problem solving division of fraction

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Dividing Fractions Word Problems Worksheets

Toss off solutions to our pdf worksheets on dividing fractions word problems to foster a sense of excellence in identifying the dividend and the divisor and solving word problems on fraction division. Equipped with answer key, our worksheets get children in grade 5, grade 6, and grade 7 rattling their way through the division of fractions and mixed numbers in regular and themed problems. A flurry of everyday scenarios, our free worksheet for dividing fractions word problems is worth a shot!

Dividing Fractions and Whole Numbers Word Problems

Prepare the child through and through so they divide fractions and whole numbers with word problems. Let them take the reciprocal of the divisor and multiply it with the dividend, and they’re good to go!

• Download the set

Dividing Fractions by Cross Cancelling Word Problems

Exceed every learning expectation with our dividing fractions word problems worksheets! Apply cross cancellation after inverting the divisor, find products of what's left, and do write the correct units.

Dividing Mixed Numbers Word Problems

Give 5th grade and 6th grade students a good round of practice to hone their skills in fraction division. Convert mixed numbers into improper fractions, and proceed to divide them as usual.

Dividing Mixed Numbers and Fractions Word Problems

The road to mastery in word problems on dividing mixed numbers and fractions is made smooth with our printable worksheets. Read the problems, identify the dividends and divisors, and find the answers.

Themed Fraction Division Word Problems

If grade 6 and grade 7 learners are bent on proving they're real gifted at tackling fraction division, nothing can stop them! With our themed word problems pdfs, problem-solving is at its most exciting.

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Course: 6th grade   >   Unit 2

• Understanding division of fractions
• Dividing fractions: 2/5 ÷ 7/3

Dividing fractions: 3/5 ÷ 1/2

• Dividing fractions
• Dividing mixed numbers
• Divide mixed numbers
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Video transcript

Dividing Fractions

Turn the second fraction upside down, then multiply.

There are 3 Simple Steps to Divide Fractions:

Example: 1 2   ÷   1 6.

Step 1. Turn the second fraction upside down (it becomes a reciprocal ):

1 6 becomes 6 1

Step 2. Multiply the first fraction by that reciprocal :

(multiply tops ...)

1 2  ×  6 1   =   1 × 6 2 × 1   =   6 2

(... multiply bottoms)

Step 3. Simplify the fraction:

6 2   =  3

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

To help you remember:

♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify, Before it's time to say goodbye" ♫

20 divided by 5 is asking "how many 5s in 20?" (=4) and so:

1 2   ÷   1 6 is really asking:

how many 1 6 s in 1 2 ?

Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?

So now you can see why 1 2   ÷   1 6 = 3

In other words "I have half a pizza, if I divide it into one-sixth slices, how many slices is that?"

Another Example: 1 8   ÷   1 4

Step 1. Turn the second fraction upside down (the reciprocal ):

1 4   becomes   4 1

1 8   ×   4 1 = 1 × 4 8 × 1 = 4 8

4 8   =   1 2

Fractions and Whole Numbers

What about division with fractions and whole numbers?

Make the whole number a fraction, by putting it over 1.

Example: 5 is also 5 1

Then continue as before.

Example: 2 3   ÷  5

Make 5 into 5 1 :

2 3   ÷   5 1

5 1 becomes 1 5

2 3 × 1 5 = 2 × 1 3 × 5 = 2 15

The fraction is already as simple as it can be.

Answer = 2 15

Example: 3  ÷   1 4

Make 3 into 3 1 :

3 1   ÷   1 4

1 4 becomes 4 1

3 1 × 4 1 = 3 × 4 1 × 1 = 12 1

And Remember ...

You can rewrite a question like "20 divided by 5" into "how many 5s in 20"

So you can also rewrite "3 divided by ¼" into "how many ¼s in 3" (=12)

Why Turn the Fraction Upside Down?

Because dividing is the opposite of multiplying!

But for DIVISION we:

• divide by the top number
• multiply by the bottom number

Example: dividing by 5 / 2 is the same as multiplying by 2 / 5

So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.

Division of Fractions

Division means sharing an item equally. We have learned about the division of whole numbers, now let us see how to divide fractions. A fraction has two parts - a numerator and a denominator. Dividing fractions is almost the same as multiplying them. For the division of fractions, we multiply the first fraction by the reciprocal (inverse) of the second fraction. Let us learn more about the division of fractions in this article.

How to Divide Fractions?

We know that division is a method of sharing equally and putting into equal groups. We divide a whole number by the divisor to get the quotient . Now, when we do division of a fraction by another fraction, it is the same as multiplying the fraction by the reciprocal of the second fraction. The reciprocal of a fraction is a simple way of interchanging the fraction's numerator and denominator. Observe the following figure to learn a simple rule of dividing fractions.

In the subsequent sections, we will learn the division of fractions with fractions, whole numbers, decimals, and mixed numbers. In every case, we will be using the same rule of dividing fractions as given above. Let's begin!

Dividing Fractions by Fractions

We just learned how to divide fractions by taking the reciprocal. Now, let us see the method of dividing fractions by fractions with an example. Have a look at the formula of the division of a fraction by fraction given below. If x/y is divided by a/b, this implies,

⇒ x/y × b/a (reciprocal of a/b is b/a)

Now, if we need to divide: 5/8 ÷ 15/16, we will substitute the values of the given numerators and denominators .

5/8 ÷ 15/16 = 5/8 × 16/15 = 2/3

∴ The value of 5/8 ÷ 15/16 = 2/3.

Division of Fractions with Whole Numbers

For the division of fractions with whole numbers , we need to multiply the denominator of the given fraction with the given whole number . In the general form, if x/y is the fraction and a is the whole number, then x/y ÷ a = x/y × 1/a = x/ya.

Let us take an example and divide 2/3 with 4.

2/3 ÷ 4 = 2/3 × 1/4

Therefore, 2/3 ÷ 4 gives us 1/6. This is how we divide fractions with whole numbers.

Dividing Fractions with Decimals

We know that decimal numbers themselves are a fraction to base 10. We can represent the decimal in the fractional form and then perform the division. For dividing fractions with decimals, follow the steps given below:

• Convert the given decimal to a fraction.
• Divide both the fractions.

Consider the example, 4/5 ÷ 0.5. Here, 0.5 can be written in fractional form as 5/10 or 1/2. Now, divide 4/5 by 1/2. This implies, 4/5 ÷ 1/2 = 4/5 × 2/1 = 8/5. This is how we perform the division of fractions with decimals. Now let us learn how to divide fractions with mixed numbers.

Division of Fractions and Mixed Numbers

We have learned how to convert mixed fractions to improper fractions . For the division of fractions with mixed numbers , we have to convert the mixed fraction to an improper fraction first and then divide them as we divide two fractions. Consider the following example.

3/4 ÷ \(1\dfrac{1}{2}\)

So, the first step is to convert \(1\dfrac{1}{2}\) to an improper fraction . \(1\dfrac{1}{2}\) is the same as 3/2. Now, it can be solved in the following way:

⇒ 3/4 × 2/3

⇒ 6/12 = 1/2

Therefore, 3/4 ÷ \(1\dfrac{1}{2}\) = 1/2. If you want to divide a mixed number with a fraction, first convert the mixed number to an improper fraction and follow the same steps as shown above.

Division of Fractions Related Articles

Check these interesting articles related to the concept of division of fractions in math.

• Dividing Fractions Calculator
• Formula for Dividing Fractions
• Dividing Fractions with Whole Numbers

Division of Fractions Examples

Example 1: Find the value of 3/16 ÷ 15/32.

To divide 3/16 ÷ 15/32, we will be using the steps of the division of fractions. The first step is to keep the first fraction as it is. Then change the division sign to multiplication sign and at last, flip the second fraction to its reciprocal. This implies 3/16 × 32/15. After simplifying, we get (3 × 32) / (16 × 15) = 2/5.

∴ The value of 3/16 ÷ 15/32 = 2/5

Example 2: Tim has \(1\frac{1}{2}\) liters of juice in a jug. He has to pour the juice into cups. Each cup can hold 1/4 liters of juice. How many cups will he need to pour all the juice?

To solve this question, we will be using the concept of the division of fractions.

Number of cups needed = Total quantity of juice ÷ Capacity of 1 cup

= 3/2 ÷ 1/4 (as \(1\frac{1}{2}\) = 3/2)

= 3/2 × 4/1

Therefore, the number of cups required to pour the juice is 6.

Example 3: Use the steps of dividing fractions with whole numbers to find the value of 8/5 ÷ 5.

To divide a fraction with a whole number, we multiply the given whole number with the denominator of the fraction. Here, 8/5 ÷ 5 = 8/5 × 1/5 = 8/25.

Therefore, 8/5 ÷ 5 = 8/25.

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Practice Questions on Dividing Fractions

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FAQs on Dividing Fractions

What does division of fractions mean.

The division of fractions means breaking down a fraction into further parts. For example, if you take half (1/2) of a pizza and you further divide it into 2 equal parts, then each portion will be 1/4th of the whole pizza. Mathematically, we can express this reasoning as 1/2 ÷ 2 = 1/4.

What is Multiplication and Division of Fractions?

The multiplication of fractions means to add a fraction to itself repeatedly a specific number of times. The following steps are used to multiply fractions:

• Step 1: Multiply the numerators of both the fractions.
• Step 2: Multiply the denominators of both the fractions.
• Step 3: Simplify the fraction obtained after multiplication.

On the other hand, the division of fractions means to do equal grouping or equal sharing of a fraction. Dividing fractions is related to multiplication, as while dividing two fractions, we multiply the reciprocal of the second fraction to the first.

How to Visualize Division of Fractions?

To visualize the division of fractions, take a piece of paper and fold it into two equal parts. Cut 1/2 of the paper with scissors. Now, you will be left with 1/2 of the paper. Now, again divide that 1/2 portion into 2 equal parts. After this, you will be left with 1/4th of the paper. That is the answer of 1/2 ÷ 2. This is how you can visualize the concept of dividing fractions.

What is the Rule for Dividing Fractions?

The basic rule of dividing fractions is to keep, change, and flip. It means we have to keep the first fraction as it is, change the division sign to the multiplication sign, and flip the second fraction to its reciprocal. By following this simple rule, you can divide any two fractions.

What are the Steps to Dividing Fractions?

The following steps have to be followed in order to divide fractions:

• Step 1: Take the reciprocal of the second fraction.
• Step 2: Multiply it with the first fraction.
• Step 3: Reduce the resultant fraction to its lowest terms.

How to Teach Division of Fractions?

Division of fractions can be taught in many ways such as, by using models or applying the concept of multiplication of fractions. Some of the ways to teach how to divide fractions are listed below:

• Take circular or rectangular fraction models to demonstrate the concept of division of fractions to your learners.
• Use worksheets including pictures and word problems.
• Use materials from day-to-day lives like beans, leaves, pebbles, etc to show learners how to divide fractions.

How to Divide a Number by a Fraction?

To divide a whole number by a fraction, we multiply the whole number with the reciprocal of the given fraction. To divide a fraction by a fraction, we multiply the reciprocal of the second fraction with the first fraction.

How to do Division of Fractions with Whole Numbers?

Dividing fractions with whole numbers is a three-step process:

• Step 1: Keep the fraction as it is. For example, 3/4 ÷ 6.
• Step 2: Flip the whole number, which will make it a fraction of the format 1/a. In this case, 6 will become 1/6.
• Step 3: Change the sign into multiplication. We will get 3/4 × 1/6 = 3/24 = 1/8.

How to do Division of Fractions With Mixed Numbers?

For the division of fractions and mixed numbers, the following steps are used:

• Step 1: Keep the fraction as it is.
• Step 2: Convert the mixed number into an improper fraction and flip the second fraction.
• Step 3: Change the sign into multiplication between the fractions. Multiply and simplify them.

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Fraction Division Word Problems Worksheets

• Pre-Algebra >
• Fractions >
• Division >

Trawl through a host of free printable dividing fractions word problems worksheets and add life to division of fractions with fascinating everyday scenarios. Equal parts exciting and challenging, our practice word problems pdfs with included answer keys ensure children's computation skills will substantially improve. With diverse exercises like solving division word problems involving fractions, mixed numbers, and whole numbers, students will be spoiled for choice!

Our pdf fraction division word problem worksheets are suitable for grade 5, grade 6, and grade 7.

Dividing Fractions and Whole Numbers Word Problems - Customary

Let the young prodigies stir up enthusiasm as they grasp the concept of dividing fractions and whole numbers by solving word problems. The trick is for them to multiply with the reciprocal of the fractions and get the solutions.

Dividing Fractions and Whole Numbers Word Problems - Metric

Master dividing fractions and whole numbers by getting your hands on these pdf worksheets. Take the reciprocal of the divisor and multiply it with the dividend, and you're all set. Remember, this set involves metric units.

Dividing Fractions Word Problems - Customary

Watch how this printable set impels 6th grade and 7th grade students to solve real-world problems involving division of fractions. Deceptively simple, these pdfs get children to perform fraction division with wit and skill.

Dividing Fractions Word Problems - Metric

Looking for ways to make dividing fractions a fun-filled affair? Grade 6 and grade 7 students will work their way through interactive word problems on dividing fractions, featuring metric units of measurement.

Dividing Mixed Numbers Word Problems - Customary

Excitement never ends in our dividing mixed numbers word problems worksheets pdf. Each situation involves a pair of mixed numbers with customary units of measurement. Convert them to fractions and obtain the answers.

Dividing Mixed Numbers Word Problems - Metric

Introduce 5th grade kids to these word problems worksheet pdfs on dividing mixed fractions to hone their skills in fraction division. Convert mixed numbers into improper fractions, and proceed to divide.

Dividing Mixed Numbers and Fractions Word Problems - Customary

Run out of resources to divide fractions and mixed numbers on and on? Grab our pdf toolkit on dividing mixed numbers and fractions, brimming with exciting word problems involving customary units of measurement.

Dividing Mixed Numbers and Fractions Word Problems - Metric

Assess the division chops of learners in grade 5 by putting into use this set of pdf worksheets on dividing fractions and mixed numbers involving metric units of measurements.

Related Printable Worksheets

▶ Dividing Fractions Using Visual Models

▶ Dividing Fractions and Whole Numbers

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Dividing fractions word problems

Dividing fractions word problems arise in numerous situations. We will show you some examples. I recommend that you review the lesson about division of fractions before starting this lesson.

Dividing fractions word problems: a few examples

3 friends share 4/5 of a pizza. what fraction of pizza does each person get?

The amount to share is 4/5

Since the amount will be shared between 3 friends, the amount must be divided between 3 people.

So each person must get 4/5 divided by 3

(4 / 5) / 3 = (4 / 5) / (3 / 1) = 4 / 5 × 1 / 3 = (4 × 1) / (5 × 3)= 4 / 15

Each person will eat 4/15.

Indeed, 4 / 15 + 4 / 15 + 4 / 15 = 12 / 15 = 4/5 (divide 12 and 15 by 3 to get 4 / 5)

On June 21st, there were 9/10 of a billion stars visible to the naked eye.

On December 21st, there were 4/5 of a billion stars out visible to the naked eye.

How many times more stars were there visible on June 21st than December 21st?

9/10 divided by 4/5 = ?

Find the reciprocal of the divisor.

9/10 × 5/4 = ?

9/10 × 5/4 = 9/2 × 1/4

Solve 9/2 × 1/4 = 9/8

Simplify again

Did you have a hard time understanding the problems above? Did you not understand them at all? Take a look at this figure !

Do you need to master fractions once and for all? Check out my book about fractions .

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4.9: Dividing Fractions- Problems

• Last updated
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• Page ID 10584

• Michelle Manes
• University of Hawaii

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

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

Think / Pair / Share

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

When to Multiply, When to Divide?

A common answer to

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

Is something like this:

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

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

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

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

Example \(\PageIndex{1}\):

Here are some division problems involving whole numbers:

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

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

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

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

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

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

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

You try it.

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

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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80 Educational Children's Math Picture Books

R eady for the biggest list of math picture books ? Because I’ve found SO MANY amazing math books, I can’t wait to tell you about them!

Use these at home, in the classroom, or with your homeschool. You’ll discover books about counting , addition, subtraction, number sense, the 100th day, sorting, fractions, division, geometry, problem-solving, money, telling time, multiplication , and algebra.

Three cheers for math!

Table of Contents

Counting math books, the 100th day books, number sense books, telling time books, addition and subtraction books, sorting and pattern books, measurement books, multiplication books, division and fraction books, geometry books, money books, algebra books, math problem-solving books, best math books.

One Big Pair of Underwear

HAHA — this is the silliest “counting” picture book you’ll read! It’s counting, subtracting, and patterns silliness that your kids will adore.

Count your way from one to ten as this family gets ready for dinner including shopping and cooking the food.

You’ll love the clever creations Medina makes with vegetables — 1 avocado deer and 2 radish mice, just to name a few.

Anno’s Counting Book by Mitsumasa Anno

Bold graphic images help children find the black dots from one to ten in different images. Fun!

What’s more relatable than candy? And brightly colored illustrations? This tasty book about large numbers is pitch-perfect.

How Many Bugs in a Box

We love this engaging book. Lift the flaps and see what pops out!

Rhyme and count with these naughty monkeys.

The snake wants to count the mice — for his dinner. Count up and count down.

Monkey counts to ten and back as she bravely faces the crocodile-infested waters in order to get to a banana tree.

Alice needs to find 100 things to bring for the 100th day — but she’s having lots of trouble deciding what.

What is he going to bring for the 100th day of school? You’ll love this delightful rhyming book.

It’s not only the kids that get to bring 100 things to school, Miss Bindergarten is getting together 100 things, too.

Learn about counting by tens as the queens plan a special birthday surprise for the king.

Grapes of Math

Fun and rhyming riddles to help kids learn problem-solving strategies.

Learn about odd and even numbers with this silly story about a boy who discovers that everything in his life is ODD! (Also read: My Even Day and My Half Day .)

Hungry for Math: Poems to Munch On

Just like numbers, ideas are infinite. This is fun story of making the challenging concept of infinity more understandable.

Even Steven is all about, you guessed it, even-numbered things. Then one day, his cousin Odd Todd comes to visit. Which terrifies Even Steven. Because even Odd Todd knocks in odd numbers. . .

Learn about a boy who loved numbers and was known as The Magician from Budapest in this playful mathematical biography.

Go Figure!: A Totally Cool Book About Numbers

Learn more about the numbers in our everyday life, their purpose, and history. Then try some of the fun number magic tricks, puzzles, and activities.

365 Penguins

Penguins are arriving every single day at their doorstep. What are they doing to do?!

How Much Is a Million?

David M. Schwartz, illustrated by  Steven Kellogg

Marvelosissimo the mathematical magician will teach you about really BIG numbers.

Skip count and estimate with pumpkins.

This is a fun book that offers 100 math riddles, each with adaptations for young kids and bigger kids.

Telling Time with Big Mama Cat by Dan Harper

Follow along with the daily schedule and use the movable hands to practice telling time.

Learn about the different measurements of time (seconds, minutes), go through a day and take mini-quizzes to figure out how much you’re learning.

This funny book is all about Mr. Crocodile’s schedule which includes finding and catching some pesky monkeys.

Pigeon Math

Hilarious! Addition and subtraction never felt so fun!!  An increasingly exasperated narrator is TRYING to tell the story about ten pigeons but it’s not going well. Visual support, goofy humor, and plenty of kid-appeal make this a 100% must-own, must-read STEM picture book.

Add the baby animals with the grown-ups to see how many all together.

Jen Arena   (Author),  Stephen Gilpin

A winter addition adventure of snowmen that will get you to 100 total.

A loving family shares a favorite cultural sweet treat and practices counting and subtracting in this beautifully written, Indian-flavored math story! Mama makes 10 gulab jamuns for guests. But, one child eats three. Now there are only 7 for the guests. And another child eats 3 more. Now there are only 4 left. Mamma wonders how she will have time to make more treats for her guests. The kids will help her make them! 

The Chicken Problem

This is a Peg and Cat picture book story their perfect picnic that goes totally crazy with runaway chickens.  Peg is “ totally freaking out ” and needs to get the one hundred chickens back in the coop. Peg and Cat must solve the chicken problem fast. I love the illustrations, the problem-solving characters, and the silly story.

Comic Book Math ~ Fun-Schooling Journal: Adding, Writing & Subtracting Games

by Sarah Janisse Brown

Use your imagination and practice math skills in a fun way. 

Subtract your way through this goofy story about an elevator going down.

Animal stories help kids learn the basics of putting numbers in groups and taking numbers away.

Count and add the animals on the back of the trucks.

Go on a butterfly addition hunt and see who will win.

When the music stops, someone is out. Subtract to see how many are left.

The O’Malleys pass the time on a long car trip by counting up different color cars using tally marks. The winner is the one who tallies the most.

Arithmechicks Take Away: A Math Story

This gives kids photographs from which they can make decisions about sorting. Use with actual physical objects to make the lessons more concrete.

Blockhead: The Life of Fibonacci

Fibonacci sees patterns in nature and develops the Fibonacci Sequence.

Learn about all the spirals in nature.

What groups can you sort out of Packy the Packrat’s stuff?

Fannie in the Kitchen: The Whole Story from Soup to Nuts of How Fannie Farmer Invented Recipes with Precise Measurements  

Kitchen measurements equal delicious foods.

Measuring Penny by Loreen Leedy

Lisa loves measurement, so she starts measuring her dog, Penny.

How Big is a Foot by Rolf Myller

The king needs to figure out how big of a bed to make for his queen. This introduces standardizing measurements.

An inchworm shows the bird why he shouldn’t be eaten — because he can measure anything!

Amanda learns that multiplication is the fastest way to count.

Kings Chessboard

Multiplying Menace: The Revenge of Rumpelstiltskin

by Pam Calvert and Wayne Geehan

This is a fun multiplication story about mischievous Rumplestiltskin and his multiplication stick.

Gorgeous illustrations illustrate this fable about a smart girl who outsmarts a king.

This is an introduction to multiplication and factorals.

Spaghetti And Meatballs For All!  

by Marilyn Burns and Debbie Tilley

Yummy! It’s time for spaghetti. But how much does everyone get to eat?

Equal Shmequal

, illustrated by  Philomena O’Neill

Mouse helps her friends how to equally divide up teams for a game of tug of war.

Elinor J Pinczes , illustrated by  Bonnie MacKain

If 100 ants are marching to a picnic, how should they sort themselves into a line? 1 line of 100? 2 lines of 50? 

The Doorbell Rang

Pat Hutchins

More and more friends arrive to share Ma’s cookies. How many cookies should each person get?

The Lion’s Share

The shared meal keeps getting divided in half leaving only a crumb for the ant. So she and the other guests bake cakes for the king. Which they have to divide.

Using the illustrations, readers get to answer division and fraction questions. What fraction of the cow is blue? Fun farm math!

Fractions in Disguise

by Edward Einhorn and David Clark

This is a mystery story about finding a missing fraction — clever!

by  Dayle Ann Dodds , illustrated by  Abby Carter

The Strawberry Inn is filled with five visitors who all want a piece of one cake. How will Miss Blue solve this problem?

Learn the basic shapes with this cute introductory book.

Lia and Luis Puzzled by Ana Crespo, illustrated by Giovana Medeiros

Grandma gives the twins a puzzle they must complete to discover what the surprise is. What will it be? First, the twins will have to collaborate and use geometry and sorting to put the puzzle together. Lia and Luis are Brazilian American and the story includes words in Portuguese like the word for Grandma and yay.

by Cindy Neuschwander and Bryan Langdo

To get to the pharaoh’s burial tomb, the kids must decode the geometric hieroglyphics.

Sir Cumference and the First Round Table by

Cindy Neuschwander and Wayne Geehan

The king needs a place for his your knights to sit and discuss battle and peace plans. Luckily Sir Cumference, Lady Di of Ameter, and their son Radius can help.

by Cindy Neuschwander and Wayne Geehan

Radius must use his wits and math skills to rescue the missing king.

When a Line Bends . . . A Shape Begins

Learn about shapes in this brightly illustrated beginning circus story.

The name says it all — learn about perimeter, area, and volume with this crew of monsters.

What’s Your Angle, Pythagoras?

Pythagoras discovered through experimentation that there are mathematical principles that always stay the same — like with right triangles.

The Greedy Triangle

Marilyn Burns , illustrated by  Gordon Silveria

This triangle doesn’t just want to have three angles, he is greedy for more angles which changes his shape completely. 

To successfully journey back to earth, Captain Invincible must use his knowledge of 3D shapes.

Grandfather Tang’s Story by Ann Tonpert, illustrated by Robert Andrew Parker

Moving the tangram shapes, help narrate the story of two fox fairies.

In this Three Little Pigs math story, the pigs must learn geometric shapes and tangrams.

Alexander trades his one dollar for many coins because he misses the point of how much things are worth, placing importance on the number of monies he has more than the value. Hilarious.

Count five pennies, count two nickels, and add them up.

Grandma’s birthday is coming. Watch as Max and Ruby learn about how much things cost and what the best presents really are.

From the history of bartering things to the creation of different types of money, this is a great informational math book about money.

Little Critter needs to earn money so that he can buy a skateboard.

Interesting information about collecting coins, plus a place to start collecting.

Pigs Will Be Pigs: Fun with Math and Money by Amy Axelrod, illustrated by Sharon McGinley-Nally

David A. Adler ,  Edward Miller

Find the unknown number of creepy things by using addition, subtraction, multiplication, and division.

The Deductive Detective

This entertaining math picture book incorporates math with the mystery genre. Detective Duck needs to use his deductive reasoning to figure out which of the twelve animal bakers stole the cake from the cake contest.  He follows the clues, subtracting each suspect as he rules them out.  Until only one animal is left! Can you use your thinking skills to figure out the culprit before Detective Duck?

Frank adopts Lucky from a shelter. Together, they have fun, educational adventures around the neighborhood. For example, Frank learns about math and puzzles, thinking about how much hair Lucky sheds and dividing up and sharing the bed with Lucky. The author makes the duo’s learning fun and embedded throughout the day, whether it’s geography, science, or math. Love it.

One Minute Mysteries: 65 Short Mysteries You Solve With Math!

by Eric Yoder and Natalie Yoder

Real-world math brainteasers. 

by Jon Scieszka and Lane Smith

If you’ve ever been a victim of a MATH CURSE, you know how horrible it can be. Because you can break the curse. FUN and funny!

by Greg Tang and Greg Paprocki

Using real artwork, this is a math picture book where kids solve math problems and appreciate famous art.

The Book of Perfectly Perilous Math: 24 Death-Defying Challenges for Young Mathematicians

Math for All Seasons

Put on your thinking caps. Look closely at the illustrations to solve the math problems.

Find books about place value , too!

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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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Geometry Problems from IMOs

olympiad geometry problems with aops links geometry articles, books, magazines, shortlists for Juniors and Seniors problem collections with solutions from National, Regional and International Mathematical Olympiads latest added on the right menu

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The company said Grok 1.5 , the first release since open-sourcing the model , performed significantly better in coding and math-related reasoning than its previous version. xAI’s testing showed Grok 1.5 outdid models like Claude 2, Gemini Pro 1.5, and GPT-4 in some problem-solving benchmarks.

Grok 1.5 will be available to early users of the model on X soon.

AT&T confirms data breach and resets millions of customer passcodes

Sam bankman-fried is still gambling, the world needs more gadgets like lg’s briefcase tv, 20 years of gmail, openai’s voice cloning ai model only needs a 15-second sample to work, more from this stream the entire story of twitter / x under elon musk, elon musk has a new pay-to-play x premium gambit., judge tosses elon musk’s x lawsuit against anti-hate group, is this what x will look like on a smart tv, here’s the elon musk interview that got don lemon’s show canceled.

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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