How to add improper fractions with different denominators. How to learn to subtract fractions with different denominators

Note! 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 obtain 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 reverse 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 a unit 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 the same 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:

One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics and even biology, is mathematics. Studying this science allows you to develop some mental qualities and improve your ability to concentrate. One of the topics that deserve special attention in the Mathematics course is adding and subtracting fractions. Many students find it difficult to study. Perhaps our article will help you better understand this topic.

How to subtract fractions whose denominators are the same

Fractions are the same numbers with which you can perform various operations. Their difference from whole numbers lies in the presence of a denominator. That is why, when performing operations with fractions, you need to study some of their features and rules. The simplest case is the subtraction of ordinary fractions whose denominators are represented as the same number. Performing this action will not be difficult if you know a simple rule:

In order to subtract a second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the fraction being reduced. We write this number into the numerator of the difference, and leave the denominator the same: k/m - b/m = (k-b)/m.

Examples of subtracting fractions whose denominators are the same

7/19 - 3/19 = (7 - 3)/19 = 4/19.

From the numerator of the fraction “7” we subtract the numerator of the fraction “3” to be subtracted, we get “4”. We write this number in the numerator of the answer, and in the denominator we put the same number that was in the denominators of the first and second fractions - “19”.

The picture below shows several more similar examples.

Let's consider a more complex example where fractions with like denominators are subtracted:

29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7)/47 = 9/47.

From the numerator of the fraction “29” being reduced by subtracting in turn the numerators of all subsequent fractions - “3”, “8”, “2”, “7”. As a result, we get the result “9”, which we write down in the numerator of the answer, and in the denominator we write down the number that is in the denominators of all these fractions - “47”.

Adding fractions that have the same denominator

Adding and subtracting ordinary fractions follows the same principle.

In order to add fractions whose denominators are the same, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator will remain the same: k/m + b/m = (k + b)/m.

Let's see what this looks like using an example:

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

To the numerator of the first term of the fraction - “1” - add the numerator of the second term of the fraction - “2”. The result - “3” - is written into the numerator of the sum, and the denominator is left the same as that present in the fractions - “4”.

Fractions with different denominators and their subtraction

We have already considered the operation with fractions that have the same denominator. As you can see, knowing simple rules, solving such examples is quite easy. But what if you need to perform an operation with fractions that have different denominators? Many secondary school students are confused by such examples. But even here, if you know the principle of the solution, the examples will no longer be difficult for you. There is also a rule here, without which solving such fractions is simply impossible.

To subtract fractions with different denominators, they must be reduced to the same smallest denominator.

We will talk in more detail about how to do this.

Property of a fraction

In order to bring several fractions to the same denominator, you need to use the main property of a fraction in the solution: after dividing or multiplying the numerator and denominator by the same number, you get a fraction equal to the given one.

So, for example, the fraction 2/3 can have denominators such as “6”, “9”, “12”, etc., that is, it can have the form of any number that is a multiple of “3”. After we multiply the numerator and denominator by “2”, we get the fraction 4/6. After we multiply the numerator and denominator of the original fraction by “3”, we get 6/9, and if we perform a similar operation with the number “4”, we get 8/12. One equality can be written as follows:

2/3 = 4/6 = 6/9 = 8/12…

How to convert multiple fractions to the same denominator

Let's look at how to reduce multiple fractions to the same denominator. For example, let's take the fractions shown in the picture below. First you need to determine which number can become the denominator for all of them. To make things easier, let's factorize the existing denominators.

The denominator of the fraction 1/2 and the fraction 2/3 cannot be factorized. The denominator 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now we need to determine which factors will be the smallest for all these four fractions. Since the first fraction has the number “2” in the denominator, it means that it must be present in all denominators; in the fraction 7/9 there are two triplets, which means that both of them must also be present in the denominator. Taking into account the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.

Let's consider the first fraction - 1/2. There is a “2” in its denominator, but there is not a single “3” digit, but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of a fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.

We perform the same operations with the remaining fractions.

2/3 - one three and one two are missing in the denominator:
2/3 = (2 x 3 x 2)/(3 x 3 x 2) = 12/18.
7/9 or 7/(3 x 3) - the denominator is missing a two:
7/9 = (7 x 2)/(9 x 2) = 14/18.
5/6 or 5/(2 x 3) - the denominator is missing a three:
5/6 = (5 x 3)/(6 x 3) = 15/18.

All together it looks like this:

How to subtract and add fractions that have different denominators

As mentioned above, in order to add or subtract fractions that have different denominators, they must be reduced to the same denominator, and then use the rules for subtracting fractions that have the same denominator, which have already been discussed.

Let's look at this as an example: 4/18 - 3/15.

Finding the multiple of numbers 18 and 15:

The number 18 is made up of 3 x 2 x 3.
The number 15 is made up of 5 x 3.
The common multiple will be the following factors: 5 x 3 x 3 x 2 = 90.

After the denominator has been found, it is necessary to calculate the factor that will be different for each fraction, that is, the number by which it will be necessary to multiply not only the denominator, but also the numerator. To do this, divide the number that we found (the common multiple) by the denominator of the fraction for which additional factors need to be determined.

90 divided by 15. The resulting number “6” will be a multiplier for 3/15.
90 divided by 18. The resulting number “5” will be a multiplier for 4/18.

The next stage of our solution is to reduce each fraction to the denominator “90”.

We have already talked about how this is done. Let's see how this is written in an example:

(4 x 5)/(18 x 5) - (3 x 6)/(15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.

If fractions have small numbers, then you can determine the common denominator, as in the example shown in the picture below.

The same is true for those with different denominators.

Subtraction and having integer parts

We have already discussed in detail the subtraction of fractions and their addition. But how to subtract if a fraction has an integer part? Again, let's use a few rules:

Convert all fractions that have an integer part to improper ones. In simple words, remove an entire part. To do this, multiply the number of the integer part by the denominator of the fraction, and add the resulting product to the numerator. The number that comes out after these actions is the numerator of the improper fraction. The denominator remains unchanged.
If fractions have different denominators, they should be reduced to the same denominator.
Perform addition or subtraction with the same denominators.
When receiving an improper fraction, select the whole part.

There is another way in which you can add and subtract fractions with whole parts. To do this, actions are performed separately with whole parts, and actions with fractions separately, and the results are recorded together.

The example given consists of fractions that have the same denominator. In the case when the denominators are different, they must be brought to the same value, and then perform the actions as shown in the example.

Subtracting fractions from whole numbers

Another type of operation with fractions is the case when a fraction must be subtracted from. At first glance, such an example seems difficult to solve. However, everything is quite simple here. To solve it, you need to convert the integer into a fraction, and with the same denominator that is in the subtracted fraction. Next, we perform a subtraction similar to subtraction with identical denominators. In an example it looks like this:

7 - 4/9 = (7 x 9)/9 - 4/9 = 53/9 - 4/9 = 49/9.

The subtraction of fractions (grade 6) presented in this article is the basis for solving more complex examples that are covered in subsequent grades. Knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the operations with fractions discussed above.

You can perform various operations with fractions, for example, adding fractions. Addition of fractions can be divided into several types. Each type of addition of fractions has its own rules and algorithm of actions. Let's look at each type of addition in detail.

Adding fractions with like denominators.

Let's look at an example of how to add fractions with a common denominator.

The tourists went on a hike from point A to point E. On the first day they walked from point A to B or $\frac(1)(5)$ of the entire path. On the second day they walked from point B to D or $\frac(2)(5)$ the whole way. How far did they travel from the beginning of the journey to point D?

To find the distance from point A to point D, you need to add the fractions $\frac(1)(5) + \frac(2)(5)$.

Adding fractions with like denominators means that you need to add the numerators of these fractions, but the denominator will remain the same.

$\frac(1)(5) + \frac(2)(5) = \frac(1 + 2)(5) = \frac(3)(5)$

In literal form, the sum of fractions with the same denominators will look like this:

$\bf \frac(a)(c) + \frac(b)(c) = \frac(a + b)(c)$

Answer: the tourists walked $\frac(3)(5)$ the entire way.

Adding fractions with different denominators.

Let's look at an example:

You need to add two fractions $\frac(3)(4)$ and $\frac(2)(7)$.

To add fractions with different denominators, you must first find, and then use the rule for adding fractions with like denominators.

For denominators 4 and 7, the common denominator will be the number 28. The first fraction $\frac(3)(4)$ must be multiplied by 7. The second fraction $\frac(2)(7)$ must be multiplied by 4.

$\frac(3)(4) + \frac(2)(7) = \frac(3 \times \color(red) (7) + 2 \times \color(red) (4))(4 \ times \color(red) (7)) = \frac(21 + 8)(28) = \frac(29)(28) = 1\frac(1)(28)$

In literal form we get the following formula:

$\bf \frac(a)(b) + \frac(c)(d) = \frac(a \times d + c \times b)(b \times d)$

Adding mixed numbers or mixed fractions.

Addition occurs according to the law of addition.

For mixed fractions, we add the whole parts with the whole parts and the fractional parts with the fractions.

If the fractional parts of mixed numbers have the same denominators, then we add the numerators, but the denominator remains the same.

Let's add the mixed numbers $3\frac(6)(11)$ and $1\frac(3)(11)$.

$3\frac(6)(11) + 1\frac(3)(11) = (\color(red) (3) + \color(blue) (\frac(6)(11))) + ( \color(red) (1) + \color(blue) (\frac(3)(11))) = (\color(red) (3) + \color(red) (1)) + (\color( blue) (\frac(6)(11)) + \color(blue) (\frac(3)(11))) = \color(red)(4) + (\color(blue) (\frac(6 + 3)(11))) = \color(red)(4) + \color(blue) (\frac(9)(11)) = \color(red)(4) \color(blue) (\frac (9)(11))$

If the fractional parts of mixed numbers have different denominators, then we find the common denominator.

Let's perform the addition of mixed numbers $7\frac(1)(8)$ and $2\frac(1)(6)$.

The denominator is different, so we need to find the common denominator, it is equal to 24. Multiply the first fraction $7\frac(1)(8)$ by an additional factor of 3, and the second fraction $2\frac(1)(6)$ by 4.

$7\frac(1)(8) + 2\frac(1)(6) = 7\frac(1 \times \color(red) (3))(8 \times \color(red) (3) ) = 2\frac(1\times \color(red) (4))(6\times \color(red) (4)) =7\frac(3)(24) + 2\frac(4)(24 ) = 9\frac(7)(24)$

Related questions:
How to add fractions?
Answer: first you need to decide what type of expression it is: fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we proceed to the solution algorithm.

How to solve fractions with different denominators?
Answer: you need to find the common denominator, and then follow the rule of adding fractions with the same denominators.

How to solve mixed fractions?
Answer: we add integer parts with integers and fractional parts with fractions.

Example #1:
Can the sum of two result in a proper fraction? Improper fraction? Give examples.

$\frac(2)(7) + \frac(3)(7) = \frac(2 + 3)(7) = \frac(5)(7)$

The fraction $\frac(5)(7)$ is a proper fraction, it is the result of the sum of two proper fractions $\frac(2)(7)$ and $\frac(3)(7)$.

$\frac(2)(5) + \frac(8)(9) = \frac(2 \times 9 + 8 \times 5)(5 \times 9) =\frac(18 + 40)(45) = \frac(58)(45)$

The fraction $\frac(58)(45)$ is an improper fraction, it is the result of the sum of the proper fractions $\frac(2)(5)$ and $\frac(8)(9)$.

Answer: The answer to both questions is yes.

Example #2:
Add the fractions: a) $\frac(3)(11) + \frac(5)(11)$ b) $\frac(1)(3) + \frac(2)(9)$.

a) $\frac(3)(11) + \frac(5)(11) = \frac(3 + 5)(11) = \frac(8)(11)$

b) $\frac(1)(3) + \frac(2)(9) = \frac(1 \times \color(red) (3))(3 \times \color(red) (3)) + \frac(2)(9) = \frac(3)(9) + \frac(2)(9) = \frac(5)(9)$

Example #3:
Write the mixed fraction as the sum of a natural number and a proper fraction: a) $1\frac(9)(47)$ b) $5\frac(1)(3)$

a) $1\frac(9)(47) = 1 + \frac(9)(47)$

b) $5\frac(1)(3) = 5 + \frac(1)(3)$

Example #4:
Calculate the sum: a) $8\frac(5)(7) + 2\frac(1)(7)$ b) $2\frac(9)(13) + \frac(2)(13) $ c) $7\frac(2)(5) + 3\frac(4)(15)$

a) $8\frac(5)(7) + 2\frac(1)(7) = (8 + 2) + (\frac(5)(7) + \frac(1)(7)) = 10 + \frac(6)(7) = 10\frac(6)(7)$

b) $2\frac(9)(13) + \frac(2)(13) = 2 + (\frac(9)(13) + \frac(2)(13)) = 2\frac(11 )(13) $

c) $7\frac(2)(5) + 3\frac(4)(15) = 7\frac(2\times 3)(5\times 3) + 3\frac(4)(15) = 7\frac(6)(15) + 3\frac(4)(15) = (7 + 3)+(\frac(6)(15) + \frac(4)(15)) = 10 + \frac (10)(15) = 10\frac(10)(15) = 10\frac(2)(3)$

Task #1:
At lunch we ate $\frac(8)(11)$ from the cake, and in the evening at dinner we ate $\frac(3)(11)$. Do you think the cake was completely eaten or not?

Solution:
The denominator of the fraction is 11, it indicates how many parts the cake was divided into. At lunch we ate 8 pieces of cake out of 11. At dinner we ate 3 pieces of cake out of 11. Let’s add 8 + 3 = 11, we ate pieces of cake out of 11, that is, the whole cake.

$\frac(8)(11) + \frac(3)(11) = \frac(11)(11) = 1$

Answer: the whole cake was eaten.

Consider the fraction $\frac63$. Its value is 2, since $\frac63 =6:3 = 2$. What happens if the numerator and denominator are multiplied by 2? $\frac63 \times 2=\frac(12)(6)$. Obviously, the value of the fraction has not changed, so $\frac(12)(6)$ as y is also equal to 2. You can multiply numerator and denominator by 3 and get $\frac(18)(9)$, or by 27 and get $\frac(162)(81)$, or by 101 and get $\frac(606)(303)$. In each of these cases, the value of the fraction that we get by dividing the numerator by the denominator is 2. This means that it has not changed.

The same pattern is observed in the case of other fractions. If the numerator and denominator of the fraction $\frac(120)(60)$ (equal to 2) are divided by 2 (the result is $\frac(60)(30)$), or by 3 (the result is $\frac(40)(20) $), or by 4 (result $\frac(30)(15)$) and so on, then in each case the value of the fraction remains unchanged and equal to 2.

This rule also applies to fractions that are not equal whole number.

If the numerator and denominator of the fraction $\frac(1)(3)$ are multiplied by 2, we get $\frac(2)(6)$, that is, the value of the fraction has not changed. And in fact, if you divide the pie into 3 parts and take one of them, or divide it into 6 parts and take 2 parts, you will get the same amount of pie in both cases. Therefore, the numbers $\frac(1)(3)$ and $\frac(2)(6)$ are identical. Let us formulate a general rule.

The numerator and denominator of any fraction can be multiplied or divided by the same number without changing the value of the fraction.

This rule turns out to be very useful. For example, it allows in some cases, but not always, to avoid operations with large numbers.

For example, we can divide the numerator and denominator of the fraction $\frac(126)(189)$ by 63 and get the fraction $\frac(2)(3)$, which is much easier to calculate with. One more example. We can divide the numerator and denominator of the fraction $\frac(155)(31)$ by 31 and get the fraction $\frac(5)(1)$ or 5, since 5:1=5.

In this example, we first encountered a fraction whose denominator is 1. Such fractions play an important role in calculations. It should be remembered that any number can be divided by 1 and its value will not change. That is, $\frac(273)(1)$ is equal to 273; $\frac(509993)(1)$ equals 509993 and so on. Therefore, we don't have to divide numbers by , since every integer can be represented as a fraction with a denominator of 1.

With such fractions, the denominator of which is 1, you can perform the same arithmetic operations as with all other fractions: $\frac(15)(1)+\frac(15)(1)=\frac(30)(1) $, $\frac(4)(1) \times \frac(3)(1)=\frac(12)(1)$.

You may ask what good is it if we represent an integer as a fraction with a unit under the line, since it is more convenient to work with an integer. But the point is that representing an integer as a fraction gives us the opportunity to perform various operations more efficiently when we are dealing with both integers and fractions at the same time. For example, to learn add fractions with different denominators. Suppose we need to add $\frac(1)(3)$ and $\frac(1)(5)$.

We know that we can only add fractions whose denominators are equal. This means that we need to learn how to reduce fractions to a form where their denominators are equal. In this case, we will again need the fact that we can multiply the numerator and denominator of a fraction by the same number without changing its value.

First, multiply the numerator and denominator of the fraction $\frac(1)(3)$ by 5. We get $\frac(5)(15)$, the value of the fraction has not changed. Then we multiply the numerator and denominator of the fraction $\frac(1)(5)$ by 3. We get $\frac(3)(15)$, again the value of the fraction has not changed. Therefore, $\frac(1)(3)+\frac(1)(5)=\frac(5)(15)+\frac(3)(15)=\frac(8)(15)$.

Now let's try to apply this system to adding numbers containing both integer and fractional parts.

We need to add $3 + \frac(1)(3)+1\frac(1)(4)$. First, let's convert all the terms into fractions and get: $\frac31 + \frac(1)(3)+\frac(5)(4)$. Now we need to bring all the fractions to a common denominator, for this we multiply the numerator and denominator of the first fraction by 12, the second by 4, and the third by 3. As a result, we get $\frac(36)(12) + \frac(4 )(12)+\frac(15)(12)$, which is equal to $\frac(55)(12)$. If you want to get rid of improper fraction, it can be turned into a number consisting of an integer and a fraction: $\frac(55)(12) = \frac(48)(12)+\frac(7)(12)$ or $4\frac(7)( 12)$.

All the rules that allow operations with fractions, which we just studied, are also valid in the case of negative numbers. So, -1: 3 can be written as $\frac(-1)(3)$, and 1: (-3) as $\frac(1)(-3)$.

Since both dividing a negative number by a positive number and dividing a positive number by a negative result in negative numbers, in both cases the answer will be a negative number. That is

$(-1) : 3 = \frac(1)(3)$ or $1: (-3) = \frac(1)(-3)$. The minus sign when written this way refers to the entire fraction, and not separately to the numerator or denominator.

On the other hand, (-1) : (-3) can be written as $\frac(-1)(-3)$, and since dividing a negative number by a negative number gives a positive number, then $\frac(-1 )(-3)$ can be written as $+\frac(1)(3)$.

Addition and subtraction of negative fractions is carried out according to the same scheme as addition and subtraction of positive fractions. For example, what is $1- 1\frac13$? Let's represent both numbers as fractions and get $\frac(1)(1)-\frac(4)(3)$. Let's bring the fractions to a common denominator and get $\frac(1 \times 3)(1 \times 3)-\frac(4)(3)$, that is, $\frac(3)(3)-\frac(4) (3)$, or $-\frac(1)(3)$.

Adding and subtracting fractions with like denominators
Adding and subtracting fractions with different denominators
Concept of NOC
Reducing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with like denominators

To add fractions with the same denominators, you need to add their numerators, but leave the denominator the same, for example:

To subtract fractions with the same denominators, 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:

To add mixed fractions, you need to separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,

If, when adding fractional parts, you get an improper fraction, select the whole part from it and add it to the whole part, for example:

2 Adding and subtracting fractions with different denominators

In order to add or subtract fractions with different denominators, you must first reduce them to the same denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the LCM (least common multiple). For the numerator of each fraction, additional factors are found by dividing the LCM by the denominator of this fraction. We will look at an example later, after we understand what an NOC is.

3 Least common multiple (LCM)

The least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both numbers without leaving a remainder. Sometimes the LCM can be found orally, but more often, especially when working with large numbers, you have to find the LCM in writing, using the following algorithm:

In order to find the LCM of several numbers, you need:

Factor these numbers into prime factors
Take the largest expansion and write these numbers as a product
Select in other decompositions the numbers that do not appear in the largest decomposition (or occur fewer times in it), and add them to the product.
Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of the numbers 28 and 21:

4Reducing fractions to the same denominator

Let's return to adding fractions with different denominators.

When we reduce fractions to the same denominator, equal to the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:

Thus, to reduce fractions to the same exponent, you must first find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then put additional factors to the numerators of the fractions. You can find them by dividing the common denominator (CLD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.

5How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number before the fraction, which will result in a mixed fraction, for example.