Students are introduced to fractions in 5th grade. Before people who knew how to perform actions with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, and so on. For several centuries, the examples were considered too complex. Now detailed rules have been developed for converting fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the solution will be given easily.
An ordinary fraction, which is called a simple fraction, is written as a division of two numbers: m and n.
M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.
Select proper fractions (m< n) а также неправильные (m >n).
A proper fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). An improper fraction is equal to or greater than 1 (8/7 - the unit will be 7/7 and one more part is taken as a plus).
So, a unit is when the numerator and denominator matched (3/3, 12/12, 100/100 and others).
Actions with ordinary fractions Grade 6
With simple fractions, you can do the following:
- Expand fraction. If you multiply the upper and lower parts of the fraction by any identical number (but not by zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
- Reducing fractions is similar to expanding, but here they are divided by a number.
- Compare. If two fractions have the same numerator, then the fraction with the smaller denominator will be larger. If the denominators are the same, then the fraction with the largest numerator will be larger.
- Perform addition and subtraction. With the same denominators, this is easy to do (we sum the upper parts, and the lower part does not change). For different ones, you will have to find a common denominator and additional factors.
- Multiply and divide fractions.
Examples of operations with fractions are considered below.
Reduced fractions Grade 6
To reduce means to divide the top and bottom of a fraction by some equal number.
The figure shows simple examples of reduction. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.
On a note! If the number is even, then it is divisible by 2 in any way. Even numbers are 2, 4, 6 ... 32 8 (ends in even), etc.
In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is also divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then 6 will come out. It turns out that the fraction was divided by six. This gradual division is called successive reduction of a fraction by common divisors.
Someone will immediately divide by 6, someone will need division by parts. The main thing is that at the end there is a fraction that cannot be reduced in any way.
Note that if the number consists of digits, the addition of which will result in a number divisible by 3, then the original can also be reduced by 3. Example: the number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, so the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divided by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.
In addition to the method of successive reduction of a fraction by common divisors, there are other ways.
GCD is the largest divisor for a number. Having found the GCD for the denominator and numerator, you can immediately reduce the fraction by the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors match, if there are several of them (as in the picture below), then you need to multiply.
Mixed fractions grade 6
All improper fractions can be converted into mixed fractions by isolating the whole part in them. The integer is written on the left.
Often you have to make a mixed number from an improper fraction. The conversion process in the example below: 22/4 = 22 divided by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be reduced, we divide the upper and lower parts by 2.
It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the whole number by the lower part of the fraction and add the numerator to this. Ready. The denominator does not change.
Calculations with fractions Grade 6
Mixed numbers can be added. If the denominators are the same, then this is easy to do: add up the integer parts and numerators, the denominator remains in place.
When adding numbers with different denominators, the process is more complicated. First, we bring the numbers to one smallest denominator (NOD).
In the example below, for the numbers 9 and 6, the denominator will be 18. After that, additional factors are needed. To find them, you should divide 18 by 9, so an additional number is found - 2. We multiply it by the numerator 4, we get the fraction 8/18). The same is done with the second fraction. We already add the converted fractions (whole numbers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper fraction (initially, the numerator turned out to be greater than the denominator).
Please note that with the difference of fractions, the algorithm of actions is the same.
When multiplying fractions, it is important to place both under the same line. If the number is mixed, then we turn it into a simple fraction. Next, multiply the top and bottom parts and write down the answer. If it is clear that fractions can be reduced, then we reduce immediately.
In this example, we didn’t have to cut anything, we just wrote down the answer and highlighted the whole part.
In this example, I had to reduce the numbers under one line. Though it is possible to reduce also the ready answer.
When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper one, then we write the numbers under one line, replacing the division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this is the rule for dividing fractions).
If necessary, we reduce the numbers (in the example below, they reduced it by five and two). We transform the improper fraction by highlighting the integer part.
Basic tasks for fractions Grade 6
The video shows a few more tasks. For clarity, graphic images of solutions are used to help visualize fractions.
Examples of fraction multiplication Grade 6 with explanations
Multiplying fractions are written under one line. After that, they are reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).
Comparison of fractions Grade 6
To compare fractions, you need to remember two simple rules.
Rule 1. If the denominators are different
Rule 2. When the denominators are the same
For example, let's compare the fractions 7/12 and 2/3.
- We look at the denominators, they do not match. So you need to find a common one.
- For fractions, the common denominator is 12.
- We divide 12 first by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
- Now we divide 12 by 3, we get 4 - add. multiplier of the 2nd fraction.
- We multiply the resulting numbers by numerators to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. Turned out: 7/12< 8/12.
To represent fractions better, you can use drawings for clarity, where an object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case, the cake is divided into 7 parts and 4 of them are chosen. In the second, they divide into 3 parts and take 2. With the naked eye, it will be clear that 2/3 will be more than 4/7.
Examples with fractions grade 6 for training
As an exercise, you can perform the following tasks.
- Compare fractions
- do the multiplication
Tip: if it is difficult to find the lowest common denominator of fractions (especially if their values are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, you get 72.
Solving equations with fractions Grade 6
In solving equations, you need to remember the actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the known factor, that is, the fractions are multiplied (the second is turned over).
If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor, you need to divide the dividend by the quotient.
Imagine simple examples solving equations:
Here it is only required to produce the difference of fractions, without leading to a common denominator.
The answer is an improper fraction. It can be converted to 1 whole and 3/5.
In the second method, the numerator and denominator were multiplied by 4 to shorten the bottom rather than flip the denominator.
This section deals with operations with ordinary fractions. If it is necessary to perform a mathematical operation with mixed numbers, then it is enough to convert the mixed fraction into an extraordinary one, perform the necessary operations and, if necessary, re-present the final result in the form mixed number. This operation will be described below.
Fraction reduction
mathematical operation. Fraction reduction
To reduce the fraction \frac(m)(n) you need to find the largest common divisor its numerator and denominator: GCD(m,n), then divide the numerator and denominator of the fraction by this number. If gcd(m,n)=1, then the fraction cannot be reduced. Example: \frac(20)(80)=\frac(20:20)(80:20)=\frac(1)(4)
Usually, immediately finding the greatest common divisor is a difficult task, and in practice the fraction is reduced in several stages, step by step highlighting obvious common factors from the numerator and denominator. \frac(140)(315)=\frac(28\cdot5)(63\cdot5)=\frac(4\cdot7\cdot5)(9\cdot7\cdot5)=\frac(4)(9)
Bringing fractions to a common denominator
mathematical operation. Bringing fractions to a common denominator
To reduce two fractions \frac(a)(b) and \frac(c)(d) to a common denominator, you need:
- find the least common multiple of the denominators: M=LCM(b,d);
- multiply the numerator and denominator of the first fraction by M/b (after which the denominator of the fraction becomes equal to the number M);
- multiply the numerator and denominator of the second fraction by M/d (after which the denominator of the fraction becomes equal to the number M).
Thus, we convert the original fractions to fractions with the same denominators (which will be equal to the number M).
For example, the fractions \frac(5)(6) and \frac(4)(9) have LCM(6,9) = 18. Then: \frac(5)(6)=\frac(5\cdot3)(6 \cdot3)=\frac(15)(18);\quad\frac(4)(9)=\frac(4\cdot2)(9\cdot2)=\frac(8)(18) . Thus, the resulting fractions have a common denominator.
In practice, finding the least common multiple (LCM) of denominators is not always an easy task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as a common denominator. For example, the fractions \frac(5)(6) and \frac(4)(9) are reduced to a common denominator N=6\cdot9:
\frac(5)(6)=\frac(5\cdot9)(6\cdot9)=\frac(45)(54);\quad\frac(4)(9)=\frac(4\cdot6)( 9\cdot6)=\frac(24)(54)
Fraction Comparison
mathematical operation. Fraction Comparison
To compare two common fractions:
- compare the numerators of the resulting fractions; a fraction with a larger numerator will be larger.
When comparing fractions, there are several special cases:
- From two fractions with the same denominators the greater is the fraction whose numerator is greater. For example \frac(3)(15)
- From two fractions with the same numerators the larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13)
- That fraction, which at the same time larger numerator and smaller denominator, more. For example, \frac(11)(3)>\frac(10)(8)
Attention! Rule 1 applies to any fractions if their common denominator is a positive number. Rules 2 and 3 apply to positive fractions (which have both numerator and denominator greater than zero).
Addition and subtraction of fractions
mathematical operation. Addition and subtraction of fractions
To add two fractions, you need:
- bring them to a common denominator;
- add their numerators and leave the denominator unchanged.
Example: \frac(7)(9)+\frac(4)(7)=\frac(7\cdot7)(9\cdot7)+\frac(4\cdot9)(7\cdot9)=\frac(49 )(63)+\frac(36)(63)=\frac(49+36)(63)=\frac(85)(63)
To subtract another fraction from one, you need:
- bring fractions to a common denominator;
- subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged.
Example: \frac(4)(15)-\frac(3)(5)=\frac(4)(15)-\frac(3\cdot3)(5\cdot3)=\frac(4)(15) -\frac(9)(15)=\frac(4-9)(15)=\frac(-5)(15)=-\frac(5)(3\cdot5)=-\frac(1)( 3)
If the original fractions initially have a common denominator, then point 1 (reduction to a common denominator) is skipped.
Converting a mixed number to an improper fraction and vice versa
mathematical operation. Converting a mixed number to an improper fraction and vice versa
To convert a mixed fraction to an improper one, it is enough to sum the whole part of the mixed fraction with the fractional part. The result of such a sum will be an improper fraction, the numerator of which is equal to the sum of the product of the integer part and the denominator of the fraction with the numerator of the mixed fraction, and the denominator remains the same. For example, 2\frac(6)(11)=2+\frac(6)(11)=\frac(2\cdot11)(11)+\frac(6)(11)=\frac(2\cdot11+ 6)(11)=\frac(28)(11)
To convert an improper fraction to a mixed number:
- divide the numerator of a fraction by its denominator;
- write the remainder of the division into the numerator, and leave the denominator the same;
- write the result of the division as an integer part.
For example, the fraction \frac(23)(4) . When dividing 23:4=5.75, that is, the integer part is 5, the remainder of the division is 23-5*4=3. Then the mixed number will be written: 5\frac(3)(4) . \frac(23)(4)=\frac(5\cdot4+3)(4)=5\frac(3)(4)
Converting a Decimal to a Common Fraction
mathematical operation. Converting a Decimal to a Common Fraction
To convert a decimal to a common fraction:
- take the n-th power of ten as a denominator (here n is the number of decimal places);
- as a numerator, take the number after the decimal point (if the integer part of the original number is not equal to zero, then take all leading zeros as well);
- the non-zero integer part is written in the numerator at the very beginning; the zero integer part is omitted.
Example 1: 0.0089=\frac(89)(10000) (4 decimal places, so the denominator 10 4 =10000, since the integer part is 0, the numerator is the number after the decimal point without leading zeros)
Example 2: 31.0109=\frac(310109)(10000) (in the numerator we write the number after the decimal point with all zeros: "0109", and then we add the integer part of the original number "31" before it)
If the integer part of a decimal fraction is different from zero, then it can be converted to a mixed fraction. To do this, we translate the number into an ordinary fraction as if the integer part were equal to zero (points 1 and 2), and simply rewrite the integer part before the fraction - this will be the integer part of the mixed number. Example:
3.014=3\frac(14)(100)
To convert an ordinary fraction to a decimal, it is enough to simply divide the numerator by the denominator. Sometimes you get an infinite decimal. In this case, it is necessary to round to the desired decimal place. Examples:
\frac(401)(5)=80.2;\quad \frac(2)(3)\approx0.6667
Multiplication and division of fractions
mathematical operation. Multiplication and division of fractions
To multiply two common fractions, you need to multiply the numerators and denominators of the fractions.
\frac(5)(9)\cdot\frac(7)(2)=\frac(5\cdot7)(9\cdot2)=\frac(35)(18)
To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second ( reciprocal is a fraction in which the numerator and denominator are reversed.
\frac(5)(9):\frac(7)(2)=\frac(5)(9)\cdot\frac(2)(7)=\frac(5\cdot2)(9\cdot7)= \frac(10)(63)
If one of the fractions is natural number, then the above rules for multiplication and division remain in effect. Just keep in mind that an integer is the same fraction, the denominator of which equal to one. For example: 3:\frac(3)(7)=\frac(3)(1):\frac(3)(7)=\frac(3)(1)\cdot\frac(7)(3)= \frac(3\cdot7)(1\cdot3)=\frac(7)(1)=7
Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is a detailed instruction for solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, whether it be addition, subtraction, multiplication or division, you need to know the basic rules:
- In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator at the top), you need to add their numerators, and leave the denominator the same.
- In order to subtract from one fractional expression the second (with the same denominator), you need to subtract their numerators, and leave the denominator the same.
- In order to add or subtract fractional expressions with different denominators, you need to find the smallest common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, while, if possible, reduce.
- To divide a fraction by a fraction, you need to multiply the first fraction by the reversed second.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if fractions of two (or more) have the same denominator, you just need to add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will be 1.
Answer: 3/4 + 1/4 = 4/4 = 1.
Rule 2, example 1:
Calculate: 3/4 - 1/4
Using rule number 2, to solve this equation, you need to subtract 1 from 3, and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.
Answer: 3/4 - 1/4 = 2/4 = 1/2.
Rule 3, Example 1
Calculate: 3/4 + 1/6
Solution: Using the 3rd rule, we find the least common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions in the example. Thus, we need to find such a minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. 12 is divided by the denominator of the first fraction, we get 3, we multiply by 3, we write 3 in the numerator *3 and + sign. We divide 12 by the denominator of the second fraction, we get 2, we multiply 2 by 1, we write 2 * 1 in the numerator. So, we got a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.
Answer: 11/12
Rule 3, Example 2:
Calculate 3/4 - 1/6. This example is very similar to the previous one. We do all the same actions, but in the numerator instead of the + sign, we write the minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.
Answer: 7/12
Rule 4, Example 1:
Calculate: 3/4 * 1/4
Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.
Answer: 3/16
Rule 4, Example 2:
Calculate 2/5 * 10/4.
This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are reduced.
2 is reduced from 4. 10 is reduced from 5. we get 1 * 2/2 = 1 * 1 = 1.
Answer: 2/5 * 10/4 = 1
Rule 5, Example 1:
Calculate: 3/4: 5/6
Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.
Answer: 9/10.
How to Solve Fraction Examples - Fractional Equations
Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.
Consider an example:
Solve equation 15/3x+5 = 3
Recall that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. To do this, there is ODZ (range of acceptable values).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
For x = 5/3, the equation simply has no solution.
By specifying the ODZ, in the best possible way solve this equation will get rid of the fractions. To do this, we first represent all non-fractional values as a fraction, in this case the number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions, you need to multiply each of them by the smallest common denominator. In this case, that would be (3x+5)*1. Sequencing:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Expand the brackets: 15*(3x+5) = 45x + 75.
- We do the same with the right side of the equation: 3*(3x+5) = 9x + 15.
- Equate the left and right sides: 45x + 75 = 9x +15
- Move x's to the left, numbers to the right: 36x = -50
- Find x: x = -50/36.
- We reduce: -50/36 = -25/18
Answer: ODZ x ≠ 5/3. x = -25/18.
How to solve examples with fractions - fractional inequalities
Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the numerical axis. Consider this example.
Sequencing:
- Equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
2. 2-x=0 => x=2 - We draw numerical axis, writing the resulting values on it.
- Draw a circle under the value. The circle is of two types - filled and empty. A filled circle means that this value is included in the range of solutions. An empty circle indicates that this value is not included in the range of solutions.
- Since the denominator cannot be zero, there will be an empty circle under the 2nd.
- To determine the signs, we substitute any number greater than two into the equation, for example 3. (3 * 3-5) / (2-3) \u003d -4. the value is negative, so we write a minus over the area after the deuce. Then we substitute any value of the interval from 5/3 to 2 instead of x, for example 1. The value is again negative. We write minus. We repeat the same with the area up to 5/3. We substitute any number less than 5/3, for example 1. Minus again.
- Since we are interested in x values, at which the expression will be greater than or equal to 0, and there are no such values (cons everywhere), this inequality has no solution, i.e. x = Ø (empty set).
Answer: x = Ø
Instruction
First, remember that a fraction is just a conditional notation for dividing one number by another. In addition and multiplication, dividing two integers does not always result in an integer. So call these two "divisible" numbers. The number that is being divided is the numerator, and the number that is being 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 a slash "/" 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.
If the numerator of a fraction is greater than its denominator, then such an "improper" fraction is usually written as a "mixed" fraction. To get a mixed fraction from an improper fraction, simply divide the numerator by the denominator and write down the resulting quotient. Then put the remainder of the division in the numerator of the fraction and write this fraction to the right of the quotient (do not touch the denominator). For example, 7/3 = 2⅓.
To add two fractions with the same denominator, simply add their numerators (leave the denominators). For example, 2/7 + 3/7 = (2+3)/7 = 5/7. Similarly, subtract two fractions (the numerators are subtracted). For example, 6/7 - 2/7 = (6-2)/7 = 4/7.
To add two fractions with different denominators, multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first. As a result, you will get the sum of two fractions with the same denominators, the addition of which is described in the previous paragraph.
For example, 3/4 + 2/3 = (3*3)/(4*3) + (2*4)/(3*4) = 9/12 + 8/12 = (9+8)/12 = 17/12 = 15/12.
If the denominators of fractions have common divisors, that is, they are divisible by the same number, select as the common denominator smallest number divisible by the first and second denominators simultaneously. So, for example, if the first denominator is 6 and the second 8, then take as a common denominator not their product (48), but the number 24, which is divisible by both 6 and 8. The numerators of the fractions are then multiplied by the quotient of dividing the common denominator by the denominator of each fraction. For example, for the denominator 6, this number will be 4 - (24/6), and for the denominator 8 - 3 (24/8). This process is more clearly seen in a specific example:
5/6 + 3/8 = (5*4)/24 + (3*3)/24 = 20/24 + 9/24 = 29/24 = 1 5/24.
Subtraction of fractions with different denominators is done in exactly the same way.
Let's agree that "actions with fractions" in our lesson will be understood as actions with ordinary fractions. A fraction is a fraction that has attributes such as a numerator, a fractional bar, and a denominator. This distinguishes an ordinary fraction from a decimal fraction, which is obtained from an ordinary one by reducing the denominator to a multiple of 10. A decimal fraction is written with a comma separating the integer part from the fractional one. We will talk about operations with ordinary fractions, since it is they that cause the greatest difficulties for students who have forgotten the basics of this topic, covered in the first half of the school mathematics course. At the same time, when transforming expressions in higher mathematics, it is mainly operations with ordinary fractions that are used. Some abbreviations of fractions are worth something! Decimal fractions do not cause much difficulty. So go ahead!
Two fractions and are called equal if .
For example, because
The fractions and (since ), and (since ) are also equal.
Obviously, both fractions and are equal. This means that if the numerator and denominator of a given fraction are multiplied or divided by the same natural number, then a fraction equal to the given one will be obtained:.
This property is called the basic property of a fraction.
The basic property of a fraction can be used to change the signs of the numerator and denominator of a fraction. If the numerator and denominator of the fraction are multiplied by -1, then we get. This means that the value of a fraction will not change if the signs of the numerator and denominator are changed at the same time. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:
Fraction reduction
Using the basic property of a fraction, you can replace a given fraction with another fraction equal to the given one, but with a smaller numerator and denominator. This substitution is called fraction reduction.
Let, for example, be given a fraction. The numbers 36 and 48 have the greatest common divisor 12. Then
.
In the general case, fraction reduction is always possible if the numerator and denominator are not coprime numbers. If the numerator and denominator are mutual prime numbers, then the fraction is called irreducible.
So, reducing a fraction means dividing the numerator and denominator of a fraction by a common factor. All of the above applies to fractional expressions containing variables.
Example 1 Reduce fraction
Solution. To factorize the numerator into factors, having previously presented the monomial - 5 xy as a sum - 2 xy - 3xy, we get
To factorize the denominator, we use the difference of squares formula:
As a result
.
Bringing fractions to a common denominator
Let two fractions and be given. They have different denominators: 5 and 7. Using the basic property of a fraction, you can replace these fractions with others equal to them, and such that the resulting fractions will have the same denominators. Multiplying the numerator and denominator of the fraction by 7, we get
Multiplying the numerator and denominator by 5, we get
So, the fractions are reduced to a common denominator:
.
But this is not the only solution to the problem: for example, these fractions can also be reduced to a common denominator of 70:
,
and in general to any denominator divisible by both 5 and 7.
Let's consider one more example: let's reduce the fraction and to a common denominator. Arguing as in the previous example, we get
,
.
But in this case, you can bring the fractions to a common denominator, less than the product of the denominators of these fractions. Find the least common multiple of 24 and 30: LCM(24, 30) = 120 .
Since 120:4=5, in order to write a fraction with a denominator of 120, both the numerator and the denominator must be multiplied by 5, this number is called an additional factor. Means 
.
Further, we get 120:30=4. Multiplying the numerator and denominator of the fraction by an additional factor of 4, we get 
.
So, these fractions are reduced to a common denominator.
The least common multiple of the denominators of these fractions is the smallest possible common denominator.
For fractional expressions that include variables, the common denominator is a polynomial that is divisible by the denominator of each fraction.
Example 2 Find the common denominator of fractions and .
Solution. The common denominator of these fractions is a polynomial, since it is divisible by both and by. However, this polynomial is not the only one that can be a common denominator of these fractions. It can also be a polynomial 
, and polynomial 
, and polynomial 
etc. Usually they take such a common denominator that any other common denominator is divisible by the chosen one without a remainder. Such a denominator is called the least common denominator.
In our example, the least common denominator is . Got:
;
.
We managed to bring fractions to the lowest common denominator. This happened by multiplying the numerator and denominator of the first fraction by , and the numerator and denominator of the second fraction by . Polynomials and are called additional factors, respectively, for the first and second fractions.
Addition and subtraction of fractions
The addition of fractions is defined as follows:
.
For example,
.
If b = d, That
.
This means that to add fractions with the same denominator, it is enough to add the numerators, and leave the denominator the same. For example,
.
If fractions with different denominators are added, then the fractions are usually reduced to the lowest common denominator, and then the numerators are added. For example,
.
Now consider an example of adding fractional expressions with variables.
Example 3 Convert expression to one fraction
.
Solution. Let's find the least common denominator. To do this, we first factorize the denominators.
