Reducing expressions with fractions online. Reducing algebraic fractions

To understand how to reduce fractions, let's first look at an example.

To reduce a fraction means to divide the numerator and denominator by the same thing. Both 360 and 420 end in a digit, so we can reduce this fraction by 2. In the new fraction, both 180 and 210 are also divisible by 2, so we reduce this fraction by 2. In the numbers 90 and 105, the sum of the digits is divisible by 3, so both these numbers are divisible by 3, we reduce the fraction by 3. In the new fraction, 30 and 35 end in 0 and 5, which means both numbers are divisible by 5, so we reduce the fraction by 5. The resulting fraction of six-sevenths is irreducible. This is the final answer.

We can arrive at the same answer in a different way.

Both 360 and 420 end in zero, which means they are divisible by 10. We reduce the fraction by 10. In the new fraction, both the numerator 36 and the denominator 42 are divided by 2. We reduce the fraction by 2. In the next fraction, both the numerator 18 and the denominator 21 are divided by 3, which means we reduce the fraction by 3. We came to the result - six sevenths.

And one more solution.

Next time we'll look at examples of reducing fractions.

Convenient and simple online calculator fractions with detailed solutions Maybe:

• Add, subtract, multiply and divide fractions online,
• Receive ready-made solution fractions with a picture and it is convenient to transfer it.

The result of solving fractions will be here...

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Fraction sign "/" + - * :
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Our online fraction calculator has quick input. To solve fractions, for example, simply write 1/2+2/7 into the calculator and press the " Solve fractions". The calculator will write to you detailed solution fractions and will issue an easy-to-copy image.

Signs used for writing in a calculator

Features of the online fraction calculator

If you need to solve negative fractions, simply use the properties of minus. When multiplying and dividing negative fractions two negatives make an affirmative. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplying or dividing, then simply remove the minus and then add it to the answer. When adding negative fractions, the result will be the same as if you were adding the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were swapped and made positive. That is, minus by minus in in this case gives a plus, but rearranging the terms does not change the sum. We use the same rules when subtracting fractions, one of which is negative.

To solve mixed fractions (fractions in which the whole part is isolated), simply fit the whole part into the fraction. To do this, multiply the whole part by the denominator and add to the numerator.

If you need to solve 3 or more fractions online, you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer you get, and so on. Perform the operations one by one, 2 fractions at a time, and eventually you will get the correct answer.

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 the numerator and denominator of a fraction into the same thing 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 expand the fraction into prime factors and reduce 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 initial 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 largest common divisor(NOD) 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.

1. You need to find the greatest common divisor for the numerator and denominator.
2. 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: \(\frac(19)(21)\) irreducible fraction.

Reducing improper fractions.

How to reduce an 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 follow the same rules as ordinary fractions. The only difference is that we can do not touch the whole part, but reduce the fractional part or Convert a mixed fraction to an improper fraction, reduce it and convert it 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 abbreviating same numbers In our case, the numerator and denominator have 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 common multiplier 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) \(\frac(100)(250)=\frac(\color(red) (2 \times 5 \times 5) \times 2)(\color(red) (2 \times 5 \times 5) \ times 5)=\frac(2)(5)\)

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 is 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 number ends with 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 will not encounter any great 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 it to 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.

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, - 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 equal to 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 partial 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 suggests that the part should always be less than the whole, but then what about fractions such as, for example, \(\frac(5)(5)\) 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 is less than the denominator, are 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.

With fractional numbers, as with natural numbers, you can do arithmetic operations. Let's look at adding fractions first. Easily add fractions with same 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, commutative and associative properties addition.

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 it is said that improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction 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, 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 mutually 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 mixed fraction, then in order to use the rule for dividing fractions, it must first be represented as an improper fraction.