Fractions
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.
Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?
Types of fractions. Transformations.
There are three types of fractions.
1. Common fractions , For example:
Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)
The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.
When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.
32/8 = 32: 8 = 4
I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.
2. Decimals , For example:
It is in this form that you will need to write down the answers to tasks “B”.
3. Mixed numbers , For example:
Mixed numbers are practically not used in high school. In order to work with them, they must be translated into common fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.
Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!
The main property of a fraction.
So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:
It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.
Do we need it, all these transformations? And how! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.
How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.
A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where it lurks typical mistake, a blooper, if you will.
For example, you need to simplify the expression:
There’s nothing to think about here, cross out the letter “a” on top and the two on the bottom! We get:
Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then in a hurry you can cross out the “a” in the expression
and get it again
Which would be categorically untrue. Because here all the numerator on "a" is already doesn't share! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!
Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?
The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?
How to convert fractions from one type to another.
WITH decimals it's simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.
What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .
But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.
What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...
Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's all.
However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.
And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction does not translate. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !
By the way, this helpful information for self-test. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.
So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It is not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.
Suppose you were horrified to see the number in the problem:
Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's all. It looks even simpler in mathematical notation:
Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.
Reverse operation - no translation proper fraction in a mixed number - rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.
Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do it? Where and when to apply this deep knowledge?
I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !
If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?
0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We easily square it (in our minds!) and get 1/64. All!
Let's summarize this lesson.
1. There are three types of fractions. Common, decimal and mixed numbers.
2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always available.
3. The choice of the type of fractions to work with a task depends on the task itself. In the presence of different types fractions in one task, the most reliable thing is to move on to ordinary fractions.
Now you can practice. First, convert these decimal fractions to ordinary fractions:
3,8; 0,75; 0,15; 1,4; 0,725; 0,012
You should get answers like this (in a mess!):
Let's finish here. In this lesson we refreshed our memory key points by fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.
Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.
The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:
The main property of a fraction
If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Reducing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- Multiply the numerator of the first fraction by the denominator of the second
- Multiply the numerator of the second fraction by the denominator of the first
- Replace the denominators of both fractions with their product
Operations with fractions
Addition. To add two fractions you need
- Add the new numerators of both fractions and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Reduce fractions to a common denominator
- Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged
Example:
Multiplication. To multiply one fraction by another, multiply their numerators and denominators:
Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:
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Ordinary(or simple) fraction - writing a rational number in the form ± m n (\displaystyle \pm (\frac (m)(n))) or ± m / n , (\displaystyle \pm m/n,) Where n ≠ 0. (\displaystyle n\neq 0.) A horizontal or slash indicates a division sign, resulting in a quotient. The dividend is called numerator fractions, and the divisor is denominator.
Notation for common fractions
There are several types of writing ordinary fractions in printed form:
Proper and improper fractions
Correct A fraction whose numerator is less than its denominator is called a fraction. A fraction that is not proper is called wrong, and represents rational number, modulo greater than or equal to one.
For example, fractions 3 5 (\displaystyle (\frac (3)(5))), 7 8 (\displaystyle (\frac (7)(8))) and are proper fractions, while 8 3 (\displaystyle (\frac (8)(3))), 9 5 (\displaystyle (\frac (9)(5))), 2 1 (\displaystyle (\frac (2)(1))) And 1 1 (\displaystyle (\frac (1)(1)))- improper fractions. Any non-zero integer can be represented as an improper fraction with a denominator of 1.
Mixed fractions
A fraction written as a whole number and a proper fraction is called mixed fraction and is understood as the sum of this number and a fraction. Any rational number can be written as mixed fraction. In contrast to a mixed fraction, a fraction containing only a numerator and a denominator is called simple.
For example, 2 3 7 = 2 + 3 7 = 14 7 + 3 7 = 17 7 (\displaystyle 2(\frac (3)(7))=2+(\frac (3)(7))=(\frac (14 )(7))+(\frac (3)(7))=(\frac (17)(7))). In strict mathematical literature, they prefer not to use such a notation due to the similarity of the notation for a mixed fraction with the notation for the product of an integer by a fraction, as well as because of the more cumbersome notation and less convenient calculations.
Compound fractions
A multi-story, or compound, fraction is an expression containing several horizontal (or, less commonly, oblique) lines:
1 2 / 1 3 (\displaystyle (\frac (1)(2))/(\frac (1)(3))) or 1 / 2 1 / 3 (\displaystyle (\frac (1/2)(1/3))) or 12 3 4 26 (\displaystyle (\frac (12(\frac (3)(4)))(26)))Decimals
A decimal is a positional representation of a fraction. It looks like this:
± a 1 a 2 … a n , b 1 b 2 … (\displaystyle \pm a_(1)a_(2)\dots a_(n)(,)b_(1)b_(2)\dots )Example: 3.141 5926 (\displaystyle 3(,)1415926).
The part of the record that comes before the positional decimal point is the integer part of the number (fraction), and the part that comes after the decimal point is the fractional part. Any ordinary fraction can be converted to a decimal, which in this case either has a finite number of decimal places or is a periodic fraction.
Generally speaking, for positional notation of numbers, you can use not only decimal system notation, but also others (including specific ones, such as Fibonacci).
The meaning of a fraction and the main property of a fraction
A fraction is just a representation of a number. The same number can correspond different fractions, both ordinary and decimal.
0 , 999... = 1 (\displaystyle 0,999...=1)- two different fractions correspond to the same number.Operations with fractions
This section covers operations on ordinary fractions. For operations with decimal fractions, see Decimal fraction.
Reduction to a common denominator
To compare, add and subtract fractions, they must be converted ( bring) to a form with the same denominator. Let two fractions be given: a b (\displaystyle (\frac (a)(b))) And c d (\displaystyle (\frac (c)(d))). Procedure:
After this, the denominators of both fractions coincide (equal M). Instead of the least common multiple, you can use simple cases take as M any other common multiple, such as the product of denominators. For an example, see the Comparison section below.
Comparison
To compare two common fractions, you need to bring them to a common denominator and compare the numerators of the resulting fractions. A fraction with a larger numerator will be larger.
Example. Let's compare 3 4 (\displaystyle (\frac (3)(4))) And 4 5 (\displaystyle (\frac (4)(5))). LCM(4, 5) = 20. We reduce the fractions to the denominator 20.
3 4 = 15 20 ; 4 5 = 16 20 (\displaystyle (\frac (3)(4))=(\frac (15)(20));\quad (\frac (4)(5))=(\frac (16)( 20)))Hence, 3 4 < 4 5 {\displaystyle {\frac {3}{4}}<{\frac {4}{5}}}
Addition and subtraction
To add two ordinary fractions, you must reduce them to a common denominator. Then add the numerators and leave the denominator unchanged:
1 2 (\displaystyle (\frac (1)(2))) + = + = 5 6 (\displaystyle (\frac (5)(6)))The LCM of the denominators (here 2 and 3) is equal to 6. We give the fraction 1 2 (\displaystyle (\frac (1)(2))) to the denominator 6, for this the numerator and denominator must be multiplied by 3.
1 2 (\displaystyle (\frac (1)(2))) - = - 1 4 (\displaystyle (\frac (1)(4))) = 1 4 (\displaystyle (\frac (1)(4)))
Happened 3 6 (\displaystyle (\frac (3)(6))). We give the fraction 1 3 (\displaystyle (\frac (1)(3))) to the same denominator, for this the numerator and denominator must be multiplied by 2. It turned out 2 6 (\displaystyle (\frac (2)(6))).
To get the difference between fractions, they also need to be brought to a common denominator, and then subtract the numerators, leaving the denominator unchanged:The LCM of the denominators (here 2 and 4) is equal to 4. We present the fraction 1 2 (\displaystyle (\frac (1)(2))) to the denominator 4, for this you need to multiply the numerator and denominator by 2. We get 2 4 (\displaystyle (\frac (2)(4))).
Multiplication and division
To multiply two ordinary fractions, you need to multiply their numerators and denominators:
a b ⋅ c d = a c b d . (\displaystyle (\frac (a)(b))\cdot (\frac (c)(d))=(\frac (ac)(bd)).)In particular, to multiply a fraction by a natural number, you need to multiply the numerator by the number, and leave the denominator the same:
2 3 ⋅ 3 = 6 3 = 2 (\displaystyle (\frac (2)(3))\cdot 3=(\frac (6)(3))=2)In general, the numerator and denominator of the resulting fraction may not be coprime, and the fraction may need to be reduced, for example:
5 8 ⋅ 2 5 = 10 40 = 1 4 . (\displaystyle (\frac (5)(8))\cdot (\frac (2)(5))=(\frac (10)(40))=(\frac (1)(4)).)To divide one ordinary fraction by another, you need to multiply the first by the reciprocal of the second:
a b: c d = a b ⋅ d c = a d b c , c ≠ 0. (\displaystyle (\frac (a)(b)):(\frac (c)(d))=(\frac (a)(b))\ cdot (\frac (d)(c))=(\frac (ad)(bc)),\quad c\neq 0.)For example,
1 2: 1 3 = 1 2 ⋅ 3 1 = 3 2. (\displaystyle (\frac (1)(2)):(\frac (1)(3))=(\frac (1)(2))\cdot (\frac (3)(1))=(\ frac (3)(2)).)Convert between different recording formats
To convert a fraction to a decimal, divide the numerator by the denominator. The result can have a finite number of decimal places, but it can also have an infinite number
While studying the queen of all sciences - mathematics, at some point everyone comes across fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not at all complicated, you need to treat it carefully, because in real life outside of school it will be very useful. So, let's refresh our knowledge about fractions: what they are, what they are for, what types they are and how to perform various arithmetic operations with them.
Her Majesty fraction: what is it
In mathematics, fractions are numbers, each of which consists of one or more parts of a unit. Such fractions are also called ordinary or simple. As a rule, they are written in the form of two numbers that are separated by a horizontal or slash line, it is called a “fractional” line. For example: ½, ¾.
The upper, or first, of these numbers is the numerator (shows how many parts are taken from the number), and the lower, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fraction bar actually functions as a division sign. For example, 7:9=7/9
Traditionally, common fractions are less than one. While decimals can be larger than it.
What are fractions for? Yes, for everything, because in the real world, not all numbers are integers. For example, two schoolgirls in the cafeteria bought one delicious chocolate bar together. When they were about to share dessert, they met a friend and decided to treat her too. However, now it is necessary to correctly divide the chocolate bar, considering that it consists of 12 squares.
At first, the girls wanted to divide everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their friend, not 1/3, but 1/4 of the chocolate. And since the schoolgirls did not study fractions well, they did not take into account that in such a situation they would end up with 9 pieces, which are very difficult to divide into two. This fairly simple example shows how important it is to be able to correctly find a part of a number. But in life there are many more such cases.
Types of fractions: ordinary and decimal
All mathematical fractions are divided into two large categories: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it’s worth paying attention to the second.
Decimal is a positional notation of a fraction of a number, which is written in writing separated by a comma, without a dash or slash. For example: 0.75, 0.5.
In fact, a decimal fraction is identical to an ordinary fraction, however, its denominator is always one followed by zeros - hence its name.
The number preceding the comma is an integer part, and everything after it is a fraction. Any simple fraction can be converted to a decimal. Thus, the decimal fractions indicated in the previous example can be written as usual: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be either positive or negative. If they are preceded by a “-” sign, this fraction is negative, if “+” is a positive fraction.
Subtypes of ordinary fractions
There are these types of simple fractions.
Subtypes of decimal fraction
Unlike a simple fraction, a decimal fraction is divided into only 2 types.
- Final - received this name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25.
- An infinite fraction is a number with an infinite number of digits after the decimal point. For example, when dividing 10 by 3, the result will be an infinite fraction 3.333...
Adding Fractions
Carrying out various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you understand the basic rules, solving any example with them will not be difficult.
For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect unit because the numerator is greater than the denominator. It can and should be transformed into a correct mixed one by dividing 17:12 = 1 and 5/12.
When mixed fractions are added, operations are performed first with whole numbers, and then with fractions.
If the example contains a decimal fraction and a regular fraction, it is necessary to make both simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as a mixed fraction of 3 and 1/10 or as an improper fraction - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator of 1/2 by 5 alternately, you get 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper reducible fraction, we bring it into normal form, reducing it by 5: 7/2 = 3 and 1/2, or decimal - 3.5.
When adding 2 decimal fractions, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add the required number of zeros, because in a decimal fraction this can be done painlessly. For example, 3.5+3.005. To solve this problem, you need to add 2 zeros to the first number and then add one by one: 3.500+3.005=3.505.
Subtracting Fractions
When subtracting fractions, you should do the same as when adding: reduce to a common denominator, subtract one numerator from another, and, if necessary, convert the result to a mixed fraction.
For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator by multiplying both its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both sides by 2 and the result is 3/10.
Multiplying fractions
Dividing and multiplying fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.
To multiply fractions, you simply need to multiply both numerators one by one, and then both denominators. Reduce the resulting result if the fraction is a reducible quantity.
For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. This fraction can be reduced by 4, so the final answer in the example is 5/18.
How to divide fractions
Dividing fractions is also a simple operation; in fact, it still comes down to multiplying them. To divide one fraction by another, you need to invert the second and multiply by the first.
For example, dividing the fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.
If you need to divide a fraction by a prime number, the technique is slightly different. Initially, you should write this number as an improper fraction, and then divide according to the same scheme. For example, 2/13:5 should be written as 2/13: 5/1. Now you need to turn over 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to treat them as you would with whole numbers: turn them into improper fractions, reverse the divisor and multiply everything. For example, 8 ½: 3. Convert everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you should convert the improper fraction to the correct one - 2 whole and 5/6.
So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it correctly.
Fraction in mathematics, a number consisting of one or more parts (fractions) of a unit. Fractions are part of the field of rational numbers. Based on the way they are written, fractions are divided into 2 formats: ordinary type and decimal .
Numerator of fraction- a number showing the number of shares taken (located at the top of the fraction - above the line). Fraction denominator- a number showing how many shares the unit is divided into (located below the line - at the bottom). , in turn, are divided into: correct And incorrect, mixed And composite are closely related to units of measurement. 1 meter contains 100 cm. Which means that 1 m is divided into 100 equal parts. Thus, 1 cm = 1/100 m (one centimeter is equal to one hundredth of a meter).
or 3/5 (three fifths), here 3 is the numerator, 5 is the denominator. If the numerator is less than the denominator, then the fraction is less than one and is called correct:
If the numerator is equal to the denominator, the fraction is equal to one. If the numerator is greater than the denominator, the fraction is greater than one. In both last cases the fraction is called wrong:
To isolate the largest whole number contained in an improper fraction, you divide the numerator by the denominator. If the division is performed without a remainder, then the improper fraction taken is equal to the quotient:
If division is performed with a remainder, then the (incomplete) quotient gives the desired integer, and the remainder becomes the numerator of the fractional part; the denominator of the fractional part remains the same.
A number containing an integer and a fractional part is called mixed. Fraction mixed number maybe improper fraction. Then you can select the largest integer from the fractional part and represent the mixed number in such a way that the fractional part becomes a proper fraction (or disappears altogether).