What can 143 be reduced by? Reducing algebraic fractions: rules, examples

If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).

The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or completely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.

The remainder is always less than the divisor.

Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.

Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the incomplete quotient, r is the remainder.

The quotient of natural numbers can be written as a fraction.

The numerator of a fraction is the dividend, and the denominator is the divisor.

Since the numerator of a fraction is the dividend, and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.

The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n) \), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n)\)

The following rules are true:

To get the fraction \(\frac(m)(n)\), you need to divide the unit into n equal parts (shares) and take m such parts.

To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.

To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.

To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.

If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)

If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.

The last two transformations are called reducing a fraction.

If fractions need to be represented as fractions with the same denominator, then this action is called reducing fractions to common denominator .

Proper and improper fractions. Mixed numbers

You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense dictates that the part should always be less than the whole, but what about fractions such as \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator less than the denominator, called correct fractions.

As you know, any common fraction, both correct and incorrect, can be considered as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.

If a number consists of an integer part and a fraction, then such fractions are called mixed.

For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.

If the numerator of the fraction \(\frac(a)(b) \) is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)

If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)

Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.

Actions with fractions. Adding fractions.

You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)

If you need to add fractions with different denominators, then they must first be brought to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)

For fractions, as for natural numbers, the commutative and associative properties of addition are valid.

Adding mixed fractions

Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”

When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)

Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction of fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)

The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.

Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)

Multiplying fractions

To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.

Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)

Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, and a mixed fraction as an improper fraction.

The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.

For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.

Division of fractions

Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).

If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.

For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).

Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)

It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)

Using reciprocal fractions, you can reduce division of fractions to multiplication.

The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.

Using letters, the rule for dividing fractions can be written as follows:
\(\large \frac(a)(b) : \frac(c)(d) = \frac(a)(b) \cdot \frac(d)(c) \)

If the dividend or divisor is natural number or a mixed fraction, then, in order to use the rule for dividing fractions, it must first be represented as an improper fraction.

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 to their largest common divisor, that is, on greatest 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.

We reduce the degrees by the degree with the lowest index. 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. From c³º what remains 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. According to 1 property of algebraic fractions, the fraction is equal to 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.

In the numerator - perfect square sums, 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):

Without knowing how to reduce a fraction and having a stable skill in solving such examples, it is very difficult to study algebra in school. The further you go, the more it interferes with your basic knowledge of reducing fractions. new information. First, powers appear, then factors, which later become polynomials.

How can you avoid getting confused here? Thoroughly consolidate skills in previous topics and gradually prepare for knowledge of how to reduce a fraction, which becomes more complex from year to year.

Basic knowledge

Without them, you will not be able to cope with tasks of any level. To understand, you need to understand two simple points. First: you can only reduce factors. This nuance turns out to be very important when polynomials appear in the numerator or denominator. Then you need to clearly distinguish where the factor is and where the addend is.

The second point says that any number can be represented in the form of factors. Moreover, the result of reduction is a fraction whose numerator and denominator can no longer be reduced.

Rules for reducing common fractions

First, you should check whether the numerator is divisible by the denominator or vice versa. Then it is precisely this number that needs to be reduced. This is the simplest option.

The second is analysis appearance numbers. If both end in one or more zeros, then they can be shortened by 10, 100 or a thousand. Here you can notice whether the numbers are even. If yes, then you can safely cut it by two.

The third rule for reducing a fraction is to factor it into prime factors numerator and denominator. At this time, you need to actively use all your knowledge about the signs of divisibility of numbers. After this decomposition, all that remains is to find all the repeating ones, multiply them and reduce them by the resulting number.

What if there is an algebraic expression in a fraction?

This is where the first difficulties appear. Because this is where terms appear that can be identical to factors. I really want to reduce them, but I can’t. Before you can reduce an algebraic fraction, it must be converted so that it has factors.

To do this, you will need to perform several steps. You may need to go through all of them, or maybe the first one will provide a suitable option.

Check whether the numerator and denominator or any expression in them differ by sign. In this case, you just need to put minus one out of brackets. This produces equal factors that can be reduced.

See if it is possible to remove the common factor from the polynomial out of brackets. Perhaps this will result in a parenthesis, which can also be shortened, or it will be a removed monomial.

Try to group the monomials in order to then add a common factor to them. After this, it may turn out that there will be factors that can be reduced, or again the bracketing of common elements will be repeated.

Try to consider abbreviated multiplication formulas in writing. With their help, you can easily convert polynomials into factors.

Sequence of operations with fractions with powers

In order to easily understand the question of how to reduce a fraction with powers, you need to firmly remember the basic operations with them. The first of these is related to the multiplication of powers. In this case, if the bases are the same, the indicators must be added.

The second is division. Again, for those that have the same reasons, the indicators will need to be subtracted. Moreover, you need to subtract from the number that is in the dividend, and not vice versa.

The third is exponentiation. In this situation, the indicators are multiplied.

Successful reduction will also require the ability to reduce degrees to on the same grounds. That is, to see that four is two squared. Or 27 - the cube of three. Because reducing 9 squared and 3 cubed is difficult. But if we transform the first expression as (3 2) 2, then the reduction will be successful.

Division and the numerator and denominator of the fraction on their common divisor, different from one, is called reducing a fraction.

To reduce a common fraction, you need to divide its numerator and denominator by the same natural number.

This number is the greatest common divisor of the numerator and denominator of the given fraction.

The following are possible decision recording forms Examples for reducing common fractions.

The student has the right to choose any form of recording.

Examples. Simplify fractions.

Reduce the fraction by 3 (divide the numerator by 3;

divide the denominator by 3).

Reduce the fraction by 7.

We perform the indicated actions in the numerator and denominator of the fraction.

The resulting fraction is reduced by 5.

Let's reduce this fraction 4) on 5·7³- the greatest common divisor (GCD) of the numerator and denominator, which consists of the common factors of the numerator and denominator, taken to the power with the smallest exponent.

Let's factor the numerator and denominator of this fraction into prime factors.

We get: 756=2²·3³·7 And 1176=2³·3·7².

Determine the GCD (greatest common divisor) of the numerator and denominator of the fraction 5) .

This is the product of common factors taken with the lowest exponents.

gcd(756, 1176)= 2²·3·7.

We divide the numerator and denominator of this fraction by their gcd, i.e. by 2²·3·7 we get an irreducible fraction 9/14 .

Or it was possible to write the decomposition of the numerator and denominator in the form of a product of prime factors, without using the concept of power, and then reduce the fraction by crossing out the same factors in the numerator and denominator. When there are no identical factors left, we multiply the remaining factors separately in the numerator and separately in the denominator and write out the resulting fraction 9/14 .

And finally, it was possible to reduce this fraction 5) gradually, applying signs of dividing numbers to both the numerator and denominator of the fraction. We reason like this: numbers 756 And 1176 end in an even number, which means both are divisible by 2 . We reduce the fraction by 2 . The numerator and denominator of the new fraction are numbers 378 And 588 also divided into 2 . We reduce the fraction by 2 . We notice that the number 294 - even, and 189 is odd, and reduction by 2 is no longer possible. Let's check the divisibility of numbers 189 And 294 on 3 .

(1+8+9)=18 is divisible by 3 and (2+9+4)=15 is divisible by 3, hence the numbers themselves 189 And 294 are divided into 3 . We reduce the fraction by 3 . Further, 63 is divisible by 3 and 98 - No. Let's look at other prime factors. Both numbers are divisible by 7 . We reduce the fraction by 7 and we get the irreducible fraction 9/14 .

So, we already know that the numerator and denominator of a fraction can be multiplied and divided by the same number, the fraction will not change. Let's consider three approaches:

Approach one.

To reduce, divide the numerator and denominator by a common divisor. Let's look at examples:

Let's shorten:

In the examples given, we immediately see which divisors to take for reduction. The process is simple - we go through 2,3,4,5 and so on. In most school course examples, this is quite enough. But if it’s a fraction:

Here the process of selecting divisors can take a long time;). Of course, such examples are outside the school curriculum, but you need to be able to cope with them. Below we will look at how this is done. For now, let's get back to the downsizing process.

As discussed above, in order to reduce a fraction, we divided by the common divisor(s) we determined. Everything is correct! One has only to add signs of divisibility of numbers:

- if the number is even, then it is divisible by 2.

- if a number from the last two digits is divisible by 4, then the number itself is divisible by 4.

— if the sum of the digits that make up the number is divisible by 3, then the number itself is divisible by 3. For example, 125031, 1+2+5+0+3+1=12. Twelve is divisible by 3, so 123031 is divisible by 3.

- if the end of a number is 5 or 0, then the number is divisible by 5.

— if the sum of the digits that make up the number is divisible by 9, then the number itself is divisible by 9. For example, 625032 =.> 6+2+5+0+3+2=18. Eighteen is divisible by 9, which means 623032 is divisible by 9.

Second approach.

To put it briefly, in fact, the whole action comes down to factoring the numerator and denominator and then reducing equal factors in the numerator and denominator (this approach is a consequence of the first approach):

Visually, in order to avoid confusion and mistakes, equal factors are simply crossed out. Question - how to factor a number? It is necessary to determine all divisors by searching. This is a separate topic, it is not complicated, look up the information in a textbook or on the Internet. You won't encounter any big problems with factoring numbers that are present in school fractions.

Formally, the reduction principle can be written as follows:

Approach three.

Here is the most interesting thing for the advanced and those who want to become one. Let's reduce the fraction 143/273. Try it yourself! Well, how did it happen quickly? Now look!

We turn it over (we change places of the numerator and denominator). Divide the resulting fraction with a corner and convert it to mixed number, that is, we select the whole part:

It's already easier. We see that the numerator and denominator can be reduced by 13:

Now don’t forget to turn the fraction back over again, let’s write down the whole chain:

Checked - it takes less time than searching through and checking divisors. Let's return to our two examples:

First. Divide with a corner (not on a calculator), we get:

This fraction is simpler, of course, but the reduction is again a problem. Now we separately analyze the fraction 1273/1463 and turn it over:

It's easier here. We can consider a divisor such as 19. The rest are not suitable, this is clear: 190:19 = 10, 1273:19 = 67. Hurray! Let's write down:

Next example. Let's shorten 88179/2717.

Divide, we get:

Separately, we analyze the fraction 1235/2717 and turn it over:

We can consider a divisor such as 13 (up to 13 is not suitable):

Numerator 247:13=19 Denominator 1235:13=95

*During the process we saw another divisor equal to 19. It turns out that:

Now we write down the original number:

And it doesn’t matter what is larger in the fraction - the numerator or the denominator, if it is the denominator, then we turn it over and act as described. This way we can reduce any fraction; the third approach can be called universal.

Of course, the two examples discussed above are not simple examples. Let's try this technology on the “simple” fractions we have already considered:

Two quarters.

Seventy-two sixties. The numerator is greater than the denominator; there is no need to reverse it:

Of course, the third approach was applied to such simple examples just as an alternative. The method, as already said, is universal, but not convenient and correct for all fractions, especially for simple ones.

The variety of fractions is great. It is important that you understand the principles. There is simply no strict rule for working with fractions. We looked, figured out how it would be more convenient to act, and moved forward. With practice, skill will come and you will crack them like seeds.

Conclusion:

If you see a common divisor(s) for the numerator and denominator, use them to reduce.

If you know how to quickly factor a number, then factor the numerator and denominator, then reduce.

If you can’t determine the common divisor, then use the third approach.

*To reduce fractions, it is important to master the principles of reduction, understand the basic property of a fraction, know approaches to solving, and be extremely careful when making calculations.

And remember! It is customary to reduce a fraction until it stops, that is, reduce it as long as there is a common divisor.

Sincerely, Alexander Krutitskikh.