Online calculator for reducing fractions. Reducing fractions, rules and examples of reducing fractions
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 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.
- You 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 as a result of division.
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 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 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 in detail at how reducing fractions. First, let's discuss what is called reducing a fraction. After this, let's talk about reducing a reducible fraction to an irreducible form. Next we will obtain the rule for reducing fractions and, finally, consider examples of the application of this rule.
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What does it mean to reduce a fraction?
We know that ordinary fractions are divided into reducible and irreducible fractions. You can guess from the names that reducible fractions can be reduced, but irreducible fractions cannot.
What does it mean to reduce a fraction? Reduce fraction- this means dividing its numerator and denominator by their positive and different from unity. It is clear that as a result of reducing a fraction, a new fraction is obtained with a smaller numerator and denominator, and, due to the basic property of the fraction, the resulting fraction is equal to the original one.
For example, let's reduce common fraction 8/24 by dividing its numerator and denominator by 2. In other words, let's reduce the fraction 8/24 by 2. Since 8:2=4 and 24:2=12, this reduction results in the fraction 4/12, which is equal to the original fraction 8/24 (see equal and unequal fractions). As a result, we have .
Reducing ordinary fractions to irreducible form
Typically, the ultimate goal of reducing a fraction is to obtain an irreducible fraction that is equal to the original reducible fraction. This goal can be achieved by reducing the original reducible fraction into its numerator and denominator. As a result of such a reduction, an irreducible fraction is always obtained. Indeed, a fraction 
is irreducible, since it is known that 
And 
- . Here we will say that the greatest common divisor of the numerator and denominator of a fraction is the largest number by which this fraction can be reduced.
So, reducing a common fraction to an irreducible form consists of dividing the numerator and denominator of the original reducible fraction by their gcd.
Let's look at an example, for which we return to the fraction 8/24 and reduce it by the greatest common divisor of the numbers 8 and 24, which is equal to 8. Since 8:8=1 and 24:8=3, we come to the irreducible fraction 1/3. So, .
Note that the phrase “reduce a fraction” often means reducing the original fraction to its irreducible form. In other words, reducing a fraction very often refers to dividing the numerator and denominator by their greatest common factor (rather than by any common factor).
How to reduce a fraction? Rules and examples of reducing fractions
All that remains is to look at the rule for reducing fractions, which explains how to reduce a given fraction.
Rule for reducing fractions consists of two steps:
- firstly, the gcd of the numerator and denominator of the fraction is found;
- secondly, the numerator and denominator of the fraction are divided by their gcd, which gives an irreducible fraction equal to the original one.
Let's sort it out example of reducing a fraction according to the stated rule.
Example.
Reduce the fraction 182/195.
Solution.
Let's carry out both steps prescribed by the rule for reducing a fraction.
First we find GCD(182, 195) . It is most convenient to use the Euclid algorithm (see): 195=182·1+13, 182=13·14, that is, GCD(182, 195)=13.
Now we divide the numerator and denominator of the fraction 182/195 by 13, and we get the irreducible fraction 14/15, which is equal to the original fraction. This completes the reduction of the fraction.
Briefly, the solution can be written as follows: .
Answer:
This is where we can finish reducing fractions. But to complete the picture, let's look at two more ways to reduce fractions, which are usually used in easy cases.
Sometimes the numerator and denominator of the fraction being reduced is not difficult. Reducing a fraction in this case is very simple: you just need to remove all common factors from the numerator and denominator.
It is worth noting that this method follows directly from the rule of reducing fractions, since the product of all common prime factors of the numerator and denominator is equal to their greatest common divisor.
Let's look at the solution to the example.
Example.
Reduce the fraction 360/2 940.
Solution.
Let's factor the numerator and denominator into simple factors: 360=2·2·2·3·3·5 and 2,940=2·2·3·5·7·7. Thus, 
.
Now we get rid of the common factors in the numerator and denominator; for convenience, we simply cross them out: 
.
Finally, we multiply the remaining factors: , and the reduction of the fraction is completed.
Here is a short summary of the solution: 
.
Answer:
Let's consider another way to reduce a fraction, which consists of sequential reduction. Here, at each step, the fraction is reduced by some common divisor of the numerator and denominator, which is either obvious or easily determined using
Convenient and simple online calculator fractions with detailed solutions Maybe:
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The result of solving fractions will be here...
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Fraction sign "/" + - * :
_erase Clear
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
You can type an example for a solution either from the keyboard or using buttons.
Features of the online fraction calculator
The fraction calculator can only perform operations on 2 simple fractions. They can be either correct (the numerator is less than the denominator) or incorrect (the numerator is greater than the denominator). The numbers in the numerator and denominators cannot be negative or greater than 999.Our online calculator solves fractions and gives the answer to the right kind- reduces the fraction and selects the whole part, if necessary.
If you need to solve negative fractions, just use the properties of minus. When multiplying and dividing negative fractions, minus by minus gives plus. 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.
For solutions mixed fractions(fractions in which whole part) just drive the whole part into a 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.
So we got to the reduction. The basic property of a fraction is applied here. BUT! Not so simple. With many fractions (including from school course) it is quite possible to get by with them. What if we take fractions that are “more abrupt”? Let's take a closer look! I recommend looking at materials with fractions.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 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 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). We divide the resulting fraction with a corner and convert it into a 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 flip the fraction back 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.
