Fractions. Multiplication and division of fractions

Fraction calculator designed for quick calculation of operations with fractions, it will help you easily add, multiply, divide or subtract fractions.

Modern schoolchildren begin to study fractions already in the 5th grade, and every year the exercises with them become more complicated. Mathematical terms and quantities that we learn in school are rarely useful to us in adulthood. However, fractions, unlike logarithms and degrees, are quite common in everyday life (measuring distance, weighing goods, etc.). Our calculator is designed for quick operations with fractions.

First, let's define what fractions are and what they are. Fractions are the ratio of one number to another; this is a number consisting of a whole number of fractions of a unit.

Fraction types:

Ordinary
Decimals
mixed

Example ordinary fractions:

The top value is the numerator, the bottom is the denominator. The dash shows us that the top number is divisible by the bottom number. Instead of a similar writing format, when the dash is horizontal, you can write differently. You can put a slanted line, for example:

1/2, 3/7, 19/5, 32/8, 10/100, 4/1

Decimals are the most popular type of fractions. They consist of an integer part and a fractional part, separated by a comma.

Decimal example:

0.2 or 6.71 or 0.125

It consists of an integer and a fractional part. To find out the value of this fraction, you need to add the whole number and the fraction.

Example of mixed fractions:

The fraction calculator on our website is able to quickly perform any mathematical operations with fractions online:

Addition
Subtraction
Multiplication
Division

To carry out the calculation, you need to enter the numbers in the fields and select the action. For fractions, you need to fill in the numerator and denominator, an integer may not be written (if the fraction is ordinary). Don't forget to click on the "equal" button.

It is convenient that the calculator immediately provides a process for solving an example with fractions, and not just a ready-made answer. It is thanks to the detailed solution that you can use this material in solving school problems and for better mastering the material covered.

You need to calculate the example:

After entering the indicators in the form fields, we get:

To make an independent calculation, enter the data in the form.

Fraction calculator

Enter two fractions:

		+ - * :

related sections.

Instruction

Reduction to a common denominator.

Let fractions a/b and c/d be given.

The numerator and denominator of the first fraction is multiplied by LCM / b

The numerator and denominator of the second fraction is multiplied by LCM/d

An example is shown in the figure.

To compare fractions, they need to have a common denominator, then compare the numerators. For example, 3/4< 4/5, см. .

Addition and subtraction of fractions.

To find the sum of two ordinary fractions, they must be reduced to a common denominator, and then add the numerators, the denominator is unchanged. An example of adding fractions 1/2 and 1/3 is shown in the figure.

The difference of fractions is found in a similar way, after finding the common denominator, the numerators of the fractions are subtracted, see the figure.

When multiplying ordinary fractions, the numerators and denominators are multiplied together.

In order to divide two fractions, you need a fraction of the second fraction, i.e. change its numerator and denominator, and then multiply the resulting fractions.

Basic property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Bringing fractions to a common denominator

To bring two fractions to a common denominator, you need:

Multiply the numerator of the first fraction by the denominator of the second
Multiply the numerator of the second fraction by the denominator of the first
Replace the denominators of both fractions with their product

Actions with fractions

Addition. To add two fractions, you need

Add new numerators of both fractions, and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another,

Bring fractions to a common denominator
Subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators:

Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:

This article deals with operations on fractions. Rules for addition, subtraction, multiplication, division or exponentiation of fractions of the form A B will be formed and justified, where A and B can be numbers, numeric expressions or expressions with variables. In conclusion, examples of solutions with a detailed description will be considered.

Yandex.RTB R-A-339285-1

Rules for performing operations with numerical fractions of a general form

Numerical fractions of a general form have a numerator and a denominator, in which there are natural numbers or numerical expressions. If we consider such fractions as 3 5 , 2 , 8 4 , 1 + 2 3 4 (5 - 2) , 3 4 + 7 8 2 , 3 - 0 , 8 , 1 2 2 , π 1 - 2 3 + π , 2 0 , 5 ln 3 , then it is clear that the numerator and denominator can have not only numbers, but also expressions of a different plan.

Definition 1

There are rules by which actions are performed with ordinary fractions. It is also suitable for fractions of a general form:

When subtracting fractions with the same denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d \u003d a ± c d, the values a, c and d ≠ 0 are some numbers or numerical expressions.
When adding or subtracting fractions with different denominators, it is necessary to reduce to a common one, and then add or subtract the resulting fractions with the same indicators. Literally, it looks like this a b ± c d = a p ± c r s , where the values a , b ≠ 0 , c , d ≠ 0 , p ≠ 0 , r ≠ 0 , s ≠ 0 are real numbers, and b p = d r = s. When p = d and r = b, then a b ± c d = a d ± c d b d.
When multiplying fractions, an action is performed with numerators, after which with denominators, then we get a b c d \u003d a c b d, where a, b ≠ 0, c, d ≠ 0 act as real numbers.
When dividing a fraction by a fraction, we multiply the first by the second reciprocal, that is, we swap the numerator and denominator: a b: c d \u003d a b d c.

Rationale for the rules

Definition 2

There are the following mathematical points that you should rely on when calculating:

a fractional bar means a division sign;
division by a number is treated as a multiplication by its reciprocal;
application of the property of actions with real numbers;
application of the basic property of a fraction and numerical inequalities.

With their help, you can make transformations of the form:

a d ± c d = a d - 1 ± c d - 1 = a ± c d - 1 = a ± c d ; a b ± c d = a p b p ± c r d r = a p s ± c e s = a p ± c r s ; a b c d = a d b d b c b d = a d a d - 1 b c b d - 1 = = a d b c b d - 1 b d - 1 = a d b c b d b d - 1 = = (a c) (b d) - 1 = a c b d

Examples

In the previous paragraph, it was said about actions with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the section on converting fractions.

First, consider the example of adding and subtracting fractions with the same denominator.

Example 1

Given fractions 8 2 , 7 and 1 2 , 7 , then according to the rule it is necessary to add the numerator and rewrite the denominator.

Decision

Then we get a fraction of the form 8 + 1 2 , 7 . After performing the addition, we get a fraction of the form 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 . So 8 2 , 7 + 1 2 , 7 = 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 .

Answer: 8 2 , 7 + 1 2 , 7 = 3 1 3

There is another way to solve. To begin with, a transition is made to the form of an ordinary fraction, after which we perform a simplification. It looks like this:

8 2 , 7 + 1 2 , 7 = 80 27 + 10 27 = 90 27 = 3 1 3

Example 2

Let us subtract from 1 - 2 3 log 2 3 log 2 5 + 1 fractions of the form 2 3 3 log 2 3 log 2 5 + 1 .

Since equal denominators are given, it means that we are calculating a fraction with the same denominator. We get that

1 - 2 3 log 2 3 log 2 5 + 1 - 2 3 3 log 2 3 log 2 5 + 1 = 1 - 2 - 2 3 3 log 2 3 log 2 5 + 1

There are examples of calculating fractions with different denominators. An important point is the reduction to a common denominator. Without this, we will not be able to perform further actions with fractions.

The process is remotely reminiscent of reduction to a common denominator. That is, a search is made for the least common divisor in the denominator, after which the missing factors are added to the fractions.

If the added fractions do not have common factors, then their product can become one.

Example 3

Consider the example of adding fractions 2 3 5 + 1 and 1 2 .

Decision

In this case, the common denominator is the product of the denominators. Then we get that 2 · 3 5 + 1 . Then, when setting additional factors, we have that to the first fraction it is equal to 2, and to the second 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 3 5 + 1. The general cast 1 2 will be 3 5 + 1 2 · 3 5 + 1 . We add the resulting fractional expressions and get that

2 3 5 + 1 + 1 2 = 2 2 2 3 5 + 1 + 1 3 5 + 1 2 3 5 + 1 = = 4 2 3 5 + 1 + 3 5 + 1 2 3 5 + 1 = 4 + 3 5 + 1 2 3 5 + 1 = 5 + 3 5 2 3 5 + 1

Answer: 2 3 5 + 1 + 1 2 = 5 + 3 5 2 3 5 + 1

When we are dealing with fractions of a general form, then the least common denominator is usually not the case. It is unprofitable to take the product of numerators as a denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Consider the example 1 6 2 1 5 and 1 4 2 3 5 when their product is equal to 6 2 1 5 4 2 3 5 = 24 2 4 5 . Then we take 12 · 2 3 5 as a common denominator.

Consider examples of multiplications of fractions of a general form.

Example 5

To do this, it is necessary to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Decision

Following the rule, it is necessary to rewrite and write the product of numerators as a denominator. We get that 2 + 1 6 2 5 3 2 + 1 2 + 1 2 5 6 3 2 + 1 . When the fraction is multiplied, reductions can be made to simplify it. Then 5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10 .

Using the rule of transition from division to multiplication by a reciprocal, we get the reciprocal of the given one. To do this, the numerator and denominator are reversed. Let's look at an example:

5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10

After that, they must perform multiplication and simplify the resulting fraction. If necessary, get rid of the irrationality in the denominator. We get that

5 3 3 2 + 1: 10 9 3 = 5 3 3 9 3 10 2 + 1 = 5 2 10 2 + 1 = 3 2 2 + 1 = = 3 2 - 1 2 2 + 1 2 - 1 = 3 2 - 1 2 2 2 - 1 2 = 3 2 - 1 2

Answer: 5 3 3 2 + 1: 10 9 3 = 3 2 - 1 2

This paragraph is applicable when a number or numerical expression can be represented as a fraction with a denominator equal to 1, then the operation with such a fraction is considered a separate paragraph. For example, the expression 1 6 7 4 - 1 3 shows that the root of 3 can be replaced by another 3 1 expression. Then this record will look like a multiplication of two fractions of the form 1 6 7 4 - 1 3 = 1 6 7 4 - 1 3 1 .

Performing an action with fractions containing variables

The rules discussed in the first article are applicable to operations with fractions containing variables. Consider the subtraction rule when the denominators are the same.

It is necessary to prove that A , C and D (D not equal to zero) can be any expressions, and the equality A D ± C D = A ± C D is equivalent to its range of valid values.

It is necessary to take a set of ODZ variables. Then A, C, D must take the corresponding values a 0 , c 0 and d0. A substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, according to the addition rule, we obtain a formula of the form a 0 ± c 0 d 0 . If we substitute the expression A ± C D , then we get the same fraction of the form a 0 ± c 0 d 0 . From this we conclude that the chosen value that satisfies the ODZ, A ± C D and A D ± C D are considered equal.

For any value of the variables, these expressions will be equal, that is, they are called identically equal. This means that this expression is considered to be a provable equality of the form A D ± C D = A ± C D .

Examples of addition and subtraction of fractions with variables

When there are the same denominators, it is only necessary to add or subtract the numerators. This fraction can be simplified. Sometimes you have to work with fractions that are identically equal, but at first glance this is not noticeable, since some transformations must be performed. For example, x 2 3 x 1 3 + 1 and x 1 3 + 1 2 or 1 2 sin 2 α and sin a cos a. Most often, a simplification of the original expression is required in order to see the same denominators.

Example 6

Calculate: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 , 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) , x - 1 x - 1 + x x + 1 .

Decision

To make a calculation, you need to subtract fractions that have the same denominators. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After that, you can open the brackets with the reduction of similar terms. We get that x 2 + 1 - 5 - x x + x - 2 = x 2 + 1 - 5 + x x + x - 2 = x 2 + x - 4 x + x - 2
Since the denominators are the same, it remains only to add the numerators, leaving the denominator: l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g 2 x + 4 + 4 x (l g x + 2)
The addition has been completed. It can be seen that the fraction can be reduced. Its numerator can be folded using the sum square formula, then we get (l g x + 2) 2 from the abbreviated multiplication formulas. Then we get that
l g 2 x + 4 + 2 l g x x (l g x + 2) = (l g x + 2) 2 x (l g x + 2) = l g x + 2 x
Given fractions of the form x - 1 x - 1 + x x + 1 with different denominators. After the transformation, you can proceed to addition.

Let's consider a two way solution.

The first method is that the denominator of the first fraction is subjected to factorization using squares, and with its subsequent reduction. We get a fraction of the form

x - 1 x - 1 = x - 1 (x - 1) x + 1 = 1 x + 1

So x - 1 x - 1 + x x + 1 = 1 x + 1 + x x + 1 = 1 + x x + 1 .

In this case, it is necessary to get rid of irrationality in the denominator.

1 + x x + 1 = 1 + x x - 1 x + 1 x - 1 = x - 1 + x x - x x - 1

The second way is to multiply the numerator and denominator of the second fraction by x - 1 . Thus, we get rid of irrationality and proceed to adding a fraction with the same denominator. Then

x - 1 x - 1 + x x + 1 = x - 1 x - 1 + x x - 1 x + 1 x - 1 = = x - 1 x - 1 + x x - x x - 1 = x - 1 + x x - x x - 1

Answer: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + x - 4 x + x - 2, 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g x + 2 x, 3) x - 1 x - 1 + x x + 1 = x - 1 + x x - x x - 1.

In the last example, we found that reduction to a common denominator is inevitable. To do this, you need to simplify the fractions. To add or subtract, you always need to look for a common denominator, which looks like the product of the denominators with the addition of additional factors to the numerators.

Example 7

Calculate the values of fractions: 1) x 3 + 1 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) (2 x - 4) - sin x x 5 ln (x + 1) (2 x - 4) , 3) 1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x

Decision

The denominator does not require any complicated calculations, so you need to choose their product of the form 3 x 7 + 2 2, then to the first fraction x 7 + 2 2 is chosen as an additional factor, and 3 to the second. When multiplying, we get a fraction of the form x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 3 x 7 + 2 2 + 3 1 3 x 7 + 2 2 = = x x 7 + 2 2 + 3 3 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2
It can be seen that the denominators are presented as a product, which means that additional transformations are unnecessary. The common denominator will be the product of the form x 5 · ln 2 x + 1 · 2 x - 4 . From here x 4 is an additional factor to the first fraction, and ln (x + 1) to the second. Then we subtract and get:
x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x + 1 x 4 x 5 ln 2 (x + 1 ) 2 x - 4 - sin x ln x + 1 x 5 ln 2 (x + 1) (2 x - 4) = = x + 1 x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) )
This example makes sense when working with denominators of fractions. It is necessary to apply the formulas of the difference of squares and the square of the sum, since they will make it possible to pass to an expression of the form 1 cos x - x · cos x + x + 1 (cos x + x) 2 . It can be seen that the fractions are reduced to a common denominator. We get that cos x - x cos x + x 2 .

Then we get that

1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = = 1 cos x - x cos x + x + 1 cos x + x 2 = = cos x + x cos x - x cos x + x 2 + cos x - x cos x - x cos x + x 2 = = cos x + x + cos x - x cos x - x cos x + x 2 = 2 cos x cos x - x cos x + x2

Answer:

1) x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) ( 2 x - 4) , 3) 1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = 2 cos x cos x - x cos x + x 2 .

Examples of multiplying fractions with variables

When multiplying fractions, the numerator is multiplied by the numerator and the denominator by the denominator. Then you can apply the reduction property.

Example 8

Multiply fractions x + 2 x x 2 ln x 2 ln x + 1 and 3 x 2 1 3 x + 1 - 2 sin 2 x - x.

Decision

You need to do the multiplication. We get that

x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = = x - 2 x 3 x 2 1 3 x + 1 - 2 x 2 ln x 2 ln x + 1 sin (2 x - x)

The number 3 is transferred to the first place for the convenience of calculations, and you can reduce the fraction by x 2, then we get an expression of the form

3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x)

Answer: x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = 3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x) .

Division

Division of fractions is similar to multiplication, since the first fraction is multiplied by the second reciprocal. If we take, for example, the fraction x + 2 x x 2 ln x 2 ln x + 1 and divide by 3 x 2 1 3 x + 1 - 2 sin 2 x - x, then this can be written as

x + 2 x x 2 ln x 2 ln x + 1: 3 x 2 1 3 x + 1 - 2 sin (2 x - x) , then replace with a product of the form x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x)

Exponentiation

Let's move on to consider the action with fractions of a general form with exponentiation. If there is a degree with a natural exponent, then the action is considered as a multiplication of identical fractions. But it is recommended to use a general approach based on the properties of powers. Any expressions A and C, where C is not identically equal to zero, and any real r on the ODZ for an expression of the form A C r, the equality A C r = A r C r is true. The result is a fraction raised to a power. For example, consider:

x 0 , 7 - π ln 3 x - 2 - 5 x + 1 2 , 5 = = x 0 , 7 - π ln 3 x - 2 - 5 2 , 5 x + 1 2 , 5

The order of operations with fractions

Actions on fractions are performed according to certain rules. In practice, we notice that an expression can contain several fractions or fractional expressions. Then it is necessary to perform all actions in a strict order: raise to a power, multiply, divide, then add and subtract. If there are brackets, the first action is performed in them.

Example 9

Calculate 1 - x cos x - 1 c o s x · 1 + 1 x .

Decision

Since we have the same denominator, then 1 - x cos x and 1 c o s x , but it is impossible to subtract according to the rule, first the actions in brackets are performed, then the multiplication, and then the addition. Then, when calculating, we get that

1 + 1 x = 1 1 + 1 x = x x + 1 x = x + 1 x

When substituting the expression into the original one, we get that 1 - x cos x - 1 cos x · x + 1 x. When multiplying fractions, we have: 1 cos x x + 1 x = x + 1 cos x x . Having made all the substitutions, we get 1 - x cos x - x + 1 cos x · x . Now you need to work with fractions that have different denominators. We get:

x 1 - x cos x x - x + 1 cos x x = x 1 - x - 1 + x cos x x = = x - x - x - 1 cos x x = - x + 1 cos x x

Answer: 1 - x cos x - 1 c o s x 1 + 1 x = - x + 1 cos x x .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Lesson content

Adding fractions with the same denominators

Adding fractions is of two types:

Adding fractions with the same denominators
Adding fractions with different denominators

Let's start with adding fractions with the same 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 . We add the numerators, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2 Add fractions and .

The answer is an improper fraction. If the end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the whole part is easily distinguished - two divided by two is equal to one:

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

Example 3. Add fractions and .

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

This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:

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 picture. If you add pizzas to pizza and add more pizzas, you get 1 whole and more pizzas.

As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:

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

Adding fractions with different denominators

Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those 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 at once, because 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 consider only one of them, since the rest of the methods may seem complicated for a beginner.

The essence of this method lies in the fact that the (LCM) denominators of both fractions are first searched. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and the second additional factor is obtained.

Then the numerators and denominators of the fractions are 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. Add 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 back to fractions and . First, we 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 factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor 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 factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:

Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:

Look closely 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 complete this example to the end:

Thus the example ends. To add it turns out.

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

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

The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).

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

But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “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:

Find the LCM of the denominators of fractions;
Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
Multiply the numerators and denominators of fractions by their additional factors;
Add fractions that have the same denominators;
If the answer turned out to be an improper fraction, then select its whole part;

Example 2 Find the value of an expression .

Let's use the instructions above.

Step 1. Find the LCM of the denominators of fractions

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

Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier 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 over 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 got the second additional factor 4. We write it over 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 got the third additional factor 3. We write it over the third fraction:

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

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions that have the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add 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 carried over 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 a 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 turned out to be an improper fraction, then select the whole part in it

Our answer is an improper fraction. We must single out the whole part of it. We highlight:

Got an answer

Subtraction of fractions with the same denominators

There are two types of fraction subtraction:

Subtraction of fractions with the same denominators
Subtraction of fractions with different denominators

First, let's learn how to subtract fractions with the same 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, and leave the denominator the same.

For example, let's find the value of the expression . To solve this example, it is necessary 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 think of a pizza that is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2 Find the value of the expression .

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

This example can be easily understood if we think of a pizza that 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 in subtracting fractions with the same denominators. It is enough to understand the following rules:

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;
If the answer turned out to be an improper fraction, then you need to select the whole part in it.

Subtraction of fractions with different denominators

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

The common denominator is found according to 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 over 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 over the second fraction.

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

Example 1 Find the value of an expression:

These fractions have different denominators, so you need to bring 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 back to fractions and

Let's find an additional factor for the first fraction. To do this, we 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. We write the four over the first fraction:

We do the same with the second fraction. 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 4. Divide 12 by 4, we get 3. Write a triple over the second fraction:

Now we are all set 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 complete this example to the end:

Got an answer

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

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

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

The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off 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 you first need to bring them to the same (common) denominator.

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, we 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 over 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 over 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 over 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 have 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 correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it easier. What can be done? You can reduce this fraction.

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

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

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

Got an answer

Multiplying a fraction by a number

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

Example 1. Multiply the fraction by the number 1.

Multiply the numerator of the fraction by the number 1

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

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

This entry can be understood as taking half of the unit. 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 is an improper fraction. Let's take a whole part of it:

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

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

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.

Example 1 Find the value of the expression .

Got an answer. It is desirable 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 get pizza. Remember what a pizza looks like divided into three parts:

One slice from this pizza and the two slices 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 is an improper fraction. Let's take a 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 correct fraction, but it will be good if it is reduced. 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 the numbers 105 and 450:

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

Representing an integer as a fraction

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

Reverse numbers

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

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

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

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

Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent 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 inverted:

What will be the 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 5 is multiplied by one, one is obtained.

The reciprocal can also be found for any other integer.

You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.

Division of a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How many pizzas will each get?

It can be seen that after splitting half of the pizza, two equal slices were obtained, each of which makes up a pizza. So everyone gets a pizza.

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

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

Portal for the student. Self-training

Fraction calculator

Basic property of a fraction

Bringing fractions to a common denominator

Actions with fractions

Rules for performing operations with numerical fractions of a general form

Rationale for the rules

Examples

Performing an action with fractions containing variables

Examples of addition and subtraction of fractions with variables

Examples of multiplying fractions with variables

Division

Exponentiation

The order of operations with fractions

Adding fractions with the same denominators

Adding fractions with different denominators

Subtraction of fractions with the same denominators

Subtraction of fractions with different denominators

Multiplying a fraction by a number

Multiplication of fractions

Representing an integer as a fraction

Reverse numbers

Division of a fraction by a number

RELATED ARTICLES