In this article we will talk about mixed numbers. First, let's define mixed numbers and give examples. Next, let's dwell on the relationship between mixed numbers and improper fractions. After that, we will show how to convert a mixed number into an improper fraction. Finally, let's explore reverse process, which is called extracting the integer part from improper fraction.
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Mixed numbers, definition, examples
Mathematicians have agreed that the sum n + a / b, where n is a natural number, a / b is a regular fraction, can be written without an addition sign in the form. For example, the sum 28+5/7 can be briefly written as . Such an entry was called mixed, and the number that corresponds to this mixed entry was called a mixed number.
So we come to the definition mixed number.
Definition.
mixed number is a number equal to the sum of a natural number n and a proper ordinary fraction a/b, and written as . In this case, the number n is called integer part of a number, and the number a/b is called fractional part of a number.
By definition, a mixed number is equal to the sum of its integer and fractional parts, that is, the equality is true, which can also be written like this:.
Let's bring examples of mixed numbers. The number is a mixed number, the natural number 5 is the integer part of the number, and is the fractional part of the number. Other examples of mixed numbers are 
.
Sometimes you can find numbers in mixed notation, but having a fractional part of an improper fraction, for example, or. These numbers are understood as the sum of their whole and fractional parts, for example, 
and 
. But such numbers do not fit the definition of a mixed number, since the fractional part of mixed numbers must be a proper fraction.
A number is also not a mixed number, since 0 is not a natural number.
Relationship between mixed numbers and improper fractions
trace relationship between mixed numbers and improper fractions best with examples.
Let there be a cake on the tray and another 3/4 of the same cake. That is, according to the meaning of addition, there are 1 + 3/4 cakes on the tray. Having written the last amount as a mixed number, we state that there is a cake on the tray. Now we will cut the whole cake into 4 equal parts. As a result, 7/4 of the cake will be on the tray. It is clear that the "quantity" of the cake has not changed, therefore.
From the considered example, the following connection is clearly visible: any mixed number can be represented as an improper fraction.
Now let there be 7/4 of the cake on the tray. Having added a whole cake out of four shares, there will be 1 + 3/4 on the tray, that is, a cake. From here it is clear that .
From this example it is clear that An improper fraction can be represented as a mixed number. (In the special case when the numerator of an improper fraction is divided by the denominator, the improper fraction can be represented as a natural number, for example, since 8:4=2).
Converting a mixed number to an improper fraction
For execution various activities with mixed numbers, the skill of representing mixed numbers as improper fractions is helpful. In the previous paragraph, we found out that any mixed number can be converted to an improper fraction. It's time to figure out how such a translation is carried out.
Let's write an algorithm showing how to convert mixed number to improper fraction:
Consider an example of converting a mixed number to an improper fraction.
Example.
Express the mixed number as an improper fraction.
Decision.
Let's perform all the necessary steps of the algorithm.
A mixed number is equal to the sum of its integer and fractional parts: .
By writing the number 5 as 5/1, the last sum becomes .
To complete the translation of the original mixed number into an improper fraction, it remains to perform the addition of fractions with different denominators: 
.
A summary of the entire solution is as follows: 
.
Answer:
So, in order to translate a mixed number into an improper fraction, you need to perform the following chain of actions:. As a result received 
, which we will use in what follows.
Example.
Write the mixed number as an improper fraction.
Decision.
Let's use the formula to convert a mixed number to an improper fraction. In this example n=15 , a=2 , b=5 . Thus, 
.
Answer:
Extracting the integer part from an improper fraction
It is not customary to write an improper fraction in the answer. Improper fraction is preliminarily replaced by either equal to it natural number(when the numerator is divided entirely by the denominator), or the so-called separation of the integer part from an improper fraction is carried out (when the numerator is not divided entirely by the denominator).
Definition.
Extracting the integer part from an improper fraction is the replacement of a fraction by its equal mixed number.
It remains to find out how you can select the whole part from an improper fraction.
It's very simple: an improper fraction a/b is equal to a mixed number of the form , where q is an incomplete quotient, and r is the remainder of dividing a by b. That is, the integer part is equal to the incomplete quotient of dividing a by b, and the remainder is equal to the numerator of the fractional part.
Let's prove this statement.
To do this, it suffices to show that . Let's translate the mixed into an improper fraction as we did in the previous paragraph:. Since q is an incomplete quotient and r is the remainder of dividing a by b , then the equality a=b q+r is true (if necessary, see
At the word "fractions" many goosebumps run. Because I remember the school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. But what if we treat tasks containing proper and improper fractions as a puzzle? After all, many adults solve digital and Japanese crosswords. Understand the rules and that's it. Same here. One has only to delve into the theory - and everything will fall into place. And examples will turn into a way to train the brain.
What types of fractions are there?
Let's start with what it is. A fraction is a number that has some fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique stroke. It equates to the division sign.
In such a notation, the number above the dash is called the numerator, and below it is called the denominator.
Among ordinary fractions, right and wrong fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called that because they have the opposite. The value of a proper fraction is always less than one. While the wrong one is always greater than this number.
There are also mixed numbers, that is, those that have an integer and a fractional part.
The second type of notation is decimal. About her separate conversation.
What is the difference between improper fractions and mixed numbers?
Basically, nothing. It's just a different notation of the same number. Improper fractions after simple operations easily become mixed numbers. And vice versa.
It all depends on specific situation. Sometimes in tasks it is more convenient to use an improper fraction. And sometimes it is necessary to translate it into a mixed number, and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers - depends on the observation of the solver of the problem.
The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than unity.
How to represent a mixed number as an improper fraction?
If you want to perform some action with several numbers that are written in different types, then you need to make them the same. One method is to represent numbers as improper fractions.
For this purpose, you will need to follow the following algorithm:
- multiply the denominator by the integer part;
- add the value of the numerator to the result;
- write the answer above the line;
- leave the denominator the same.
Here are examples of how to write improper fractions from mixed numbers:
- 17 ¼ \u003d (17 x 4 + 1): 4 \u003d 69/4;
- 39 ½ \u003d (39 x 2 + 1): 2 \u003d 79/2.
How to write an improper fraction as a mixed number?
The next method is the opposite of the one discussed above. That is, when all mixed numbers are replaced with improper fractions. The algorithm of actions will be as follows:
- divide the numerator by the denominator to get the remainder;
- write the quotient in place of the integer part of the mixed;
- the remainder should be placed above the line;
- the divisor will be the denominator.
Examples of such a transformation:
76/14; 76:14 = 5 with a remainder of 6; the answer is 5 integers and 6/14; the fractional part in this example needs to be reduced by 2, you get 3/7; the final answer is 5 whole 3/7.
108/54; after division, the quotient 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer is an integer - 2.
How do you turn an integer into an improper fraction?
There are situations when such action is necessary. To get improper fractions with a predetermined denominator, you will need to perform the following algorithm:
- multiply an integer by the desired denominator;
- write this value above the line;
- place a denominator below it.
The simplest option is when the denominator is equal to one. Then there is no need to multiply. It is enough just to write an integer, which is given in the example, and place a unit under the line.
Example: Make 5 an improper fraction with a denominator of 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the task is a fraction: 15/3.
Two approaches to solving tasks with different numbers
In the example, it is required to calculate the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.
In the first approach the mixed number will be represented as an improper fraction.
After performing the steps described above, you get the following value: 13/5.
In order to find out the sum, you need to reduce the fractions to the same denominator. 13/5 multiplied by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is the answer to the problem.
When finding the difference, these same numbers are subtracted: 143 - 70 = 73. The answer is a fraction: 73/55.
When multiplying 13/5 and 14/11, you do not need to reduce to a common denominator. Just multiply the numerators and denominators in pairs. The answer will be: 182/55.
Likewise with division. For right decision you need to replace division with multiplication and flip the divisor: 13/5: 14/11 \u003d 13/5 x 11/14 \u003d 143/70.
In the second approach An improper fraction becomes a mixed number.
After performing the actions of the algorithm, 14/11 will turn into a mixed number with an integer part of 1 and a fractional part of 3/11.
When calculating the sum, you need to add the integer and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 whole 48/55. In the first approach there was a fraction 213/55. You can check the correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.
When subtracting, the "+" sign is replaced by "-". 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check the answer from the previous approach, you need to convert it to a mixed number: 73 is divided by 55 and you get a quotient of 1 and a remainder of 18.
To find the product and the quotient, it is inconvenient to use mixed numbers. Here it is always recommended to switch to improper fractions.
Decimal numbers such as 0.2; 1.05; 3.017 etc. as they are heard, so they are written. Zero point two, we get a fraction. One whole five hundredths, we get a fraction. Three whole seventeen thousandths, we get a fraction. The digits before the decimal point in a decimal number are the integer part of the fraction. The number after the decimal point is the numerator of the future fraction. If after the comma single digit- the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some of the resulting fractions can be reduced. In our examples 
Converting a fraction to a decimal number
This is the reverse of the previous transformation. What is a decimal fraction? Her denominator is always 10, or 100, or 1000, or 10,000, and so on. If your usual fraction has such a denominator, there is no problem. For example, or
If a fraction, for example . In this case, you need to use the basic property of a fraction and convert the denominator to 10 or 100, or 1000 ... In our example, if we multiply the numerator and denominator by 4, we get a fraction that can be written as decimal number 0,12.
Some fractions are easier to divide than to convert the denominator. For example,
Some fractions cannot be converted to decimal numbers!
For example,
Converting a mixed fraction to an improper
A mixed fraction, such as , is easily converted to an improper fraction. To do this, you need to multiply the integer part by the denominator (bottom) and add it to the numerator (top), leaving the denominator (bottom) unchanged. I.e
When converting mixed fraction into the wrong one, you can remember that you can use the addition of fractions
Converting an improper fraction to a mixed one (highlighting the whole part)
An improper fraction can be converted to a mixed fraction by highlighting the whole part. Consider an example, . Determine how many integer times "3" fit in "23". Or we divide 23 by 3 on the calculator, the whole number up to the decimal point is the desired one. This is "7". Next, we determine the numerator of the future fraction: we multiply the resulting "7" by the denominator "3" and subtract the result from the numerator "23". 
How would we find the excess that remains from the numerator "23", if we remove the maximum number of "3". The denominator is left unchanged. Everything is done, write down the result
Instruction
Find the numerator of the resulting fraction, which should remain after extracting the integer part from it. To do this, multiply the calculated integer part (20) by the denominator (23) and subtract the result (20*23=460) from the numerator of the original fraction (475). This operation can also be done in the mind, in a column or using a calculator (475-460=15).
Collect the calculated data in one record in the form of a mixed fraction - first write the whole part (20), then, then put the correct one with the numerator (15) and (23). For the example used as a sample, the transformation of an improper fraction into a proper one (more precisely, into a mixed one) can be written as follows: 475/23=20 15/23.
Often you have to divide something into parts, and those parts into which the whole is divided are fractions. In mathematics, there are several types of fractions: decimal (0.1; 2.5, and so on) and ordinary (1/3; 5/9; 67/89, and so on). Common fractions are right and wrong.
Instruction
ordinary fraction is called correct if the number in its numerator is less than number, which is in the denominator. Fraction reduction is done to work with the smallest numbers.
You can convert an improper fraction to a proper one by dividing the numerator of such a fraction by the denominator - this way we get the correct fraction. Otherwise, an improper fraction can be written as a simple decimal number.
An improper fraction is a fraction whose numerator is greater than the denominator. correct - that fraction, in which, accordingly, the numerator is less than the denominator. there is no way to turn an improper fraction into a proper one, but it can be represented as a mixed number consisting of two parts (one part will be an integer, and the other will be just a proper fraction).
for example 5/2=2+1/2 (only a fraction is usually written immediately after an integer without a plus sign)
here you need to divide the numerator of the improper fraction by the denominator. write down the integer part of the division (in our case 2). then the remainder of the division (that is, 1) is written as the numerator of the fraction, which we write next to the two.
We know from the school mathematics course. An improper fraction is a fraction whose numerator is greater than its denominator. To convert it to a proper fraction, you need to divide the numerator of such a fraction by its denominator. Everything is very simple, so it will become a correct, or a decimal fraction.
An improper fraction, for example: 9/5, we select its integer part, it will be: 1 4/5 is now a bit like the correct one, only with the integer part being one.
You can also turn it into decimal in our case it will be 1.8
To solve the problem, you first need to clearly understand for yourself what is a proper fraction and what is an incorrect one.
Let's start with the statement
true not for all numbers on the number line.
numerator is (-10), denominator is (-4)
similar statement
also not always true
numerator is 2, denominator is (-3)
An improper fraction can be written using the sum of an integer and a proper fraction (mixed fraction) and for this you need:
divide the numerator by the denominator, write the resulting integer in the integer part, the remainder in the numerator, leave the denominator unchanged
in the numerator (-15), in the denominator 2, we take the minus outside the fraction - (15/2), divide 15 by 2, put the integer 7 in the integer part of the fraction, write the remainder of division 1 in the numerator, and leave the denominator 2 without changes.
In order to convert an improper fraction to a proper one, you first need to say:
In an improper fraction, the numerator (the top number in the fraction) is greater than or equal to the denominator;
For a proper fraction, the opposite is true.
We will analyze the conversion process using the example of a fraction 260/7:
1) First we divide 260 by 7, we get the number 37.14 ..
2) The number 37 will come before the fraction as an integer
3) Now 37 * 7 = 259
4) From the numerator we subtract the resulting number 260 - 259 \u003d 1 - this number will be in the numerators of our regular fraction.
5) When writing a new fraction, the denominator remains unchanged. In this case, it is 7. The correct fraction will look like this:
Checking the converted fraction:
We multiply the whole number by the denominator and add the numerator 37 * 7 + 1 = 260.
A proper fraction is a fraction whose denominator is greater than the numerator. This suggests that this fraction shows some part of the whole. For example, the fraction 1/2 indicates that we have half, for example, a watermelon, and the fraction 7/9 indicates that we have seven pieces of watermelon cut into 9 parts. Someone ate two.
If the fraction is incorrect, that is, the numerator is greater than the denominator, then it is completely incomprehensible what part of the whole, but cut watermelon we have and how many more whole watermelons are available. Therefore, you have to convert the improper fraction to the correct one. in this case, we will get some integer and the remainder - exactly the correct fraction.
To translate, we divide the numerator by the denominator into a column. Example: 7/4. Seven by four gives one and the remainder is 3/4. So we converted the fraction to the correct one - the answer is 1 and 3/4.
Improper fraction called a fraction that has numerator greater than denominator. So a proper fraction is one whose numerator is less than the denominator. To turn an improper fraction into a proper one, you can represent it as a decimal number. For example, 17/8 can be written like this: 2.125. Or write it like this: 2 1/8.
A proper fraction is considered to be one in which the denominator is higher than the numerator. In order to convert an improper fraction into a correct one, it is necessary to divide the numerator of the improper fraction by its denominator, the result will be a number with a remainder.
For example, 4 integers and three elevenths, we multiply 4 by 11 and +3, then we divide by 11, it turns out 44 +3 and divide by 11, and we get the fraction 47/11. An improper fraction is when there is an integer such as 5.10, that is, five integers and 10/100, five we multiply 100 and +10, it turns out 10/500. Also, if for example 6.6, it’s easier here, we multiply 6 by 6 and +6 it turns out 12/6, we cut by two, we get six thirds, we cut six thirds by three, we get the first two, two divided by one, we get two. That is, 6.6 = 2.
