] m,    [L.C.M. of 1, 2 is 2]

            = [\(\frac{20}{2}\) - \(\frac{7}{2}\)] m,    [\(\frac{10}{1}\) × \(\frac{2}{2}\)]

            = [\(\frac{20 - 7}{2}\)] m

            = \(\frac{13}{2}\) m

            = 6\(\frac{1}{2}\) m

4. One half of the students in a school are girls, 3/5 of these girls are studying in lower classes. What fraction of girls are studying in lower classes?

Fraction of girls studying in school = 1/2

Fraction of girls studying in lower classes = 3/5 of 1/2

                                                            = 3/5 × 1/2

                                                            = (3 × 1)/(5 × 2)

                                                            = 3/10

Therefore, 3/10 of girls studying in lower classes.

5.  Maddy reads three-fifth of 75 pages of his lesson. How many more pages he need to complete the lesson? Solution: Maddy reads = 3/5 of 75 = 3/5 × 75

= 45 pages. Maddy has to read = 75 – 45. = 30 pages. Therefore, Maddy has to read 30 more pages. 6.  A herd of cows gives 4 litres of milk each day. But each cow gives one-third of total milk each day. They give 24 litres milk in six days. How many cows are there in the herd? Solution: A herd of cows gives 4 litres of milk each day. Each cow gives one-third of total milk each day = 1/3 of 4 Therefore, each cow gives 4/3 of milk each day. Total no. of cows = 4 ÷ 4/3                          = 4 × ¾                          = 3 Therefore there are 3 cows in the herd.

Questions and Answers on Word problems on Fractions:

1. Shelly walked \(\frac{1}{3}\) km. Kelly walked \(\frac{4}{15}\) km. Who walked farther? How much farther did one walk than the other?

2. A frog took three jumps. The first jump was \(\frac{2}{3}\) m long, the second was \(\frac{5}{6}\) m long and the third was \(\frac{1}{3}\) m long. How far did the frog jump in all?

3. A vessel contains 1\(\frac{1}{2}\) l of milk. John drinks \(\frac{1}{4}\) l of milk; Joe drinks \(\frac{1}{2}\) l of milk. How much of milk is left in the vessel?

●   Multiplication is Repeated Addition.

●  Multiplication of Fractional Number by a Whole Number.

●  Multiplication of a Fraction by Fraction.

●  Properties of Multiplication of Fractional Numbers.

●  Multiplicative Inverse.

●  Worksheet on Multiplication on Fraction.

●  Division of a Fraction by a Whole Number.

●  Division of a Fractional Number.

●  Division of a Whole Number by a Fraction.

●  Properties of Fractional Division.

●  Worksheet on Division of Fractions.

●  Simplification of Fractions.

●  Worksheet on Simplification of Fractions.

●  Word Problems on Fraction.

●  Worksheet on Word Problems on Fractions.

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Fractions  - Adding and Subtracting Fractions

Fractions  -, adding and subtracting fractions, fractions adding and subtracting fractions.

Fractions: Adding and Subtracting Fractions

Lesson 3: adding and subtracting fractions.

/en/fractions/comparing-and-reducing-fractions/content/

Adding and subtracting fractions

In the previous lessons, you learned that a fraction is part of a whole. Fractions show how much you have of something, like 1/2 of a tank of gas or 1/3 of a cup of water.

In real life, you might need to add or subtract fractions. For example, have you ever walked 1/2 of a mile to work and then walked another 1/2 mile back? Or drained 1/4 of a quart of gas from a gas tank that had 3/4 of a quart in it? You probably didn't think about it at the time, but these are examples of adding and subtracting fractions.

Click through the slideshow to learn how to set up addition and subtraction problems with fractions.

Let's imagine that a cake recipe tells you to add 3/5 of a cup of oil to the batter.

You also need 1/5 of a cup of oil to grease the pan. To see how much oil you'll need total, you can add these fractions together.

When you add fractions, you just add the top numbers, or numerators .

That's because the bottom numbers, or denominators , show how many parts would make a whole.

We don't want to change how many parts make a whole cup ( 5 ). We just want to find out how many parts we need total.

So we only need to add the numerators of our fractions.

We can stack the fractions so the numerators are lined up. This will make it easier to add them.

And that's all we have to do to set up an addition example with fractions. Our fractions are now ready to be added.

We'll do the same thing to set up a subtraction example. Let's say you had 3/4 of a tank of gas when you got to work.

If you use 1/4 of a tank to drive home, how much will you have left? We can subtract these fractions to find out.

Just like when we added, we'll stack our fractions to keep the numerators lined up.

This is because we want to subtract 1 part from 3 parts.

Now that our example is set up, we're ready to subtract!

Try setting up these addition and subtraction problems with fractions. Don't try solving them yet!

You run 4/10 of a mile in the morning. Later, you run for 3/10 of a mile.

You had 7/8 of a stick of butter and used 2/8 of the stick while cooking dinner.

Your gas tank is 2/5 full, and you put in another 2/5 of a tank.

Solving addition problems with fractions

Now that we know how to write addition problems with fractions, let's practice solving a few. If you can add whole numbers , you're ready to add fractions.

Click through the slideshow to learn how to add fractions.

Let's continue with our previous example and add these fractions: 3/5 of cup of oil and 1/5 of a cup of oil.

Remember, when we add fractions, we don't add the denominators.

This is because we're finding how many parts we need total. The numerators show the parts we need, so we'll add 3 and 1 .

3 plus 1 equals 4 . Make sure to line up the 4 with the numbers you just added.

The denominators will stay the same, so we'll write 5 on the bottom of our new fraction.

3/5 plus 1/5 equals 4/5 . So you'll need 4/5 of a cup of oil total to make your cake.

Let's try another example: 7/10 plus 2/10 .

Just like before, we're only going to add the numerators. In this example, the numerators are 7 and 2 .

7 plus 2 equals 9 , so we'll write that to the right of the numerators.

Just like in our earlier example, the denominator stays the same.

So 7/10 plus 2/10 equals 9/10 .

Try solving some of the addition problems below.

Solving subtraction problems with fractions

Subtracting fractions is a lot like regular subtraction. If you can subtract whole numbers , you can subtract fractions too!

Click through the slideshow to learn how to subtract fractions.

Let's use our earlier example and subtract 1/4 of a tank of gas from 3/4 of a tank.

Just like in addition, we're not going to change the denominators.

We don't want to change how many parts make a whole tank of gas. We just want to know how many parts we'll have left.

We'll start by subtracting the numerators. 3 minus 1 equals 2 , so we'll write 2 to the right of the numerators.

Just like when we added, the denominator of our answer will be the same as the other denominators.

So 3/4 minus 1/4 equals 2/4 . You'll have 2/4 of a tank of gas left when you get home.

Let's try solving another problem: 5/6 minus 3/6 .

We'll start by subtracting the numerators.

5 minus 3 equals 2 . So we'll put a 2 to the right of the numerators.

As usual, the denominator stays the same.

So 5/6 minus 3/6 equals 2/6 .

Try solving some of the subtraction problems below.

After you add or subtract fractions, you may sometimes have a fraction that can be reduced to a simpler fraction. As you learned in Comparing and Reducing Fractions , it's always best to reduce a fraction to its simplest form when you can. For example, 1/4 plus 1/4 equals 2/4 . Because 2 and 4 can both be divided 2 , we can reduce 2/4 to 1/2 .

Adding fractions with different denominators

On the last page, we learned how to add fractions that have the same denominator, like 1/4 and 3/4 . But what if you needed to add fractions with different denominators? For example, our cake recipe might say to blend 1/4 cup of milk in slowly and then dump in another 1/3 of a cup.

In Comparing and Reducing Fractions , we compared fractions with a different bottom number, or denominator. We had to change the fractions so their denominators were the same. To do that, we found the lowest common denominator , or LCD .

We can only add or subtract fractions if they have the same denominators. So we'll need to find the lowest common denominator before we add or subtract these fractions. Once the fractions have the same denominator, we can add or subtract as usual.

Click through the slideshow to learn how to add fractions with different denominators.

Let's add 1/4 and 1/3 .

Before we can add these fractions, we'll need to change them so they have the same denominator .

To do that, we'll have to find the LCD , or lowest common denominator, of 4 and 3 .

It looks like 12 is the smallest number that can be divided by both 3 and 4, so 12 is our LCD .

Since 12 is the LCD, it will be the new denominator for our fractions.

Now we'll change the numerators of the fractions, just like we changed the denominators.

First, let's look at the fraction on the left: 1/4 .

To change 4 into 12 , we multiplied it by 3 .

Since the denominator was multiplied by 3 , we'll also multiply the numerator by 3 .

1 times 3 equals 3 .

1/4 is equal to 3/12 .

Now let's look at the fraction on the right: 1/3 . We changed its denominator to 12 as well.

Our old denominator was 3 . We multiplied it by 4 to get 12.

We'll also multiply the numerator by 4 . 1 times 4 equals 4 .

So 1/3 is equal to 4/12 .

Now that our fractions have the same denominator, we can add them like we normally do.

3 plus 4 equals 7 . As usual, the denominator stays the same. So 3/12 plus 4/12 equals 7/12 .

Try solving the addition problems below.

Subtracting fractions with different denominators

We just saw that fractions can only be added when they have the same denominator. The same thing is true when we're subtracting fractions. Before we can subtract, we'll have to change our fractions so they have the same denominator.

Click through the slideshow to learn how to subtract fractions with different denominators.

Let's try subtracting 1/3 from 3/5 .

First, we'll change the denominators of both fractions to be the same by finding the lowest common denominator .

It looks like 15 is the smallest number that can be divided evenly by 3 and 5 , so 15 is our LCD.

Now we'll change our first fraction. To change the denominator to 15 , we'll multiply the denominator and the numerator by 3 .

5 times 3 equals 15 . So our fraction is now 9/15 .

Now let's change the second fraction. To change the denominator to 15 , we'll multiply both numbers by 5 to get 5/15 .

Now that our fractions have the same denominator, we can subtract like we normally do.

9 minus 5 equals 4 . As always, the denominator stays the same. So 9/15 minus 5/15 equals 4/15 .

Try solving the subtraction problems below.

Adding and subtracting mixed numbers

Over the last few pages, you've practiced adding and subtracting different kinds of fractions. But some problems will need one extra step. For example, can you add the fractions below?

In Introduction to Fractions , you learned about mixed numbers . A mixed number has both a fraction and a whole number . An example is 2 1/2 , or two-and-a-half . Another way to write this would be 5/2 , or five-halves . These two numbers look different, but they're actually the same.

5/2 is an improper fraction . This just means the top number is larger than the bottom number. Even though improper fractions look strange, you can add and subtract them just like normal fractions. Mixed numbers aren't easy to add, so you'll have to convert them into improper fractions first.

Let's add these two mixed numbers: 2 3/5 and 1 3/5 .

We'll need to convert these mixed numbers to improper fractions. Let's start with 2 3/5 .

As you learned in Lesson 2 , we'll multiply the whole number, 2 , by the bottom number, 5 .

2 times 5 equals 10 .

Now, let's add 10 to the numerator, 3 .

10 + 3 equals 13 .

Just like when you add fractions, the denominator stays the same. Our improper fraction is 13/5 .

Now we'll need to convert our second mixed number: 1 3/5 .

First, we'll multiply the whole number by the denominator. 1 x 5 = 5 .

Next, we'll add 5 to the numerators. 5 + 3 = 8 .

Just like last time, the denominator remains the same. So we've changed 1 3/5 to 8/5 .

Now that we've changed our mixed numbers to improper fractions, we can add like we normally do.

13 plus 8 equals 21 . As usual, the denominator will stay the same. So 13/5 + 8/5 = 21/5 .

Because we started with a mixed number, let's convert this improper fraction back into a mixed number.

As you learned in the previous lesson , divide the top number by the bottom number. 21 divided by 5 equals 4, with a remainder of 1 .

The answer, 4, will become our whole number.

And the remainder , 1, will become the numerator of the fraction.

So 2 3/5 + 1 3/5 = 4 1/5 .

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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}

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.

[FREE] Solve Equations with Fractions Worksheet (Grade 6 to 8)

Use this worksheet to check your grade 6 to 8 students’ understanding of solving equations with fractions. 15 questions with answers to identify areas of strength and support!

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.

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.

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.

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.

\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.

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.

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.

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.

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.

• 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

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

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.

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.

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.

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.

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.

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

• Inequalities
• Types of graphs
• Coordinate plane
• Number patterns
• Algebraic expressions
• Fractions operations

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11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes. 

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!” 

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

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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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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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Computer Science > Computation and Language

Title: large language models are unconscious of unreasonability in math problems.

Abstract: Large language models (LLMs) demonstrate substantial capabilities in solving math problems. However, they tend to produce hallucinations when given questions containing unreasonable errors. In this paper, we study the behavior of LLMs when faced with unreasonable math problems and further explore their potential to address these problems. First, we construct the Unreasonable Math Problem (UMP) benchmark to examine the error detection ability of LLMs. Experiments show that LLMs are able to detect unreasonable errors, but still fail in generating non-hallucinatory content. In order to improve their ability of error detection and correction, we further design a strategic prompt template called Critical Calculation and Conclusion(CCC). With CCC, LLMs can better self-evaluate and detect unreasonable errors in math questions, making them more reliable and safe in practical application scenarios.

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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

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The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

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