The numerator, and that by which it is divided is the denominator.
To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.
To calculate the product of fractions, first multiply the numerator of one fractions to another numerator. Write the result to the numerator of the new fractions. Then multiply the denominators as well. Specify the final value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).
To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the steps, first “flip” the divisor, if it’s more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 × 5 = 5; 3 × 1 = 3).
Sources:
- Basic tasks for fractions
Fractional numbers allow you to express in different form exact value quantities. With fractions, you can perform the same mathematical operations as with integers: subtraction, addition, multiplication, and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution, they require reduction of the fractional part of the result.
You will need
- - calculator
Instruction
Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. This value will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.
Rewrite them through the separator ":" and continue the usual division.
To get the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.
note
Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.
When writing fractional numbers, the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by a single whole number. In this example, dividing by 2 is possible. The result is 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.
Note! Before writing a final answer, see if you can reduce the fraction you received.
Subtraction of fractions with the same denominators examples:
,
,
Subtracting a proper fraction from one.
If it is necessary to subtract from the unit a fraction that is correct, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.
Subtraction Example proper fraction from unit:
The denominator of the fraction to be subtracted = 7 , i.e., we represent the unit as an improper fraction 7/7 and subtract according to the rule for subtracting fractions with the same denominators.
Subtracting a proper fraction from a whole number.
Rules for subtracting fractions - correct from integer (natural number):
- We translate the given fractions, which contain an integer part, into improper ones. We get the normal terms (it doesn't matter if they are different denominators), which we consider according to the rules given above;
- Next, we calculate the difference of the fractions that we received. As a result, we will almost find the answer;
- We perform the inverse transformation, that is, we get rid of the improper fraction - we select the integer part in the fraction.
Let us subtract a proper fraction from a whole number: we present natural number as a mixed number. Those. we take a unit in a natural number and translate it into the form of an improper fraction, the denominator is the same as that of the subtracted fraction.
Fraction subtraction example:
In the example, we replaced the unit with an improper fraction 7/7 and instead of 3 we wrote mixed number and the fraction was taken away from the fractional part.
Subtraction of fractions with different denominators.
Or, to put it another way, subtraction of different fractions.
Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to bring these fractions to the lowest common denominator (LCD), and only after that to subtract 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 the given fractions.
Attention! If in 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 result of the subtraction without reducing the fraction where possible is an unfinished solution to the example!
Procedure for subtracting fractions with different denominators.
- find the LCM for all denominators;
- put additional multipliers for all fractions;
- multiply all numerators by an additional factor;
- we write the resulting products in the numerator, signing a 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 in the presence of letters in the numerator.
Subtraction of fractions, examples:
Subtraction of mixed fractions.
At subtraction 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 is to subtract mixed fractions.
If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract from it) ≥ the numerator of the fractional part of the subtrahend (we subtract it).
For example:
The second option is to subtract mixed fractions.
When the fractional parts various denominators. To begin with, we reduce the fractional parts to a common denominator, and then we subtract the integer part from the integer, and the fractional from the fractional.
For example:
The third option is to subtract 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. So, we take a unit from the integer part and bring this unit to the form of an improper fraction with the same denominator and numerator = 18.
In the numerator from the right side we write the sum of the numerators, then we open the brackets in the numerator from the right side, that is, we multiply everything and give similar ones. We do not open brackets 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. The study of this science allows you to develop some mental qualities, improve the ability to concentrate. One of the topics that deserve special attention in the course "Mathematics" is the addition and subtraction of fractions. Many students find it difficult to study. Perhaps our article will help to better understand this topic.
How to subtract fractions whose denominators are the same
Fractions are the same numbers with which you can produce various activities. Their difference from integers lies 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 ordinary fractions, whose denominators are represented as the same number. It will not be difficult to perform this action if you know a simple rule:
- In order to subtract a second fraction from one, it is necessary to subtract the numerator of the fraction to be subtracted from the numerator of the reduced fraction. 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 reduced fraction "7" we subtract the numerator of the subtracted fraction "3", we get "4". We write this number in the numerator of the answer, and put in the denominator the same number that was in the denominators of the first and second fractions - "19".
The picture below shows a few 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 the number that is in the denominators of all these fractions - "47".
Adding fractions with the same denominator
Addition and subtraction of ordinary fractions is carried out according to the same principle.
- To add fractions with the same denominators, 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" - we add the numerator of the second term of the fraction - "2". The result - "3" - is written in the numerator of the amount, and the denominator is left the same as that was present 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 we 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 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 the solution of such fractions is simply impossible.
- 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 deuce:
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. - The number 18 consists of 3 x 2 x 3.
- The number 15 consists of 5 x 3.
- The common multiple will consist of the following factors 5 x 3 x 3 x 2 = 90.
- 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.
- Convert all fractions that have an integer part to improper ones. talking in simple words, remove the whole part. To do this, the number of the integer part is multiplied by the denominator of the fraction, the resulting product is added to the numerator. The number that will be obtained after these actions is the numerator of an improper fraction. The denominator remains unchanged.
- If fractions have different denominators, they should be reduced to the same.
- Perform addition or subtraction with the same denominators.
- When receiving an improper fraction, select the whole part.
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.
Fraction property
In order to reduce 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 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 look like any number that is a multiple of "3". After we multiply the numerator and denominator by "2", we get a fraction of 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 action with the number "4", we get 8/12. In one equation, this can be written as:
2/3 = 4/6 = 6/9 = 8/12…
How to bring multiple fractions to the same denominator
Consider how to reduce 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, let's decompose the available denominators into factors.
The denominator of the fraction 1/2 and the fraction 2/3 cannot be factored. The denominator of 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 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 triples, which means that they must also be present in the denominator. Given 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 "3", 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.
Similarly, we perform actions with the remaining fractions.
All 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 have already been described.
Consider this with an example: 4/18 - 3/15.
Finding multiples of 18 and 15:
After the denominator is found, it is necessary to calculate a 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, we divide the number that we found (common multiple) by the denominator of the fraction for which additional factors need to be determined.
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 fractions with small numbers, then you can determine the common denominator, as in the example shown in the picture below.
Similarly produced and having different denominators.
Subtraction and having integer parts
Subtraction of fractions and their addition, we have already analyzed in detail. But how to subtract if the fraction has an integer part? Again, let's use a few rules:
There is another way by which you can add and subtract fractions with integer parts. For this, actions are performed separately with integer parts, and separately 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 follow the steps as shown in the example.
Subtracting fractions from a whole number
Another of the varieties of actions with fractions is the case when the fraction must be subtracted from At first glance, such an example seems difficult to solve. However, everything is quite simple here. To solve it, it is necessary to convert an integer into a fraction, and with such a denominator, which is in the fraction to be subtracted. Next, we perform 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 given in this article (Grade 6) is the basis for solving more difficult examples which are discussed in later classes. Knowledge of this topic is used subsequently to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the actions with fractions discussed above.
This lesson will cover addition and subtraction. algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in 8th grade. Moreover, this topic will be found in many topics of the algebra course, which you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, as well as analyze a number of typical examples.
Consider the simplest example for common fractions.
Example 1 Add fractions: .
Solution:
Remember the rule for adding fractions. To begin with, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.
Definition
The smallest natural number that is divisible by both numbers and .
To find the LCM, it is necessary to expand the denominators into prime factors, and then choose all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two 2s and two 3s: .
After finding the common denominator, it is necessary for each of the fractions to find an additional factor (in fact, divide the common denominator by the denominator of the corresponding fraction).
Then each fraction is multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.
We get: 
.
Answer:.
Consider now the addition of algebraic fractions with different denominators. First consider fractions whose denominators are numbers.
Example 2 Add fractions: .
Solution:
The solution algorithm is absolutely similar to the previous example. It is easy to find a common denominator for these fractions: and additional factors for each of them.
.
Answer:.
So let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:
1. Find the smallest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of this fraction).
3. Multiply the numerators by the appropriate additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with the same denominators.
Consider now an example with fractions in the denominator of which there are literal expressions.
Example 3 Add fractions: .
Solution:
Since the literal expressions in both denominators are the same, you should find a common denominator for numbers. The final common denominator will look like: . So the solution this example looks like:.
Answer:.
Example 4 Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use the abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as a common denominator.
Answer:.
In general, when solving such examples, the most difficult task is to find a common denominator.
Let's look at a more complex example.
Example 5 Simplify: .
Solution:
When finding a common denominator, you must first try to factorize the denominators of the original fractions (to simplify the common denominator).
In this particular case:
Then it is easy to determine the common denominator: 
.
We determine additional factors and solve this example:
Answer:.
Now we will fix the rules for adding and subtracting fractions with different denominators.
Example 6 Simplify: .
Solution:
Answer:.
Example 7 Simplify: .
Solution:
.
Answer:.
Consider now an example in which not two, but three fractions are added (after all, the rules for addition and subtraction for more fractions remain the same).
Example 8 Simplify: .
Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.
Consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:
As you can see, nothing complicated: just add or subtract the numerators - and that's it.
But even in such simple actions people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is easy enough. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Let's analyze all this with specific examples:
A task. Find the value of the expression:
In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:
What if the denominators are different
You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:
A task. Find the value of the expression:
In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.
What if the fraction has an integer part
I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.
Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use a simple circuit below:
- Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.
Transition rules to improper fractions and selection of the integer part are described in detail in the lesson "What is a fraction". If you don't remember, be sure to repeat. Examples:
A task. Find the value of the expression:
Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:
To simplify the calculations, I skipped some obvious steps in the last examples.
A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.
Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They love to give such tasks to control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.
Summary: General Scheme of Computing
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
- Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.
