Last time we made a plan, following which you can learn how to quickly reduce fractions. Now let's consider specific examples reduction of fractions.
Examples.
Let's check whether the larger number is divisible by the smaller number (numerator by denominator or denominator by numerator)? Yes, in all three of these examples the larger number is divided by the smaller number. Thus, we reduce each fraction by the smaller of the numbers (by the numerator or by the denominator). We have:
Let's check if the larger number is divisible by the smaller number? No, it doesn't share.
Then we move on to checking the next point: does the entry of both the numerator and denominator end with one, two or more zeros? In the first example, the numerator and denominator end in zero, in the second example, two zeros, and in the third, three zeros. This means that we reduce the first fraction by 10, the second by 100, and the third by 1000:
We got irreducible fractions.
A larger number cannot be divided by a smaller number, and numbers do not end with zeros.
Now let’s check whether the numerator and denominator are in the same column in the multiplication table? 36 and 81 are both divisible by 9, 28 and 63 are divisible by 7, and 32 and 40 are divisible by 8 (they are also divisible by 4, but if there is a choice, we will always reduce by a larger one). Thus, we come to the answers:
All numbers obtained are irreducible fractions.
A larger number cannot be divided by a smaller number. But the record of both the numerator and the denominator ends in zero. So, we reduce the fraction by 10:
This fraction can still be reduced. We check the multiplication table: both 48 and 72 are divisible by 8. We reduce the fraction by 8:
We can also reduce the resulting fraction by 3:
This fraction is irreducible.
The larger number is not divisible by the smaller number. The numerator and denominator end in zero. This means we reduce the fraction by 10.
We check the numbers obtained in the numerator and denominator for and. Since the sum of the digits of both 27 and 531 is divisible by 3 and 9, this fraction can be reduced by either 3 or 9. We choose the larger one and reduce by 9. The resulting result is an irreducible fraction.
First level
Converting Expressions. Detailed theory (2019)
Converting Expressions
We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:
“It’s much simpler,” we say, but such an answer usually doesn’t work.
Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson you will simplify this example to (just!) regular number(yes, to hell with these letters).
But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.
Have you read it? If yes, then you are now ready.
Basic simplification operations
Now let's look at the basic techniques that are used to simplify expressions.
The simplest one is
1. Bringing similar
What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in the sum, similar terms are and.
Do you remember?
To bring similar means to add several similar terms to each other and get one term.
How can we put the letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .
Now try this expression: .
To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.
So, the rule for bringing similar ones is:
Examples:
Give similar ones:
Answers:
2. (and similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.
You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):
Solutions:
3. Reducing a fraction.
Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?
That's the beauty of downsizing.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).
To reduce a fraction you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be crossed out.
The principle, I think, is clear?
I would like to draw your attention to one thing typical mistake when contracting. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.
No abbreviations if the numerator or denominator is a sum.
For example: we need to simplify.
Some people do this: which is absolutely wrong.
Another example: reduce.
The “smartest” will do this: .
Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.
Here's another example: .
This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:
You can immediately divide it into:
To avoid such mistakes, remember easy way how to determine whether an expression is factorized:
The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).
To consolidate, solve a few yourself examples:
Answers:
1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be factorization:
4. Adding and subtracting fractions. Reducing fractions to a common denominator.
Addition and subtraction ordinary fractions- the operation is well known: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:
Answers:
1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. First thing here mixed fractions we turn them into incorrect ones, and then follow the usual pattern:
It's a completely different matter if the fractions contain letters, for example:
Let's start with something simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:
Now in the numerator you can give similar ones, if any, and factor them:
Try it yourself:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
· first of all, we determine the common factors;
· then we write out all the common factors one at a time;
· and multiply them by all other non-common factors.
To determine the common factors of the denominators, we first decompose them into prime factors:
Let us emphasize the common factors:
Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
· factor the denominators;
· determine common (identical) factors;
· write out all common factors once;
· multiply them by all other non-common factors.
So, in order:
1) factor the denominators:
2) determine common (identical) factors:
3) write out all the common factors once and multiply them by all other (non-underlined) factors:
So there's a common denominator here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to a degree
to a degree
to a degree
to a degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?
So, another unshakable rule:
When you reduce fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply by to get?
So multiply by. And multiply by:
We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.
What about the expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic “”).
So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.
We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).
The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:
Great! Then:
Another example:
Solution:
As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:
So let's write:
That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now let's bring it to a common denominator:
Got it? Let's check it now.
Tasks for independent solution:
Answers:
Here we need to remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .
A - this is the so-called not perfect square sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:
What to do if there are already three fractions?
Yes, the same thing! First of all, let's make sure that maximum amount the factors in the denominators were the same:
Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.
We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:
Hmm... It’s clear what to do with fractions. But what about the two?
It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Exactly what is needed!
5. Multiplication and division of fractions.
Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:
Did you count?
It should work.
So, let me remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the expression in brackets is evaluated out of turn!
If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.
What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But this is not the same as an expression with letters?
No, it's the same! Only instead of arithmetic operations you need to do algebraic ones, that is, the actions described in previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply take out common multiplier out of brackets.
Usually our goal is to represent the expression as a product or quotient.
For example:
Let's simplify the expression.
1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).
2) We get:
Multiplying fractions: what could be simpler.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:
Now I’ll show you the process, tinting the current action in red:
Finally, I will give you two useful tips:
1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.
2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And what was promised at the very beginning:
Solutions (brief):
If you have coped with at least the first three examples, then you have mastered the topic.
Now on to learning!
CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS
Basic simplification operations:
- Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
- Factorization: putting the common factor out of brackets, applying it, etc.
- Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Adding and subtracting fractions:
; - Multiplying and dividing fractions:
;
It is based on their basic property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then an equal fraction will be obtained.
You can only reduce multipliers!
Members of polynomials cannot be abbreviated!
To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factorized.
Let's look at examples of reducing fractions.
The numerator and denominator of the fraction contain monomials. They represent work(numbers, variables and their powers), multipliers we can reduce.
We reduce the numbers by their greatest common divisor, that is, by nai larger number, by which each of these numbers is divided. For 24 and 36 this is 12. After reduction, 2 remains from 24, and 3 from 36.
Degrees are reduced by degree c the lowest rate. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.
a² and a⁷ are reduced to a². In this case, one remains in the numerator of a² (we write 1 only in the case when, after reduction, there are no other factors left. From 24, 2 remains, so we do not write 1 remaining from a²). From a⁷, after reduction, a⁵ remains.
b and b are reduced by b; the resulting units are not written.
c³º and c⁵ are shortened to c⁵. What remains from c³º is c²⁵, from c⁵ is one (we don’t write it). Thus,
The numerator and denominator of this algebraic fraction are polynomials. You cannot cancel terms of polynomials! (you cannot reduce, for example, 8x² and 2x!). To reduce this fraction, you need . The numerator has a common factor of 4x. Let's take it out of brackets:
Both the numerator and denominator have the same factor (2x-3). We reduce the fraction by this factor. In the numerator we got 4x, in the denominator - 1. For 1 property algebraic fractions, the fraction is 4x.
You can only reduce factors (you cannot reduce this fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.
The numerator is the complete square of the sum, the denominator is the difference of squares. After decomposition using abbreviated multiplication formulas, we obtain:
We reduce the fraction by (5x+1) (to do this, cross out the two in the numerator as an exponent, leaving (5x+1)² (5x+1)):
The numerator has a common factor of 2, let's take it out of brackets. The denominator is the formula for the difference of cubes:
As a result of the expansion, the numerator and denominator received the same factor (9+3a+a²). We reduce the fraction by it:
The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and remove the common factor x² from the first brackets. We decompose the denominator using the sum of cubes formula:
In the numerator, let’s take the common factor (x+2) out of brackets:
Reduce the fraction by (x+2):
Let's understand what reducing fractions is, why and how to reduce fractions, and give the rule for reducing fractions and examples of its use.
Yandex.RTB R-A-339285-1
What is "reducing fractions"
Reduce fractionTo reduce a fraction is to divide its numerator and denominator by a common factor that is positive and different from one.
As a result of this action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.
For example, let's take common fraction 6 24 and shorten it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12. In this example, we reduced the original fraction by 2.
Reducing fractions to irreducible form
In the previous example, we reduced the fraction 6 24 by 2, resulting in the fraction 3 12. It is easy to see that this fraction can be further reduced. Typically, the goal of reducing fractions is to end up with an irreducible fraction. How to reduce a fraction to its irreducible form?
This can be done by reducing the numerator and denominator by their greatest common factor (GCD). Then, by the property of the greatest common divisor, the numerator and denominator will have mutually prime numbers, and the fraction will be irreducible.
a b = a ÷ N O D (a , b) b ÷ N O D (a , b)
Reducing a fraction to an irreducible form
To reduce a fraction to an irreducible form, you need to divide its numerator and denominator by their gcd.
Let's return to the fraction 6 24 from the first example and bring it to its irreducible form. The greatest common divisor of the numbers 6 and 24 is 6. Let's reduce the fraction:
6 24 = 6 ÷ 6 24 ÷ 6 = 1 4
Reducing fractions is convenient to use so as not to work with in big numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. Reducing a fraction most often means reducing it to an irreducible form, and not simply reducing it by the common divisor of the numerator and denominator.
Rule for reducing fractions
To reduce fractions, just remember the rule, which consists of two steps.
Rule for reducing fractions
To reduce a fraction you need:
- Find the gcd of the numerator and denominator.
- Divide the numerator and denominator by their gcd.
Let's look at practical examples.
Example 1. Let's reduce the fraction.
Given the fraction 182 195. Let's shorten it.
Let's find the gcd of the numerator and denominator. To do this, in this case it is most convenient to use the Euclidean algorithm.
195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13
Divide the numerator and denominator by 13. We get:
182 195 = 182 ÷ 13 195 ÷ 13 = 14 15
Ready. We have obtained an irreducible fraction that is equal to the original fraction.
How else can you reduce fractions? In some cases, it is convenient to factor the numerator and denominator into prime factors, and then remove all common factors from the upper and lower parts of the fraction.
Example 2. Reduce the fraction
Given the fraction 360 2940. Let's shorten it.
To do this, imagine the original fraction in the form:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7
Let's get rid of the common factors in the numerator and denominator, resulting in:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49
Finally, let's look at another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, in each of which the fraction is reduced by some obvious common factor.
Example 3. Reduce the fraction
Let's reduce the fraction 2000 4400.
It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:
2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44
20 44 = 20 ÷ 2 44 ÷ 2 = 10 22
We reduce the resulting result again by 2 and obtain an irreducible fraction:
10 22 = 10 ÷ 2 22 ÷ 2 = 5 11
If you notice an error in the text, please highlight it and press Ctrl+Enter
In this lesson we will study the basic property of a fraction, find out which fractions are equal to each other. We'll learn to reduce fractions, determine whether a fraction is reducible or not, practice reducing fractions, and learn when to use a contraction and when not to.
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The main property of a fraction
Imagine this situation.
At the table 3 person and 5 apples Share 5 apples for three. Everyone gets \(\mathbf(\frac(5)(3))\) apples.
And at the next table 3 person and too 5 apples Each again \(\mathbf(\frac(5)(3))\)
In total 10 apples 6 Human. Each \(\mathbf(\frac(10)(6))\)
But it's the same thing.
\(\mathbf(\frac(5)(3) = \frac(10)(6))\)
These fractions are equivalent.
You can double the number of people and double the number of apples. The result will be the same.
In mathematics it is formulated like this:
If the numerator and denominator of a fraction are multiplied or divided by the same number (not equal to 0), then the new fraction will be equal to the original.
This property is sometimes called " main property of a fraction ».
$$\mathbf(\frac(a)(b) = \frac(a\cdot c)(b\cdot c) = \frac(a:d)(b:d))$$
For example, The path from city to village - 14 km.
We walk along the road and determine the distance traveled by kilometer markers. Having walked six columns, six kilometers, we understand that we have covered \(\mathbf(\frac(6)(14))\) distance.
But if we don’t see the poles (maybe they weren’t installed), we can calculate the path using the electric poles along the road. Their 40 pieces for every kilometer. That is, in total 560 all the way. Six kilometers - \(\mathbf(6\cdot40 = 240)\) pillars. That is, we have passed 240 from 560 pillars-\(\mathbf(\frac(240)(560))\)
\(\mathbf(\frac(6)(14) = \frac(240)(560))\)
Example 1
Mark a point with coordinates ( 5; 7 ) on coordinate plane XOY. It will correspond to the fraction \(\mathbf(\frac(5)(7))\)
Connect the origin of coordinates to the resulting point. Construct another point that has coordinates twice the previous ones. What fraction did you get? Will they be equal?
Solution
A fraction on the coordinate plane can be marked with a dot. To represent the fraction \(\mathbf(\frac(5)(7))\), mark the point with the coordinate 5 along the axis Y And 7 along the axis X. Let's draw a straight line from the origin through our point.
The point corresponding to the fraction \(\mathbf(\frac(10)(14))\) will also lie on the same line
They are equivalent: \(\mathbf(\frac(5)(7) = \frac(10)(14))\)
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