How to abbreviate in large equations. Reducing algebraic fractions

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. Let's think like this: numbers 756 And 1176 end in an even number, which means both are divisible by 2 . We reduce the fraction by 2 . Numerator and denominator new fraction- 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 . Next, 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 .

Online calculator performs reduction algebraic fractions in accordance with the rule of reducing fractions: replacing the original fraction with an equal fraction, but with a smaller numerator and denominator, i.e. Simultaneously dividing the numerator and denominator of a fraction by their common greatest common factor (GCD). The calculator also displays detailed solution, which will help you understand the sequence of the reduction.

Given:

Solution:

Performing fraction reduction

checking the possibility of performing algebraic fraction reduction

1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction

determining the greatest common divisor (GCD) of the numerator and denominator of an algebraic fraction

2) Reducing the numerator and denominator of a fraction

reducing the numerator and denominator of an algebraic fraction

3) Selecting the whole part of a fraction

separating the whole part of an algebraic fraction

4) Converting an algebraic fraction to a decimal fraction

converting an algebraic fraction to decimal

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I. Procedure for reducing an algebraic fraction using an online calculator:

To reduce an algebraic fraction, enter the values of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the whole part of the fraction. If the fraction is simple, then leave the whole part field blank.
To specify a negative fraction, place a minus sign on the whole part of the fraction.
Depending on the specified algebraic fraction, the following sequence of actions is automatically performed:

determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
reducing the numerator and denominator of a fraction by gcd;
highlighting the whole part of a fraction, if the numerator of the final fraction is greater than the denominator.
converting the final algebraic fraction to a decimal fraction rounded to the nearest hundredth.

The reduction may result in an improper fraction. In this case, the whole part of the final improper fraction will be highlighted and the final fraction will be converted to a proper fraction.

II. For reference:

A fraction is a number consisting of one or more parts (fractions) of a unit. A common fraction (simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction) separated by a horizontal bar (the fraction bar) indicating the division sign. The numerator of a fraction is the number above the fraction line. The numerator shows how many shares were taken from the whole. The denominator of a fraction is the number below the fraction line. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have a whole part. A simple fraction can be proper or improper. proper fraction - a fraction whose numerator is less than the denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. improper fraction - a fraction in which the numerator is greater than or equal to the denominator, therefore an improper fraction is always more than one or equal to it. Example of improper fractions: 7/6, 8/7, 13/13. mixed fraction is a number that contains a whole number and a proper fraction, and denotes the sum of that whole number and the proper fraction. Any mixed fraction can be converted to an improper fraction simple fraction. Example of mixed fractions: 1¼, 2½, 4¾.

III. Note:

Source data block highlighted yellow , the block of intermediate calculations is highlighted in blue, the solution block is highlighted in green.
To add, subtract, multiply and divide common or mixed fractions, use the online fraction calculator with detailed solutions.

Last time we made a plan, following which you can learn how to quickly reduce fractions. Now let's look at specific examples of reducing 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.

Reducing fractions is necessary in order to reduce the fraction to more simple view, for example, in the answer obtained as a result of solving an expression.

Reducing fractions, definition and formula.

What is reducing fractions? What does it mean to reduce a fraction?

Definition:
Reducing Fractions- this is the division of a fraction's numerator and denominator by the same positive number not equal to zero and one. As a result of the reduction, a fraction with a smaller numerator and denominator is obtained, equal to the previous fraction according to.

Formula for reducing fractions main property rational numbers.

\(\frac(p \times n)(q \times n)=\frac(p)(q)\)

Let's look at an example:
Reduce the fraction \(\frac(9)(15)\)

Solution:
We can factor a fraction into prime factors and cancel common factors.

\(\frac(9)(15)=\frac(3 \times 3)(5 \times 3)=\frac(3)(5) \times \color(red) (\frac(3)(3) )=\frac(3)(5) \times 1=\frac(3)(5)\)

Answer: after reduction we got the fraction \(\frac(3)(5)\). According to the basic property of rational numbers, the original and resulting fractions are equal.

\(\frac(9)(15)=\frac(3)(5)\)

How to reduce fractions? Reducing a fraction to its irreducible form.

To get an irreducible fraction as a result, we need find the greatest common divisor (GCD) for the numerator and denominator of the fraction.

There are several ways to find GCD; in the example we will use the decomposition of numbers into prime factors.

Get the irreducible fraction \(\frac(48)(136)\).

Solution:
Let's find GCD(48, 136). Let's write the numbers 48 and 136 into prime factors.
48=2⋅2⋅2⋅2⋅3
136=2⋅2⋅2⋅17
GCD(48, 136)= 2⋅2⋅2=6

\(\frac(48)(136)=\frac(\color(red) (2 \times 2 \times 2) \times 2 \times 3)(\color(red) (2 \times 2 \times 2) \times 17)=\frac(\color(red) (6) \times 2 \times 3)(\color(red) (6) \times 17)=\frac(2 \times 3)(17)=\ frac(6)(17)\)

The rule for reducing a fraction to an irreducible form.

We need to find the greatest common divisor for the numerator and denominator.
You need to divide the numerator and denominator by the greatest common divisor to obtain an irreducible fraction.

Example:
Reduce the fraction \(\frac(152)(168)\).

Solution:
Let's find GCD(152, 168). Let's write the numbers 152 and 168 into prime factors.
152=2⋅2⋅2⋅19
168=2⋅2⋅2⋅3⋅7
GCD(152, 168)= 2⋅2⋅2=6

\(\frac(152)(168)=\frac(\color(red) (6) \times 19)(\color(red) (6) \times 21)=\frac(19)(21)\)

Answer: \(\frac(19)(21)\) is an irreducible fraction.

Reducing improper fractions.

How to cut improper fraction?
The rules for reducing fractions are the same for proper and improper fractions.

Let's look at an example:
Reduce the improper fraction \(\frac(44)(32)\).

Solution:
Let's write the numerator and denominator into simple factors. And then we’ll reduce the common factors.

\(\frac(44)(32)=\frac(\color(red) (2 \times 2 ) \times 11)(\color(red) (2 \times 2 ) \times 2 \times 2 \times 2 )=\frac(11)(2 \times 2 \times 2)=\frac(11)(8)\)

Reducing mixed fractions.

Mixed fractions using the same rules as common fractions. The only difference is that we can do not touch the whole part, but reduce the fractional part or mixed fraction convert to an improper fraction, reduce and convert back to a proper fraction.

Let's look at an example:
Cancel the mixed fraction \(2\frac(30)(45)\).

Solution:
Let's solve it in two ways:
First way:
Let's write the fractional part into simple factors, but we won't touch the whole part.

\(2\frac(30)(45)=2\frac(2 \times \color(red) (5 \times 3))(3 \times \color(red) (5 \times 3))=2\ frac(2)(3)\)

Second way:
Let's first convert it to an improper fraction, and then write it into prime factors and reduce. Let's convert the resulting improper fraction into a proper fraction.

\(2\frac(30)(45)=\frac(45 \times 2 + 30)(45)=\frac(120)(45)=\frac(2 \times \color(red) (5 \times 3) \times 2 \times 2)(3 \times \color(red) (3 \times 5))=\frac(2 \times 2 \times 2)(3)=\frac(8)(3)= 2\frac(2)(3)\)

Related questions:
Can you reduce fractions when adding or subtracting?
Answer: no, you must first add or subtract fractions according to the rules, and only then reduce them. Let's look at an example:

Evaluate the expression \(\frac(50+20-10)(20)\) .

Solution:
They often make the mistake of reducing the same numbers in the numerator and denominator, in our case the number 20, but they cannot be reduced until you have completed the addition and subtraction.

\(\frac(50+\color(red) (20)-10)(\color(red) (20))=\frac(60)(20)=\frac(3 \times 20)(20)= \frac(3)(1)=3\)

What numbers can you reduce a fraction by?
Answer: You can reduce a fraction by the greatest common factor or the common divisor of the numerator and denominator. For example, the fraction \(\frac(100)(150)\).

Let's write the numbers 100 and 150 into prime factors.
100=2⋅2⋅5⋅5
150=2⋅5⋅5⋅3
The greatest common divisor will be the number GCD(100, 150)= 2⋅5⋅5=50

\(\frac(100)(150)=\frac(2 \times 50)(3 \times 50)=\frac(2)(3)\)

We got the irreducible fraction \(\frac(2)(3)\).

But it is not necessary to always divide by gcd; an irreducible fraction is not always needed; you can reduce the fraction by a simple divisor of the numerator and denominator. For example, the number 100 and 150 have a common divisor of 2. Let's reduce the fraction \(\frac(100)(150)\) by 2.

\(\frac(100)(150)=\frac(2 \times 50)(2 \times 75)=\frac(50)(75)\)

We got the reducible fraction \(\frac(50)(75)\).

What fractions can be reduced?
Answer: You can reduce fractions in which the numerator and denominator have a common divisor. For example, the fraction \(\frac(4)(8)\). The number 4 and 8 have a number by which they are both divisible - the number 2. Therefore, such a fraction can be reduced by the number 2.

Example:
Compare the two fractions \(\frac(2)(3)\) and \(\frac(8)(12)\).

These two fractions are equal. Let's take a closer look at the fraction \(\frac(8)(12)\):

\(\frac(8)(12)=\frac(2 \times 4)(3 \times 4)=\frac(2)(3) \times \frac(4)(4)=\frac(2) (3) \times 1=\frac(2)(3)\)

From here we get, \(\frac(8)(12)=\frac(2)(3)\)

Two fractions are equal if and only if one of them is obtained by reducing the other fraction by the common factor of the numerator and denominator.

Example:
If possible, reduce the following fractions: a) \(\frac(90)(65)\) b) \(\frac(27)(63)\) c) \(\frac(17)(100)\) d) \(\frac(100)(250)\)

Solution:
a) \(\frac(90)(65)=\frac(2 \times \color(red) (5) \times 3 \times 3)(\color(red) (5) \times 13)=\frac (2 \times 3 \times 3)(13)=\frac(18)(13)\)
b) \(\frac(27)(63)=\frac(\color(red) (3 \times 3) \times 3)(\color(red) (3 \times 3) \times 7)=\frac (3)(7)\)
c) \(\frac(17)(100)\) irreducible fraction
d) \(\frac(100)(250)=\frac(\color(red) (2 \times 5 \times 5) \times 2)(\color(red) (2 \times 5 \times 5) \ times 5)=\frac(2)(5)\)

In this article we will look at basic operations with algebraic fractions:

reducing fractions
multiplying fractions
dividing fractions

Let's start with reduction of algebraic fractions.

It would seem algorithm obvious.

To reduce algebraic fractions, need to

1. Factor the numerator and denominator of the fraction.

2. Reduce equal factors.

However, schoolchildren often make the mistake of “reducing” not the factors, but the terms. For example, there are amateurs who “reduce” fractions by and get as a result , which, of course, is not true.

Let's look at examples:

1. Reduce a fraction:

1. Let us factorize the numerator using the formula of the square of the sum, and the denominator using the formula of the difference of squares

2. Divide the numerator and denominator by

2. Reduce a fraction:

1. Let's factorize the numerator. Since the numerator contains four terms, we use grouping.

2. Let's factorize the denominator. We can also use grouping.

3. Let's write down the fraction that we got and reduce the same factors:

Multiplying algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator by the numerator, and multiply the denominator by the denominator.

Important! There is no need to rush to multiply the numerator and denominator of a fraction. After we have written down the product of the numerators of the fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.

Let's look at examples:

3. Simplify the expression:

1. Let’s write the product of fractions: in the numerator the product of the numerators, and in the denominator the product of the denominators:

2. Let's factorize each bracket:

Now we need to reduce the same factors. Note that the expressions and differ only in sign: and as a result of dividing the first expression by the second we get -1.