Pay attention! Before writing your final answer, see if you can shorten the fraction you received.

Subtracting fractions with like denominators, examples:

,

,

Subtracting a proper fraction from one.

If it is necessary to subtract a fraction from a unit that is proper, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.

An example of subtracting a proper fraction from one:

Denominator of the fraction to be subtracted = 7 , i.e., we represent one as an improper fraction 7/7 and subtract it according to the rule for subtracting fractions with like denominators.

Subtracting a proper fraction from a whole number.

Rules for subtracting fractions - correct from a whole number (natural number):

• We convert given fractions that contain an integer part into improper ones. We get normal terms (it doesn’t matter if they have different denominators), which we calculate according to the rules given above;
• Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
• We perform the inverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.

Subtract a proper fraction from a whole number: represent the natural number as a mixed number. Those. We take one in a natural number and convert it to the form of an improper fraction, the denominator being the same as that of the subtracted fraction.

Example of subtracting fractions:

In the example, we replaced one with the improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.

Subtracting fractions with different denominators.

Or, to put it another way, subtracting different fractions.

Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.

The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.

Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!

Procedure for subtracting fractions with different denominators.

• find the LCM for all denominators;
• put additional factors for all fractions;
• multiply all numerators by an additional factor;
• We write the resulting products into the numerator, signing the common denominator under all fractions;
• subtract the numerators of fractions, signing the common denominator under the difference.

In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.

Subtracting fractions, examples:

Subtracting mixed fractions.

At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.

The first option for subtracting mixed fractions.

If the fractional parts identical denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).

For example:

The second option for subtracting mixed fractions.

When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.

For example:

The third option for subtracting mixed fractions.

The fractional part of the minuend is less than the fractional part of the subtrahend.

Example:

Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.

The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.

In the numerator on the right side we write the sum of the numerators, then we open the brackets in the numerator on the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:

Adding fractions with like denominators

There are two types of addition of fractions:

1. Adding fractions with like denominators
2. Adding fractions with different denominators

First, let's learn the addition of fractions with like denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged. For example, let's add the fractions and . Add the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2. Add fractions and .

The answer turned out to be an improper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:

This example can be easily understood if we remember about a pizza that is divided into two parts. If you add more pizza to the pizza, you get one whole pizza:

Example 3. Add fractions and .

Again, we add up the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a drawing. If you add pizza to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, there is nothing complicated about adding fractions with the same denominators. It is enough to understand the following rules:

1. To add fractions with the same denominator, you need to add their numerators and leave the denominator unchanged;

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the fractions must be the same. But they are not always the same.

For example, fractions can be added because they have the same denominators.

But fractions cannot be added right away, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will look at only one of them, since the other methods may seem complicated for a beginner.

The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

The numerators and denominators of the fractions are then multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

Example 1. Let's add the fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now let's return to fractions and . First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:

Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:

Look carefully at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's take this example to the end:

This completes the example. It turns out to add .

Let's try to depict our solution using a drawing. If you add pizza to a pizza, you get one whole pizza and another sixth of a pizza:

Reducing fractions to the same (common) denominator can also be depicted using a picture. Reducing the fractions and to a common denominator, we got the fractions and . These two fractions will be represented by the same pieces of pizza. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).

The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).

Please note that we have described this example in too much detail. In educational institutions it is not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:

But there is also another side to the coin. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

1. Find the LCM of the denominators of fractions;
2. Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction;
3. Multiply the numerators and denominators of fractions by their additional factors;
4. Add fractions that have the same denominators;
5. If the answer turns out to be an improper fraction, then highlight its whole part;

Example 2. Find the value of an expression .

Let's use the instructions given above.

Step 1. Find the LCM of the denominators of the fractions

Find the LCM of the denominators of both fractions. The denominators of fractions are the numbers 2, 3 and 4

Step 2. Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it above the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We get the second additional factor 4. We write it above the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We get the third additional factor 3. We write it above the third fraction:

Step 3. Multiply the numerators and denominators of the fractions by their additional factors

We multiply the numerators and denominators by their additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it up:

The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is moved to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of the new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an improper fraction, then highlight the whole part of it

Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:

We received an answer

Subtracting fractions with like denominators

There are two types of subtraction of fractions:

1. Subtracting fractions with like denominators
2. Subtracting fractions with different denominators

First, let's learn how to subtract fractions with like denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, but leave the denominator the same.

For example, let's find the value of the expression . To solve this example, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. Let's do this:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction you need to subtract the numerators of the remaining fractions:

As you can see, there is nothing complicated about subtracting fractions with the same denominators. It is enough to understand the following rules:

1. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
2. If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

Subtracting fractions with different denominators

For example, you can subtract a fraction from a fraction because the fractions have the same denominators. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found using the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written above the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written above the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators are converted into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1. Find the meaning of the expression:

These fractions have different denominators, so you need to reduce them to the same (common) denominator.

First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now let's return to fractions and

Let's find an additional factor for the first fraction. To do this, divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. Write a four above the first fraction:

We do the same with the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a three over the second fraction:

Now we are ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:

We received an answer

Let's try to depict our solution using a drawing. If you cut pizza from a pizza, you get pizza

This is the detailed version of the solution. If we were in school, we would have to solve this example shorter. Such a solution would look like this:

Reducing fractions to a common denominator can also be depicted using a picture. Reducing these fractions to a common denominator, we got the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into equal shares (reduced to the same denominator):

The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2. Find the value of an expression

These fractions have different denominators, so first you need to reduce them to the same (common) denominator.

Let's find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it above the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it above the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it above the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler. What can be done? You can shorten this fraction.

To reduce a fraction, you need to divide its numerator and denominator by (GCD) of the numbers 20 and 30.

So, we find the gcd of numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found gcd, that is, by 10

We received an answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the fraction by that number and leave the denominator the same.

Example 1. Multiply a fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The recording can be understood as taking half 1 time. For example, if you take pizza once, you get pizza

From the laws of multiplication we know that if the multiplicand and the factor are swapped, the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying a whole number and a fraction works:

This notation can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The answer was an improper fraction. Let's highlight the whole part of it:

The expression can be understood as taking two quarters 4 times. For example, if you take 4 pizzas, you will get two whole pizzas

And if we swap the multiplicand and the multiplier, we get the expression . It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplying fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.

Example 1. Find the value of the expression.

We received an answer. It is advisable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what a pizza looks like, divided into three parts:

One piece of this pizza and the two pieces we took will have the same dimensions:

In other words, we are talking about the same size pizza. Therefore the value of the expression is

Example 2. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer was an improper fraction. Let's highlight the whole part of it:

Example 3. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer turned out to be a regular fraction, but it would be good if it was shortened. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.

So, let’s find the gcd of numbers 105 and 450:

Now we divide the numerator and denominator of our answer by the gcd that we have now found, that is, by 15

Representing a whole number as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . This will not change the meaning of five, since the expression means “the number five divided by one,” and this, as we know, is equal to five:

Reciprocal numbers

Now we will get acquainted with a very interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to numbera is a number that, when multiplied bya gives one.

Let's substitute in this definition instead of the variable a number 5 and try to read the definition:

Reverse to number 5 is a number that, when multiplied by 5 gives one.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out it is possible. Let's imagine five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let’s multiply the fraction by itself, only upside down:

What will happen as a result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number , since when you multiply 5 by you get one.

The reciprocal of a number can also be found for any other integer.

You can also find the reciprocal of any other fraction. To do this, just turn it over.

Dividing a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How much pizza will each person get?

It can be seen that after dividing half the pizza, two equal pieces were obtained, each of which constitutes a pizza. So everyone gets a pizza.

Division of fractions is done using reciprocals. Reciprocal numbers allow you to replace division with multiplication.

To divide a fraction by a number, you need to multiply the fraction by the inverse of the divisor.

Using this rule, we will write down the division of our half of the pizza into two parts.

So, you need to divide the fraction by the number 2. Here the dividend is the fraction and the divisor is the number 2.

To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is the fraction. So you need to multiply by

Has your child brought homework from school and you don't know how to solve it? Then this mini lesson is for you!

How to add decimals

It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:

• The place must be under the place, the comma under the comma.

As you can see in the example, the whole units are located under each other, the tenths and hundredths digits are located under each other. Now we add the numbers, ignoring the comma. What to do with the comma? The comma is moved to the place where it stood in the integer category.

Adding fractions with equal denominators

To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction that will be the total sum.

Adding fractions with different denominators using the common multiple method

The first thing you need to pay attention to is the denominators. The denominators are different, whether one is divisible by the other, or whether they are prime numbers. First you need to bring it to one common denominator; there are several ways to do this:

• 1/3 + 3/4 = 13/12, to solve this example we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b – LCM (a;b). In this example LCM (3;4)=12. We check: 12:3=4; 12:4=3.
• We multiply the factors and add the resulting numbers, we get 13/12 - an improper fraction.

• In order to convert an improper fraction into a proper one, divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.

Adding fractions using the cross-cross multiplication method

To add fractions with different denominators, there is another method using the “cross to cross” formula. This is a guaranteed way to equalize the denominators; to do this, you need to multiply the numerators with the denominator of one fraction and vice versa. If you are just at the initial stage of learning fractions, then this method is the simplest and most accurate way to get the correct result when adding fractions with different denominators.

In the article we will show how to solve fractions using simple, understandable examples. Let's figure out what a fraction is and consider solving fractions!

Concept fractions is introduced into the mathematics course starting from the 6th grade of secondary school.

Fractions have the form: ±X/Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:

In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, of which 4 parts were taken, i.e. 4/7.

If the part of dividing one number by another is not a whole number, it is written as a fraction.

For example, the expression 4:2 = 2 gives an integer, but 4:7 is not divisible by a whole, so this expression is written as a fraction 4/7.

In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written using a fractional slash.

If the numerator is less than the denominator, the fraction is proper; if vice versa, it is an improper fraction. A fraction can contain a whole number.

For example, 5 whole 3/4.

This entry means that in order to get the whole 6, one part of four is missing.

If you want to remember, how to solve fractions for 6th grade, you need to understand that solving fractions, basically, comes down to understanding a few simple things.

• A fraction is essentially an expression of a fraction. That is, a numerical expression of what part a given value is of one whole. For example, the fraction 3/5 expresses that if we divided something whole into 5 parts and the number of shares or parts of this whole is three.
• The fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2 = 1 whole 1/2.
• Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole numbers but fractions. You can perform all the same operations with them as with numbers. Counting fractions is no more difficult, and we will show this further with specific examples.

How to solve fractions. Examples.

A wide variety of arithmetic operations are applicable to fractions.

Reducing a fraction to a common denominator

For example, you need to compare the fractions 3/4 and 4/5.

To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible without a remainder by each of the denominators of the fractions

Least common denominator(4.5) = 20

Then the denominator of both fractions is reduced to the lowest common denominator

Answer: 15/20

Adding and subtracting fractions

If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference between fractions is calculated in the same way, the only difference is that the numerators are subtracted.

For example, you need to find the sum of the fractions 1/2 and 1/3

Now let's find the difference between the fractions 1/2 and 1/4

Multiplying and dividing fractions

Here solving fractions is not difficult, everything is quite simple here:

• Multiplication - numerators and denominators of fractions are multiplied together;
• Division - first we get the fraction inverse of the second fraction, i.e. We swap its numerator and denominator, after which we multiply the resulting fractions.

For example:

That's about it how to solve fractions, All. If you still have any questions about solving fractions If something is not clear, write in the comments and we will definitely answer you.

If you are a teacher, then perhaps downloading a presentation for elementary school (http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) will be useful for you.

Online calculator.
Evaluate an expression with numerical fractions.
Multiplying, subtracting, dividing, adding and reducing fractions with different denominators.

With this online calculator you can multiply, subtract, divide, add and reduce fractions with different denominators.

The program works with regular, improper and mixed number fractions.

This program (online calculator) can:
- perform addition of mixed fractions with different denominators
- perform subtraction of mixed fractions with different denominators
- divide mixed fractions with different denominators
- multiply mixed fractions with different denominators
- reduce fractions to a common denominator
- convert mixed fractions to improper fractions
- reduce fractions

You can also enter not an expression with fractions, but one single fraction.
In this case, the fraction will be reduced and the whole part will be separated from the result.

The online calculator for calculating expressions with numerical fractions does not just give the answer to the problem, it provides a detailed solution with explanations, i.e. displays the process of finding a solution.

This program can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

If you are not familiar with the rules for entering expressions with numerical fractions, we recommend that you familiarize yourself with them.

Rules for entering expressions with numerical fractions

Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering a numerical fraction, the numerator is separated from the denominator by a division sign: /
Input: -2/3 + 7/5
Result: \(-\frac(2)(3) + \frac(7)(5)\)

The whole part is separated from the fraction by the ampersand sign: &
Input: -1&2/3 * 5&8/3
Result: \(-1\frac(2)(3) \cdot 5\frac(8)(3)\)

The division of fractions is introduced by the colon sign: :
Input: -9&37/12: -3&5/14
Result: \(-9\frac(37)(12) : \left(-3\frac(5)(14) \right) \)
Remember that you cannot divide by zero!

You can use parentheses when entering expressions with numeric fractions.
Input: -2/3 * (6&1/2-5/9) : 2&1/4 + 1/3
Result: \(-\frac(2)(3) \cdot \left(6 \frac(1)(2) - \frac(5)(9) \right) : 2\frac(1)(4) + \frac(1)(3)\)

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A little theory.

Ordinary fractions. Division with remainder

If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).

The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or completely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.

The remainder is always less than the divisor.

Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.

Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.

The quotient of natural numbers can be written as a fraction.

The numerator of a fraction is the dividend, and the denominator is the divisor.

Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.

The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n) \), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n)\)

The following rules are true:

To get the fraction \(\frac(m)(n)\), you need to divide the unit into n equal parts (shares) and take m such parts.

To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.

To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.

To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.

If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)

If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.

The last two transformations are called reducing a fraction.

If fractions need to be represented as fractions with the same denominator, then this action is called reducing fractions to a common denominator.

Proper and improper fractions. Mixed numbers

You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense dictates that the part should always be less than the whole, but what about fractions such as \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.

As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.

If a number consists of an integer part and a fraction, then fractions are called mixed.

For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.

If the numerator of the fraction \(\frac(a)(b) \) is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)

If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)

Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.

Actions with fractions. Adding fractions.

You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)

If you need to add fractions with different denominators, they must first be reduced to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)

For fractions, as for natural numbers, the commutative and associative properties of addition are valid.

Adding mixed fractions

Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”

When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)

Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction of fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)

The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.

Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)

Multiplying fractions

To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.

Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)

Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, and a mixed fraction as an improper fraction.

The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.

For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.

Division of fractions

Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).

If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.

For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).

Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)

It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)

Using reciprocal fractions, you can reduce division of fractions to multiplication.

The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.