Addition and subtraction of fractions with a common denominator. Online calculator. Evaluating an expression with numeric fractions

The rules for adding fractions with different denominators are very simple.

Consider the rules for adding fractions with different denominators in steps:

1. Find the LCM (least common multiple) of the denominators. The resulting NOC will be common denominator fractions;

2. Bring fractions to a common denominator;

3. Add the fractions reduced to a common denominator.

Using a simple example, we will learn how to apply the rules for adding fractions with different denominators.

Example

An example of adding fractions with different denominators.

Add fractions with different denominators:

We will decide in steps.

1. Find the LCM (least common multiple) of the denominators.

The number 12 is divisible by 6.

From this we conclude that 12 is the least common multiple of 6 and 12.

Answer: the number of numbers 6 and 12 is 12:

LCM (6, 12) = 12

The resulting LCM will be the common denominator of the two fractions 1/6 and 5/12.

2. Bring the fractions to a common denominator.

In our example, only the first fraction needs to be reduced to a common denominator of 12, because the second fraction has a denominator already equal to 12.

Divide the common denominator 12 by the denominator of the first fraction:

2 has an additional multiplier.

Multiply the numerator and denominator of the first fraction (1/6) by an additional factor of 2.

Mixed fractions as well as simple fractions can be subtracted. To subtract mixed numbers of fractions, you need to know several subtraction rules. Let's explore these rules with examples.

Subtraction of mixed fractions with the same denominator.

Consider an example with the condition that the reduced integer and fractional parts are greater than the subtracted whole and fractional parts, respectively. Under these conditions, the deduction takes place separately. Subtract the whole part from the whole part, and the fractional part from the fractional part.

Let's consider an example:

Subtract the mixed fractions \ (5 \ frac (3) (7) \) and \ (1 \ frac (1) (7) \).

\ (5 \ frac (3) (7) -1 \ frac (1) (7) = (5-1) + (\ frac (3) (7) - \ frac (1) (7)) = 4 \ frac (2) (7) \)

The correctness of the subtraction is checked by addition. Let's check the subtraction:

\ (4 \ frac (2) (7) +1 \ frac (1) (7) = (4 + 1) + (\ frac (2) (7) + \ frac (1) (7)) = 5 \ frac (3) (7) \)

Consider an example with the condition when the fractional part of the reduced is less, respectively, the fractional part of the subtracted. In this case, we borrow one from the whole in the decreasing one.

Let's consider an example:

Perform mixed fraction subtraction \ (6 \ frac (1) (4) \) and \ (3 \ frac (3) (4) \).

The reduced \ (6 \ frac (1) (4) \) has a fractional part less than the fractional part of the subtracted \ (3 \ frac (3) (4) \). That is, \ (\ frac (1) (4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)

\ (\ begin (align) & 6 \ frac (1) (4) -3 \ frac (3) (4) = (6 + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ color (red) (1) + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ color (red) (\ frac (4) (4)) + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ frac (5) (4)) - 3 \ frac (3) (4) = \\\\ & = 5 \ frac (5) (4) -3 \ frac (3) (4) = 2 \ frac (2) (4) = 2 \ frac (1) (4) \\\\ \ end (align) \)

Next example:

\ (7 \ frac (8) (19) -3 = 4 \ frac (8) (19) \)

Subtracting a mixed fraction from an integer.

Example: \ (3-1 \ frac (2) (5) \)

Reduced 3 does not have a fractional part, so we cannot immediately subtract it. Let's borrow one from the integer part of 3, and then perform the subtraction. We will write the unit as \ (3 = 2 + 1 = 2 + \ frac (5) (5) = 2 \ frac (5) (5) \)

\ (3-1 \ frac (2) (5) = (2 + \ color (red) (1)) - 1 \ frac (2) (5) = (2 + \ color (red) (\ frac (5 ) (5))) - 1 \ frac (2) (5) = 2 \ frac (5) (5) -1 \ frac (2) (5) = 1 \ frac (3) (5) \)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if the fractional parts of the reduced and subtracted with different denominators. You need to bring to a common denominator, and then perform the subtraction.

Subtract two mixed fractions with different denominators \ (2 \ frac (2) (3) \) and \ (1 \ frac (1) (4) \).

The common denominator is 12.

\ (2 \ frac (2) (3) -1 \ frac (1) (4) = 2 \ frac (2 \ times \ color (red) (4)) (3 \ times \ color (red) (4) ) -1 \ frac (1 \ times \ color (red) (3)) (4 \ times \ color (red) (3)) = 2 \ frac (8) (12) -1 \ frac (3) (12 ) = 1 \ frac (5) (12) \)

Questions on the topic:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and, by the type of expression, apply the solution algorithm. Subtract the whole from the whole part, subtract the fractional part from the fractional part.

How to subtract a fraction from an integer? How to subtract a fraction from an integer?
Answer: you need to take a unit from an integer and write this unit as a fraction

\ (4 = 3 + 1 = 3 + \ frac (7) (7) = 3 \ frac (7) (7) \),

and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:

\ (4-2 \ frac (3) (7) = (3 + \ color (red) (1)) - 2 \ frac (3) (7) = (3 + \ color (red) (\ frac (7 ) (7))) - 2 \ frac (3) (7) = 3 \ frac (7) (7) -2 \ frac (3) (7) = 1 \ frac (4) (7) \)

Example # 1:
Subtract the correct fraction from one: a) \ (1- \ frac (8) (33) \) b) \ (1- \ frac (6) (7) \)

Solution:
a) We represent the unit as a fraction with the denominator 33. We get \ (1 = \ frac (33) (33) \)

\ (1- \ frac (8) (33) = \ frac (33) (33) - \ frac (8) (33) = \ frac (25) (33) \)

b) We represent the unit as a fraction with the denominator 7. We get \ (1 = \ frac (7) (7) \)

\ (1- \ frac (6) (7) = \ frac (7) (7) - \ frac (6) (7) = \ frac (7-6) (7) = \ frac (1) (7) \)

Example # 2:
Subtract a mixed fraction from an integer: a) \ (21-10 \ frac (4) (5) \) b) \ (2-1 \ frac (1) (3) \)

Solution:
a) We borrow 21 units from an integer and write it like this \ (21 = 20 + 1 = 20 + \ frac (5) (5) = 20 \ frac (5) (5) \)

\ (21-10 \ frac (4) (5) = (20 + 1) -10 \ frac (4) (5) = (20 + \ frac (5) (5)) - 10 \ frac (4) ( 5) = 20 \ frac (5) (5) -10 \ frac (4) (5) = 10 \ frac (1) (5) \\\\\)

b) Let's borrow a unit from the integer 2 and write it like this \ (2 = 1 + 1 = 1 + \ frac (3) (3) = 1 \ frac (3) (3) \)

\ (2-1 \ frac (1) (3) = (1 + 1) -1 \ frac (1) (3) = (1 + \ frac (3) (3)) - 1 \ frac (1) ( 3) = 1 \ frac (3) (3) -1 \ frac (1) (3) = \ frac (2) (3) \\\\\)

Example # 3:
Subtract an integer from a mixed fraction: a) \ (15 \ frac (6) (17) -4 \) b) \ (23 \ frac (1) (2) -12 \)

a) \ (15 \ frac (6) (17) -4 = 11 \ frac (6) (17) \)

b) \ (23 \ frac (1) (2) -12 = 11 \ frac (1) (2) \)

Example No. 4:
Subtract the correct fraction from the mixed fraction: a) \ (1 \ frac (4) (5) - \ frac (4) (5) \)

\ (1 \ frac (4) (5) - \ frac (4) (5) = 1 \\\\\)

Example # 5:
Calculate \ (5 \ frac (5) (16) -3 \ frac (3) (8) \)

\ (\ begin (align) & 5 \ frac (5) (16) -3 \ frac (3) (8) = 5 \ frac (5) (16) -3 \ frac (3 \ times \ color (red) ( 2)) (8 \ times \ color (red) (2)) = 5 \ frac (5) (16) -3 \ frac (6) (16) = (5 + \ frac (5) (16)) - 3 \ frac (6) (16) = (4 + \ color (red) (1) + \ frac (5) (16)) - 3 \ frac (6) (16) = \\\\ & = (4 + \ color (red) (\ frac (16) (16)) + \ frac (5) (16)) - 3 \ frac (6) (16) = (4 + \ color (red) (\ frac (21 ) (16))) - 3 \ frac (3) (8) = 4 \ frac (21) (16) -3 \ frac (6) (16) = 1 \ frac (15) (16) \\\\ \ end (align) \)

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, since $ \ frac (12) (6) $ as y is also equal to 2. You can multiply the 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 of $ \ frac (60) (30) $), or by 3 (the result of $ \ frac (40) (20) $), or by 4 (the result of $ \ 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. Indeed, if you divide the cake into 3 pieces and take one of them, or divide it into 6 pieces and take 2 pieces, you will get the same amount of cake in both cases. Therefore, the numbers $ \ frac (1) (3) $ and $ \ frac (2) (6) $ are identical. Let's 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 $ \ frac (126) (189) $ by 63 and get $ \ frac (2) (3) $ which is much easier to calculate. 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 met fraction with denominator 1... Such fractions play an important role in calculations. It should be remembered that any number can be divided by 1 without changing its value. That is, $ \ frac (273) (1) $ is 273; $ \ frac (509993) (1) $ equals 509993 and so on. Therefore, we can not divide the numbers by, since each integer can be represented as a fraction with denominator 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 is the use of representing an integer as a fraction with one under the line, because it is more convenient to work with an integer. But the fact is that representing an integer as a fraction allows us to more efficiently produce various actions, when we deal simultaneously with wholes and fractional numbers... For example, to learn add fractions with different denominators... Suppose we want to add $ \ frac (1) (3) $ and $ \ frac (1) (5) $.

We know that you can add only those fractions whose denominators are equal. This means that we need to learn how to bring fractions to such a form when their denominators are equal. In this case, it is again useful to us that you can multiply the numerator and denominator of a fraction by the same number without changing its value.

First, multiply the numerator and denominator of $ \ 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 the addition of numbers containing both integer and fractional parts.

We need to add $ 3 + \ frac (1) (3) +1 \ frac (1) (4) $. First, we translate all the terms into fractions and get: $ \ frac31 + \ frac (1) (3) + \ frac (5) (4) $. Now we need to bring all 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 equals $ \ frac (55) (12) $. If you want to get rid of wrong fraction, it can be turned into a number consisting of integer and fractional parts: $ \ frac (55) (12) = \ frac (48) (12) + \ frac (7) (12) $ or $ 4 \ frac (7) ( 12) $.

All the rules to allow fraction operations that we have just studied are also true 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 and dividing a positive number by a negative result in negative numbers, in both cases we get the answer in the form of a negative number. That is

$ (- 1): 3 = \ frac (1) (3) $ or $ 1: (-3) = \ frac (1) (- 3) $. The minus sign with this writing refers to the entire fraction as a whole, 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 we get positive number, then $ \ frac (-1) (- 3) $ can be written as $ + \ frac (1) (3) $.

Addition and subtraction negative fractions carried out in the same way as addition and subtraction of positive fractions. For example, what is $ 1- 1 \ frac13 $? We represent both numbers as fractions and get $ \ frac (1) (1) - \ frac (4) (3) $. Reduce 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) $.

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

How to subtract fractions with the same denominators

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

• In order to subtract the second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the reduced fraction. We write this number in 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.

We subtract the numerator of the subtracted fraction “3” from the numerator of the reduced fraction “7”, 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 some more similar examples.

Consider a more complex example, where fractions with the same 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 reduced fraction "29" 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 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 with the same denominator

The addition and subtraction of ordinary fractions is carried out according to the same principle.

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

Let's see how it looks like in 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 in the numerator of the sum, and the denominator is the same as in the fractions - "4".

Fractions with different denominators and their subtraction

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

To subtract fractions with different denominators, you need to bring them to the same lowest denominator.



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

Fraction property

In order to bring several fractions to the same denominator, you need to use the main property of the 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 such denominators 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 the same action with the number "4", we get 8/12. With one equality, it can be written like this:

2/3 = 4/6 = 6/9 = 8/12…

How to convert multiple fractions to the same denominator

Let's consider how to bring several fractions to the same denominator. For example, take the fractions shown in the picture below. First, you need to determine what number can become the denominator for all of them. To make it easier, we factor the available denominators.

The denominator of 1/2 and 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 you need to determine what factors will be the smallest for all these four fractions. Since the first fraction in the denominator contains the number "2", which means that it must be present in all denominators, there are two triples in the 7/9 fraction, which means that both of them must also be present in the denominator. Considering the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.



Consider the first fraction - 1/2. Its denominator contains "2", but there is not a single digit "3", but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of the fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3) / (2 x 3 x 3) = 9/18.

Similarly, we perform actions with the remaining fractions.

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

Together, it looks like this:



How to subtract and add fractions with different denominators

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

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

Find a multiple of 18 and 15:

• 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 5 x 3 x 3 x 2 = 90.

After the denominator is found, it is necessary to calculate the factor that will be different for each fraction, that is, the number by which not only the denominator, but also the numerator will need to be multiplied. To do this, the number that we found (the common multiple) is divided 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 factor for 3/15.
• 90 divided by 18. The resulting number "5" will be a multiplier for 4/18.

The next step in our solution is to bring each fraction to the denominator "90".

We have already discussed 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 the fractions are with small numbers, then the common denominator can be determined, as in the example shown in the picture below.



Similarly, it is produced and having different denominators.

Subtraction and having whole parts

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

• All fractions that have an integer part should be converted to incorrect ones. Speaking in simple words, remove the whole part. To do this, multiply the number of the integer part by the denominator of the fraction, add the resulting product to the numerator. The number that will be obtained after these actions is the numerator of the improper fraction. The denominator remains unchanged.
• If fractions have different denominators, you should bring them to the same.
• Add or subtract with the same denominators.
• If you get an incorrect fraction, select the whole part.



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



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

Subtracting fractions from an integer

Another of the varieties of actions with fractions is the case when the fraction must be subtracted from At first glance similar example seems difficult to solve. However, everything is pretty simple here. To solve it, it is necessary to convert an integer to a fraction, and with the same denominator, which is in the fraction to be subtracted. Next, we make a subtraction, similar to subtraction with the same denominators. For 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) given in this article is the basis for solving more complex examples which are covered in later classes. The knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the actions with fractions discussed above.

Find the numerator and denominator. A fraction includes two numbers: the number above the line is called the numerator, and the number below the line is called the denominator. The denominator denotes total amount parts into which a whole is broken, and the numerator is the number of such parts under consideration.

• For example, in the fraction ½, the numerator is 1 and the denominator is 2.

Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, some whole is divided into the same number of parts. It is very easy to add fractions with a common denominator, since the denominator of the total fraction will be the same as for the added fractions. For example:

• Fractions 3/5 and 2/5 have a common denominator of 5.
• Fractions 3/8, 5/8, 17/8 have a common denominator of 8.