How to subtract a fraction with different denominators. Adding and subtracting fractions
Pay attention! Before writing your final answer, see if you can shorten the fraction you received.
Subtracting fractions from same 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 the unit in the form improper fraction 7/7 and subtract according to the rule for subtracting fractions with like denominators.
Subtracting a proper fraction from a whole number.
Rules for subtracting fractions - correct from a whole number (natural number):
- We convert given fractions that contain an integer part into improper ones. We get normal terms (it doesn’t matter if they are with different denominators), which we calculate according to the rules given above;
- Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
- We perform the inverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.
Subtract a proper fraction from a whole number: imagine 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 mixed number and a fraction was subtracted from the fractional part.
Subtracting fractions with different denominators.
Or, to put it another way, subtracting different fractions.
Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.
The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.
Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!
Procedure for subtracting fractions with different denominators.
- find the LCM for all denominators;
- put additional factors for all fractions;
- multiply all numerators by an additional factor;
- We write the resulting products into the numerator, signing the common denominator under all fractions;
- subtract the numerators of fractions, signing the common denominator under the difference.
In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.
Subtracting fractions, examples:
Subtracting mixed fractions.
At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.
The first option for subtracting mixed fractions.
If the fractional parts identical denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).
For example:
The second option for subtracting mixed fractions.
When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.
For example:
The third option for subtracting mixed fractions.
The fractional part of the minuend is less than the fractional part of the subtrahend.
Example:
Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.
The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.
In the numerator on the right side we write the sum of the numerators, then we open the brackets in the numerator on the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:
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 the 8th grade course. Moreover, this topic will appear in many topics in the algebra course that 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, and also analyze a whole series typical examples.
Let's consider simplest example for ordinary fractions.
Example 1. Add fractions: .
Solution:
Let's remember the rule for adding fractions. To begin, 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, you need to factor the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two twos and two threes: .
After finding the common denominator, you need to find an additional factor for each fraction (in fact, divide the common denominator by the denominator of the corresponding fraction).
Each fraction is then 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:.
Let us now consider the addition of algebraic fractions with different denominators. First, let's look at 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 the common denominator of these fractions: and additional factors for each of them.
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Answer:.
So, let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:
1. Find the lowest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of the given fraction).
3. Multiply the numerators by the corresponding additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with like denominators.
Let us now consider an example with fractions whose denominator contains literal expressions.
Example 3. Add fractions: .
Solution:
Since the letter expressions in both denominators are the same, you should find a common denominator for the numbers. The final common denominator will look like: . Thus, the solution to 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 abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as the common denominator.
Answer:.
In general, when deciding similar examples, the most difficult task is finding 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 factor 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 let's establish the rules for adding and subtracting fractions with different denominators.
Example 6. Simplify: .
Solution:
Answer:.
Example 7. Simplify: .
Solution:
.
Answer:.
Let us now consider an example in which not two, but three fractions are added (after all, the rules of addition and subtraction for more fractions remain the same).
Example 8. Simplify: .
This lesson will cover adding and subtracting algebraic fractions with like denominators. We already know how to add and subtract common fractions with like denominators. It turns out that algebraic fractions follow the same rules. Learning to work with fractions with like denominators is one of the cornerstones of learning how to work with algebraic fractions. In particular, understanding this topic will make it easy to master a more complex topic - adding and subtracting fractions with different denominators. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with like denominators, and also analyze a number of typical examples
Rule for adding and subtracting algebraic fractions with like denominators
Sfor-mu-li-ru-em pra-vi-lo slo-zhe-niya (you-chi-ta-niya) al-geb-ra-i-che-skih fractions from one-on-to-you -mi know-me-na-te-la-mi (it coincides with the analogous rule for ordinary shot-beats): That is for addition or calculation of al-geb-ra-i-che-skih fractions with one-to-you know-me-on-te-la-mi necessary -ho-di-mo to compose the corresponding al-geb-ra-i-che-sum of numbers, and the sign-me-na-tel leave without any.
We understand this rule both for the example of ordinary ven-draws and for the example of al-geb-ra-i-che-draws. hit.
Examples of applying the rule for ordinary fractions
Example 1. Add fractions: .
Solution
Let's add the number of fractions, and leave the sign the same. After this, we decompose the number and sign into simple multiplicities and combinations. Let's get it: 
.
Note: a standard error that is allowed when solving similar types of examples, for -klu-cha-et-sya in the following possible solution: 
. This is a gross mistake, since the sign remains the same as it was in the original fractions.
Example 2. Add fractions: .
Solution
This one is in no way different from the previous one: .
Examples of applying the rule for algebraic fractions
From ordinary dro-beats, we move to al-geb-ra-i-che-skim.
Example 3. Add fractions: .
Solution: as already mentioned above, the composition of al-geb-ra-i-che-fractions is in no way different from the word the same as usual shot-fights. Therefore, the solution method is the same: .
Example 4. You are the fraction: .
Solution
You-chi-ta-nie al-geb-ra-i-che-skih fractions from-whether from addition only by the fact that in the number pi-sy-va-et-sya difference in the number of used fractions. That's why .
Example 5. You are a fraction: .
Solution: .
Example 6. Simplify: .
Solution: .
Examples of applying the rule followed by reduction
In a fraction that has the same meaning in the result of compounding or calculating, combinations are possible nia. In addition, you should not forget about the ODZ of al-geb-ra-i-che-skih fractions.
Example 7. Simplify: .
Solution: .
At the same time. In general, if the ODZ of the initial fractions coincides with the ODZ of the total, then it can be omitted (after all, the fraction is being in the answer, will also not exist with the corresponding significant changes). But if the ODZ of the used fractions and the answer doesn’t match, then the ODZ needs to be indicated.
Example 8. Simplify: .
Solution: . At the same time, y (the ODZ of the initial fractions does not coincide with the ODZ of the result).
Adding and subtracting fractions with different denominators
To add and read al-geb-ra-i-che-fractions with different know-me-on-the-la-mi, we do ana-lo -giyu with ordinary-ven-ny fractions and transfer it to al-geb-ra-i-che-fractions.
Let's look at the simplest example for ordinary fractions.
Example 1. Add fractions: .
Solution:
Let's remember the rules for adding fractions. To begin with, a fraction needs to be brought to a common sign. In the role of a general sign for ordinary fractions, you act least common multiple(NOK) initial signs.
Definition
The smallest number, which is divided at the same time into numbers and.
To find the NOC, it is necessary to break down the knowledge into simple sets, and then select everything there are many, which are included in the division of both signs.
; . Then the LCM of numbers must include two twos and two threes: .
After finding the general knowledge, it is necessary for each of the fractions to find a complete multiplicity resident (in fact, in fact, to put the common sign on the sign of the corresponding fraction).
Then each fraction is multiplied by a half-full factor. Let's get some fractions from the same ones you know, add them up and read them out. -studied in previous lessons.
Let's eat: 
.
Answer:.
Let's now look at the composition of al-geb-ra-i-che-fractions with different signs. Now let’s look at the fractions and see if there are any numbers.
Adding and subtracting algebraic fractions with different denominators
Example 2. Add fractions: .
Solution:
Al-go-rhythm of the decision ab-so-lyut-but ana-lo-gi-chen to the previous example. It’s easy to take the common sign of the given fractions: and additional multipliers for each of them.
.
Answer:.
So, let's form al-go-rhythm of addition and calculation of al-geb-ra-i-che-skih fractions with different signs:
1. Find the smallest common sign of the fraction.
2. Find additional multipliers for each of the fractions (indeed, the common sign of the sign is given -th fraction).
3. Up-to-many numbers on the corresponding up-to-full multiplicities.
4. Add or calculate fractions, using the right-of-minor additions and calculating fractions with the same knowledge -me-na-te-la-mi.
Let's now look at an example with fractions, in the sign of which there are letters you -nia.
Fractions are ordinary numbers and can also be added and subtracted. But because they have a denominator, they require more complex rules than for integers.
Let's consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, you need 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 adding and subtracting fractions we get:
As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.
But even in such simple actions, people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Getting rid of the bad habit of adding denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) 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 to put a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Let's look at all this with specific examples:
Task. Find the meaning of the expression:
In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:
What to do if the denominators are different
You cannot add fractions with different denominators directly. 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 “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:
Task. Find the meaning of the expression:
In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.
What to do if a fraction has an integer part
I can please you: different denominators in fractions are not the biggest evil. Much more errors occur when the whole part is highlighted in the addend fractions.
Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use simple diagram, given below:
- Convert all fractions containing an integer part to improper ones. We obtain normal terms (even 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 problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.
The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:
Task. Find the meaning of the expression:
Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:
To simplify the calculations, I have skipped some obvious steps in the last examples.
A small note about the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.
Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such tasks to tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.
Summary: general calculation scheme
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If one or more fractions have an integer part, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
- If possible, shorten the result. If the fraction is incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the problem, immediately before writing down the answer.
Find the numerator and denominator. A fraction includes two numbers: the number that is located above the line is called the numerator, and the number that is located below the line is called the denominator. The denominator denotes the total number of parts into which a whole is divided, and the numerator is the number of such parts considered.
- 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, a certain whole is divided into the same number of parts. Add fractions with common denominator very simple, since the denominator of the total fraction will be the same as that of the added fractions. For example:
- The fractions 3/5 and 2/5 have a common denominator of 5.
- The fractions 3/8, 5/8, 17/8 have a common denominator of 8.
Determine the numerators. To add fractions with a common denominator, add their numerators and write the result above the denominator of the fractions being added.
- The fractions 3/5 and 2/5 have numerators 3 and 2.
- Fractions 3/8, 5/8, 17/8 have numerators 3, 5, 17.
Add up the numerators. In problem 3/5 + 2/5, add the numerators 3 + 2 = 5. In problem 3/8 + 5/8 + 17/8, add the numerators 3 + 5 + 17 = 25.
Write the total fraction. Remember that when adding fractions with a common denominator, it remains unchanged - only the numerators are added.
- 3/5 + 2/5 = 5/5
- 3/8 + 5/8 + 17/8 = 25/8
Convert the fraction if necessary. Sometimes a fraction can be written as a whole number rather than as a fraction or decimal. For example, the fraction 5/5 is easily converted to 1, since any fraction whose numerator is equal to its denominator is 1. Imagine a pie cut into three parts. If you eat all three parts, you will have eaten the whole (one) pie.
- I love it common fraction can be converted to decimal; To do this, divide the numerator by the denominator. For example, the fraction 5/8 can be written as follows: 5 ÷ 8 = 0.625.
If possible, simplify the fraction. A simplified fraction is a fraction whose numerator and denominator do not have common factors.
- For example, consider the fraction 3/6. Here both the numerator and the denominator have common divisor, equal to 3, that is, the numerator and denominator are completely divisible by 3. Therefore, the fraction 3/6 can be written as follows: 3 ÷ 3/6 ÷ 3 = ½.
If necessary, convert improper fraction to mixed fraction(mixed number). An improper fraction has a numerator greater than its denominator, for example, 25/8 (a proper fraction has a numerator less than its denominator). An improper fraction can be converted to a mixed fraction, which consists of an integer part (that is, a whole number) and a fraction part (that is, a proper fraction). To convert an improper fraction, such as 25/8, to a mixed number, follow these steps:
- Divide the numerator of an improper fraction by its denominator; write down the partial quotient (whole answer). In our example: 25 ÷ 8 = 3 plus some remainder. In this case, the whole answer is the whole part of the mixed number.
- Find the remainder. In our example: 8 x 3 = 24; subtract the resulting result from the original numerator: 25 - 24 = 1, that is, the remainder is 1. In this case, the remainder is the numerator of the fractional part of the mixed number.
- Write down the mixed fraction. The denominator does not change (that is, it is equal to the denominator of the improper fraction), so 25/8 = 3 1/8.
