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.
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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 the common fraction 6 24 and reduce 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 large 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 by 2 again and obtain an irreducible fraction:
10 22 = 10 ÷ 2 22 ÷ 2 = 5 11
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Many students make the same mistakes when working with fractions. And all because they forget the basic rules arithmetic. Today we will repeat these rules on specific tasks that I give in my classes.
Here is the task that I offer to everyone who is preparing for the Unified State Exam in mathematics:
Task. A porpoise eats 150 grams of food per day. But she grew up and began to eat 20% more. How many grams of feed does the pig eat now?
Wrong decision. This is a percentage problem that boils down to the equation:
Many (very many) reduce the number 100 in the numerator and denominator of a fraction:
This is the mistake my student made right on the day of writing this article. Numbers that have been truncated are marked in red.
Needless to say, the answer was wrong. Judge for yourself: the pig ate 150 grams, but began to eat 3150 grams. The increase is not 20%, but 21 times, i.e. by 2000%.
To avoid such misunderstandings, remember the basic rule:
Only multipliers can be reduced. The terms cannot be reduced!
Thus, the correct solution to the previous problem looks like this:
Numbers that are abbreviated in the numerator and denominator are marked in red. As you can see, the numerator is a product, the denominator is an ordinary number. Therefore, the reduction is completely legal.
Working with proportions
Another problem area is proportions. Especially when the variable is on both sides. For example:
Task. Solve the equation:
Wrong solution - some people are literally itching to shorten everything by m:
Reduced variables are shown in red. The expression 1/4 = 1/5 turns out to be complete nonsense, these numbers are never equal.
And now - the right decision. Essentially it's ordinary linear equation. It can be solved either by moving all elements to one side, or by the basic property of proportion:
Many readers will object: “Where is the mistake in the first solution?” Well, let's find out. Let's remember the rule for working with equations:
Any equation can be divided and multiplied by any number, non-zero.
Did you miss the trick? You can only divide by numbers non-zero. In particular, you can divide by a variable m only if m != 0. But what if m = 0? Let's substitute and check:
We received the correct numerical equality, i.e. m = 0 is the root of the equation. For the remaining m != 0 we obtain an expression of the form 1/4 = 1/5, which is naturally incorrect. Thus, there are no non-zero roots.
Conclusions: putting it all together
So, to solve fractional rational equations, remember three rules:
- Only multipliers can be reduced. Addends are not possible. Therefore, learn to factor the numerator and denominator;
- The main property of proportion: the product of the extreme elements is equal to the product of the middle ones;
- Equations can only be multiplied and divided by numbers k other than zero. The case k = 0 must be checked separately.
Remember these rules and don't make mistakes.
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 .
Online calculator performs reduction of 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 a detailed solution that 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 a 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.
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. A proper fraction is a fraction whose numerator is less than its denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. An improper fraction is a fraction in which the numerator is greater than or equal to the denominator, so an improper fraction is always greater than or equal to one. 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. Example of mixed fractions: 1¼, 2½, 4¾.
III. Note:
- The source data block is highlighted in 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.
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 entirely. 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 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 a 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 is less than the denominator, are called correct fractions.
As you know, any common fraction, both proper and improper, can be thought of 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, they must first be reduced 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, 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 a 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.