How to bring fractions to the smallest standard. Bringing the fraction to the smallest standard of the standard: apply the rule to the solution. Zagalny banner: vyznachennya, butt
In order to bring the fractions to the smallest common standard, it is necessary: 1) to know the least significant multiples of the signs of these fractions, it will be the least significant standard. 2) to know the additional multiplier for skin shots, which is to add a new standard to the skin shots standard. 3) multiply the number and standard of the skin shot by the additional multiplier.
apply. Bring such fractions to the smallest common standard.
We know the least significant multiple of znamennikiv: NOK (5; 4) \u003d 20, so 20 is the least number, as it divides i by 5 i by 4. We know for the 1st fraction the additive multiplier 4 (20 : 5 = 4). For the 2nd fraction, the additional multiplier is 5 (20 : 4 = 5). We multiply the number and banner of the 1st fraction by 4, and the number and banner of the 2nd fraction by 5. We brought qi fractions to the smallest zagalny banner ( 20 ).
The smallest blatant sign of these shots is the number 8, the shards of 8 are divided by 4 and itself. There will be no additional multiplier up to the 1st fraction (or you can say that there are more than one), up to the 2nd fraction the additional multiplier is more expensive 2 (8 : 4 = 2). We multiply the number and standard of the 2nd fraction by 2. We brought the given fraction to the smallest standard of the standard ( 8 ).
Given fractions are non-short.
The 1st drop is fast for 4, and the 2nd drop is fast for 2. ( div. put on fast shots of great fractions: Site map → 5.4.2. Apply fast fractions). Known NOC(16) ; 20)=2 4 · 5=16· 5 = 80. Additional multiplier for the 1st fraction is 5 (80 : 16 = 5). Additional multiplier for the 2nd fraction is 4 (80 : 20 = 4). We multiply the number and banner of the 1st fraction by 5, and the number and banner of the 2nd fraction by 4. We brought qi fractions to the smallest zagalny banner ( 80 ).
We know the smallest common banner NOZ (5 ; 6 і 15) = LCM (5 ; 6 and 15) \u003d 30. Additional multiplier up to the 1st fraction is additional 6 (30 : 5 \u003d 6), an additional multiplier up to the 2nd fraction is additional 5 (30 : 6=5), additional multiplier up to the 3rd fraction adds 2 (30 : 15 = 2). We multiply the number and banner of the 1st fraction by 6, the number and banner of the 2nd fraction by 5, the number and banner of the 3rd fraction by 2. 30 ).
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How to bring fractions to a sleeping banner
If the great fractions have the same banners, then it seems that fractions brought to the standard banner.
butt 1
For example, the fractions $\frac(3)(18)$ and $\frac(20)(18)$ have the same standard. To say that the stink of a sleeping banner of $18$. The fractions $\frac(1)(29)$, $\frac(7)(29)$ and $\frac(100)(29)$ have the same standard. To say that the stink of a sleeping banner of $29$.
As for the shots, the banners are not the same, they can be called up to a sleeping banner. For whom it is necessary to multiply their numerals and banners by songs of additional multipliers.
butt 2
How to bring two fractions $\frac(6)(11)$ and $\frac(2)(7)$ to a double standard.
Solution.
Multiplying the fractions $\frac(6)(11)$ and $\frac(2)(7)$ by the additional factors $7$ and $11$ reducibly and reducibly їх to the full standard $77$:
$\frac(6\cdot 7)(11\cdot 7)=\frac(42)(77)$
$\frac(2\cdot 11)(7\cdot 11)=\frac(22)(77)$
in such a manner, brought shots to a sleeping banner name the multiples of the numeral and the standard of given fractions on additional multipliers, as a result, it is possible to take fractions with the same standard.
Spіlny znamennik
Appointment 1
Be-yaké a positively solemn multiple of all signs in a given set of fractions to name sleeping banner.
Otherwise, it seems, the blazing banner of the tasks of the most significant fractions - be it a natural number, which can be divided into all the banners of the tasks of the fractions.
The names of the names of the anonymous commemorative commemoratives of this set of shots.
butt 3
Find the common denominators of the fractions $\frac(3)(7)$ and $\frac(2)(13)$.
Solution.
Znamenniks make fractions, equals $7$ and $13$ obviously. Positive global multiples of $2$ and $5$ are equal to $91, 182, 273, 364$ and so on.
Whether it is possible to win from these numbers as a common banner of fractions $\frac(3)(7)$ and $\frac(2)(13)$.
butt 4
Find out how the fractions $\frac(1)(2)$, $\frac(16)(7)$ and $\frac(11)(9)$ can be reduced to the double standard $252$.
Solution.
In order to signify, how to bring the drib to the double banner of $252$, it is necessary to convert є $252$ to the top multiple banners of $2, 7$ and $9$. For whom we divide the number of $252$ into skins from the banners:
$\frac(252)(2)=126,$ $\frac(252)(7)=36$, $\frac(252)(9)=28$.
The number $252$ is spread across all the banners, that is. Let's take a big multiple of the numbers $2, 7$ and $9$. Also, qi fractions $\frac(1)(2)$, $\frac(16)(7)$ and $\frac(11)(9)$ can be reduced to the double standard $252$.
Suggestion: you can.
The smallest sleeping banner
Appointment 2
In the midst of the successive znamenniks in given fractions, one can see the smallest natural number, as they call the smallest sleeping banner.
Because LCM is the smallest positive reference to the given set of numbers, then the LCM of the standard of given fractions is the smallest positive reference of these fractions.
Also, in order to know the smallest scorching standard of shots, it is necessary to know the NOC of the signs of these shots.
butt 5
Given a fraction $\frac(4)(15)$ i $\frac(37)(18)$. Know their smallest sleeping banner.
Solution.
The denominators of these shots cost $15$ and $18$. We know the smallest common banner as the LCM of the numbers $15$ and $18$. Victory for which layout of numbers on simple multipliers:
$15=3\cdot 5$, $18=2\cdot 3\cdot 3$
$NOK(15, 18) = 2 cdot 3 cdot 3 cdot 5 = $90.
Suggestion: $90.
The rule of reduction of fractions to the smallest common banner
Most of the time, the task of algebra, geometry, physics is completed. Ring out simple fractions to the smallest standard banner, but not to the smallest standard banner.
Algorithm:
- For the help of the NOC bannermen in the tasks of shots, know the smallest blazing bannerman.
- 2. Calculate the additional multiplier for given fractions. For this kind of knowledge, it is necessary to add skin shot to the banner of the smallest blazon. Otrimane number will be an additional multiplier of this fraction.
- Multiply by the knowledge of the additional multiplier, the number and the banner of the skin shot.
butt 6
Find the smallest hot number of fractions $\frac(4)(16)$ і $\frac(3)(22)$ and bring the fraction to the new one.
Solution.
Speeding up by the algorithm of reduction of fractions to the smallest common banner.
The least calculable multiple of the numbers $16$ and $22$:
Let's decompose the banners into simple multipliers: $16 = 2 \ cdot 2 \ cdot 2 \ cdot 2 $, $ 22 = 2 \ cdot 11 $.
$ LCM (16, 22) = 2 cdot 2 cdot 2 cdot 2 cdot 11 = $176.
Calculate the additive multipliers for skin shot:
$176\div 16=11$ – for the fraction $\frac(4)(16)$;
$176\div 22=8$ – for the fraction $\frac(3)(22)$.
Multiplying the numerals and the denominators of the fractions $\frac(4)(16)$ and $\frac(3)(22)$ by the additional factors $11$ and $8$ is correct. We take:
$\frac(4)(16)=\frac(4\cdot 11)(16\cdot 11)=\frac(44)(176)$
$\frac(3)(22)=\frac(3\cdot 8)(22\cdot 8)=\frac(24)(176)$
Shot of fractions brought to the smallest spilny banner of $176$.
Suggestion: $ frac (4) (16) = frac (44) (176) $, $ frac (3) (22) = frac (24) (176) $.
Sometimes, in order to make the smallest blatant banner, it is necessary to carry out a series of laborious calculations, which can really correct the meta rozv'azannya. With such a vapadka, you can speed up in the simplest way - to call fractions to a double banner, which is the same as the banner of these shots.
How to bring algebraic (rational) fractions to a double banner?
1) It is necessary to try one of the best ways to stand in the banners of shots.
2) The smallest hot banner (NOZ) is added up with all multiples taken from the largest step.
The smallest blatant banner for numbers is clearly known as the smallest number, as other numbers can be shared.
3) To know the additional multiplier to skin shot, you need to add a new banner to the old one.
4) The chiselnik and banner of the cob shot are multiplied by the additional multiplier.
Let's take a look at the given algebraic fractions to the double banner.
In order to know the common banner for numbers, we choose more and reverify, which one is divided into less. 15 to 9 is not subdivision. Multiply 15 by 2 and check if the number is divisible by 9. 30 cannot be divisible by 9. Multiplying 15 by 3 and reversing how the number is divided by 9. 45 by 9 is divisible by 9, again, the blatant banner for the numbers is more expensive 45.
The smallest blazed banner is made up of all the multipliers taken by the greatest of the world. In this rank, the holy banner of given shots is 45 bc (it is customary to write the letters in alphabetical order).
To know the additional multiplier for skin shot, you need to divide the new banner into the old one. 45bc: (15b) = 3c, 45bc: (9c) = 5b. We multiply the number and the standard of the skin shot by the additional multiplier:
On the back, it’s a big banner for numbers: 8 does not divide by 6, 8 2 \u003d 16 does not divide by 6, 8 3 \u003d 24 does not divide by 6. It is necessary to turn on the leather from the replacements to the sleeping banner once. Three steps take steps with a great show.
In this rank, the incendiary banner of the given shots is finished on 24a?bc.
In order to know the additional multiplier to the skin shot, it is necessary to divide the new banner into the old one: 24a³bc: (6a³c)=4b, 24a³bc: (8a²bc)=3a.
The additional multiplier is multiplied by the number and the banner:
Rich limbs that stand at the banners of these shots are necessary. The banner of the first shot has a new square of difference: x²-18x+81=(x-9)²; the bannerman has a different one - the difference of squares: x²-81=(x-9)(x+9):
The golden banner is made up of the necessary multipliers, taken the most, so that they are worth (x-9) ² (x + 9). We know the additive multipliers and multiply them by the number and the standard of the skin shot:
At this article, it is revealed how to bring the fractions to a double banner, and how to know the smallest double banner. Designated, given the rule of bringing shots to the standard banner and looking at the practical butt.
What is the reduction of the shot to the standard banner?
Zvichaynі fractions are added from the numeral - the upper part, and the banner - the lower part. If the fractions make the same banner, then it seems that the stench is brought to the sleeping banner. For example, fractions 11 14 17 14 9 14 to hang the same banner 14 . In other words, the stench is brought to the sleeping banner.
Well, well, fractions can be made by different banners, you can always bring them to a sleeping banner for the help of simple ones. Sob tse robiti, it is necessary to multiply the numeral and banner by the songs of the supplementary multipliers.
It is obvious that the fractions 45 and 34 are not reduced to the standard banner. Sob tse robiti, it is necessary to bring them to the banner of 20 for the vicariances of the additional multipliers 5 and 4. Multiply the numeral and znamennik of the fraction 4 5 by 4, and the numeral and znamennik of the fraction 3 4 multiply by 5. Replacement of shots 4 5 і 3 4 omitted depending on 16 20 and 15 20 .
Bringing shots to a sleeping banner
The reduction of fractions to the double standard - the multiplier of numbers and the standard of fractions are such multipliers that the result has identical fractions with the same standard.
Zagalny banner: vyznachennya, butt
What is a sleeping banner?
Spіlny znamennik
The scorching banner of shots - if it’s a positive number, like a sizzling multiple of all these shots.
Otherwise, it seems that the main standard of such a set of shots will be such a natural number, as it will be divided into all the signs of these shots.
A number of natural numbers are inexhaustible, and to that, zgіdno z nomenclature, skin typing of the most significant fractions may be impersonal spilnyh znamennikіv. Otherwise, hanging, using impersonal multiples of all signs of a set of fractions.
It is easy to know the scorching banner for a few shots, vicorist is made. Let it be the fractions 16 and 35. The common banner of the fractions will be a positively significant multiple of the numbers 6 and 5. Such positive high multiples are the numbers 30, 60, 90, 120, 150, 180, 210 and so on.
Let's look at an example.
Butt 1. Spilny banner
Can you bring shots 1 3 , 21 6 , 5 12 to a double banner, which is more expensive 150?
Schob z'yasuvati, chi tse so, it is necessary to revise, chi є 150 common multiple for znamennikіv fractions, then for the numbers 3, 6, 12. Otherwise, it seems that the number 150 may be divisible by 3, 6, 12. Revisited:
150 ÷ 3 = 50 , 150 ÷ 6 = 25 , 150 ÷ 12 = 12 , 5
Otzhe, 150 is not a common banner of the indicated fractions.
The smallest sleeping banner
The smallest natural number from the multiplier of the common banners in such a set of fractions is called the smallest common banner.
The smallest sleeping banner
The smallest blazing banner of shots is the least of the middle of the most common banners of these shots.
The smallest possible divisor of this set of numbers is the smallest possible multiple (LCM). The NOC of the used znamennikіv drobіv є the smallest zagalny znamennikіv drobіv.
How to know the smallest common banner? This value is reduced to the value of the smallest common multiple of the fraction. Let's go back for example:
Example 2
It is necessary to know the smallest standard for shots 110 and 12728.
Shukaёmo NOK numbers 10 and 28. Let's put them into simple multipliers and take them away:
10 \u003d 2 5 28 \u003d 2 2 7 N O K (15, 28) \u003d 2 2 5 7 \u003d 140
How to bring fractions to the smallest standard
I use the rule, as I explain, how to cause fractions to a double banner. The rule is made up of three points.
The rule of bringing fractions to a double banner
- Find the smallest blazing banner of shots.
- For skin shot, know the additional multiplier. In order to know the multiplier, it is necessary to divide the smallest sleeping banner into a skin shot banner.
- Multiply the number book and the banner by the knowledge of the additional multiplier.
Let's take a look at the drafting of this rule on a specific example.
Butt 3
Є fractions 3 14 and 5 18. Let's guide them to the smallest sleeping banner.
As a rule, we know the NOC of the znamenniks in fractions.
14 \u003d 2 7 18 \u003d 2 3 3 N O K (14, 18) \u003d 2 3 3 7 \u003d 126
Calculating additive multipliers for skin shot. For 3 14 the additional multiplier is 126 ÷ 14 = 9, and for the fraction 5 18 the additional multiplier is 126 ÷ 18 = 7.
We multiply the number and the standard of fractions by the additional multipliers and we take it:
3 9 14 9 \u003d 27 126, 5 7 18 7 \u003d 35 126.
Bringing a lot of shots to the smallest spilny banner
After looking at the rule, up to a sleeping banner, you can make it like a bet of fractions, and even more of them.
Let's take one more example.
Butt 4
Bring fractions 3 2 , 5 6 , 3 8 and 17 18 to the smallest double standard.
Let's count the NOC of the famous people. We know the NOC of three and more numbers:
N O C (2, 6) = 6 N O C (6, 8) = 24 N O C (24, 18) = 72 N O C (2, 6, 8, 18) = 72
For 3 2 the additional multiplier is more 72 ÷ 2 = ?
We multiply the fractions by the additional multipliers and we pass to the smallest double standard:
3 2 36 = 108 72 5 6 12 = 60 72 3 8 9 = 27 72 17 18 4 = 68 72
How did you remember the pardon in the text, be kind, see it and press Ctrl + Enter
Shotguns have different and same banners. The same banner or otherwise called sleeping banner at the fraction An example of a zagal banner:
\(\frac(17)(5), \frac(1)(5)\)
Butt of different banners at shots:
\(\frac(8)(3), \frac(2)(13)\)
How to bring a fraction to the standard banner?
The first fraction has a sign of more than 3, the other has more of 13. It is necessary to know such a number, so that it extended by 3 and by 13. The whole number is 39.
The first drib should be multiplied by additive multiplier 13. Sob drib not changing, we multiply obov'yazkovo and the number by 13 and the banner.
\(\frac(8)(3) = \frac(8 \times \color(red) (13))(3 \times \color(red) (13)) = \frac(104)(39)\)
The other drіb is multiplied by the additional multiplier 3.
\(\frac(2)(13) = \frac(2 \times \color(red) (3))(13 \times \color(red) (3)) = \frac(6)(39)\)
We brought a fraction to the standard banner:
\(\frac(8)(3) = \frac(104)(39), \frac(2)(13) = \frac(6)(39)\)
The smallest sleeping banner.
Let's take a look at the example:
Bring the fractions \(\frac(5)(8)\) і \(\frac(7)(12)\) up to the double standard.
Zagalny banner for numbers 8 and 12 can be numbers 24, 48, 96, 120, ..., accepted to choose the smallest sleeping banner at times the number is 24.
The smallest sleeping banner- Tse least number, on the yak there is a banner of the first and the other shot.
How to know the smallest common banner?
Using the method of enumeration of numbers, on the yak the banner of the first and the other fraction is divided and the least of them is selected.
We need to multiply the fraction with the standard 8 by 3, and multiply the fraction with the standard 12 by 2.
\(\begin(align)&\frac(5)(8) = \frac(5 \times \color(red) (3))(8 \times \color(red) (3)) = \frac(15 )(24)\\\\&\frac(7)(12) = \frac(7 \times \color(red) (2))(12 \times \color(red) (2)) = \frac( 14)(24)\\\end(align)\)
If you can’t afford to bring a shot to the smallest sleeping banner, there’s nothing terrible, you can’t give a butt to you
You can know the zagalny znamennik for whether there are two shots, or maybe there are two znamenniks for these shots.
For example:
Bring the fractions \(\frac(1)(4)\) and \(\frac(9)(16)\) up to the smallest double standard.
The easiest way to know the famous bannerman is the tse vitvir of the bannermen 4⋅16=64. The number 64 is the smallest zagalny banner. For orders, it is necessary to know the smallest well-known banner. So just kidding away. We need a number that dilates i by 4, i by 16, and the whole number is 16. Let us reduce the fraction to a double standard, multiplying drib from the standard 4 by 4, and drib from the standard 16 by one. We take:
\(\begin(align)&\frac(1)(4) = \frac(1 \times \color(red) (4))(4 \times \color(red) (4)) = \frac(4 ) )(16)\\\\&\frac(9)(16) = \frac(9 \times \color(red) (1))(16 \times \color(red) (1)) = \frac ( 9)(16) \\\end(align)\)
