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-the-la-mi necessary -ho-di-mo-compile a 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 the 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, it is necessary to bring it 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, you need 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. -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 composition and calculation of al-geb-ra-i-che-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-mind 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.
The numerator, and that which is divided by is the denominator.
To write a fraction, first write the numerator, then draw a horizontal line under the number, and write the denominator below the line. The horizontal line separating the numerator and denominator is called a fraction line. 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 the numerator is different. Write the result in the numerator of the new fractions. After this, multiply the denominators. Enter the total 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 actions, first “flip” the divisor, if it is more convenient for you: the denominator should appear 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 fraction problems
Fractional numbers allow you to express the exact value of a quantity in different forms. You can do the same math operations with fractions as you can with whole numbers: subtraction, addition, multiplication, and division. To learn to decide fractions, we must remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations require the fractional part of the result to be reduced after execution.
You will need
- - calculator
Instructions
Look closely at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular 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 an integer part is isolated must be converted 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 select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. 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 of separately integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.
Rewrite them using the “:” separator and continue with normal division.
To obtain 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 integers above and below the line.
Please note
Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.
Useful advice
When writing fractional numbers, the dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice as a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, the fraction is called a decimal. 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 ease of calculation, 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 one integer. In this example, you can divide by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.
Pay attention! 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 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: 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 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 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 the parentheses in the denominator. It is customary to leave the product in the denominators. We get:
Has your child brought homework from school and you don't know how to solve it? Then this mini lesson is for you!
How to add decimals
It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:
- The place must be under the place, the comma under the comma.
As you can see in the example, the whole units are located under each other, the tenths and hundredths digits are located under each other. Now we add the numbers, ignoring the comma. What to do with the comma? The comma is moved to the place where it stood in the integer category.
Adding fractions with equal denominators
To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction that will be the total sum.
Adding fractions with different denominators using the common multiple method
The first thing you need to pay attention to is the denominators. The denominators are different, whether one is divisible by the other, or whether they are prime numbers. First we need to bring it to one common denominator; there are several ways to do this:
- 1/3 + 3/4 = 13/12, to solve this example we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b – LCM (a;b). In this example LCM (3;4)=12. We check: 12:3=4; 12:4=3.
- We multiply the factors and add the resulting numbers, we get 13/12 - an improper fraction.
- In order to convert an improper fraction into a proper one, divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.
Adding fractions using the cross-cross multiplication method
To add fractions with different denominators, there is another method using the “cross to cross” formula. This is a guaranteed way to equalize the denominators; to do this, you need to multiply the numerators with the denominator of one fraction and vice versa. If you are just at the initial stage of learning fractions, then this method is the simplest and most accurate way to get the correct result when adding fractions with different denominators.
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions with different denominators are multiplied.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the existing ones.
When multiplying simple fractions with different denominators for two or more factors the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional line will be a product of different numbers and, naturally, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way of representing a mixed fraction as an improper fraction, and can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s easy to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
