In this article we will talk about mixed numbers. First, let's define mixed numbers and give examples. Next, let's look at the connection between mixed numbers and improper fractions. After that, we'll show you how to convert a mixed number to an improper fraction. Finally, let's study the reverse process, which is called separating the whole part from an improper fraction.
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Mixed numbers, definition, examples
Mathematicians agreed that the sum n+a/b, where n is a natural number, a/b is a proper fraction, can be written without the addition sign in the form. For example, the sum 28+5/7 can be briefly written as . Such a record was called mixed, and the number that corresponds to this mixed record was called a mixed number.
This is how we come to the definition of a mixed number.
Definition.
Mixed number is a number equal to the sum of the natural number n and the proper ordinary fraction a/b, and written in the form . In this case, the number n is called whole part of the 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 be written like this: .
Let's give examples of mixed numbers. A number is a mixed number, the natural number 5 is the integer part of the number, and the fractional part of the number. Other examples of mixed numbers are 
.
Sometimes you can find numbers in mixed notation, but having an improper fraction as a fraction, for example, or. These numbers are understood as the sum of their integer 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.
The number is also not a mixed number, since 0 is not a natural number.
The relationship between mixed numbers and improper fractions
Follow connection between mixed numbers and improper fractions best with examples.
Let there be a cake and another 3/4 of the same cake on the tray. That is, according to the meaning of addition, there are 1+3/4 cakes on the tray. Having written down the last amount as a mixed number, we state that there is a cake on the tray. Now cut the whole cake into 4 equal parts. As a result, there will be 7/4 of the cake on the tray. It is clear that the “quantity” of the cake has not changed, so .
From the example considered, 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 folded a whole cake from four parts, there will be 1 + 3/4 on the tray, that is, a cake. From this 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 evenly 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
To perform various operations with mixed numbers, the skill of representing mixed numbers as improper fractions is useful. In the previous paragraph, we found out that any mixed number can be converted into an improper fraction. It's time to figure out how such a translation is carried out.
Let us write an algorithm showing how to convert a mixed number to an improper fraction:
Let's look at an example of converting a mixed number to an improper fraction.
Example.
Express a mixed number as an improper fraction.
Solution.
Let's perform all the necessary steps of the algorithm.
A mixed number is equal to the sum of its integer and fractional parts: .
Having written the number 5 as 5/1, the last sum will take the form .
To finish converting the original mixed number into an improper fraction, all that remains is to add fractions with different denominators: 
.
A short summary of the entire solution is: 
.
Answer:
So, to convert a mixed number to an improper fraction, you need to perform the following chain of actions: . Finally received 
, which we will use further.
Example.
Write the mixed number as an improper fraction.
Solution.
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:
Separating the whole part from an improper fraction
It is not customary to write an improper fraction in the answer. The improper fraction is first replaced either by an equal natural number (when the numerator is divisible by the denominator), or the so-called separation of the whole part from the improper fraction is carried out (when the numerator is not divisible by the denominator).
Definition.
Separating the whole part from an improper fraction- This is the replacement of a fraction with an equal mixed number.
It remains to find out how you can isolate the whole part from an improper fraction.
It's very simple: the improper fraction a/b is equal to a mixed number of the form, where q is the partial quotient, and r is the remainder of a divided 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 is enough to show that . Let's convert 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
Every modern person during his school days, while solving mathematical problems, was often faced with a variety of problems involving fractions. There are quite a lot of them, so it makes sense to consider various options for solving the most basic similar problems.
Proper and improper fractions
The top number of any fraction is called the numerator, while the bottom number is the denominator. Ordinary fractions are quotients of two numbers, moreover, one of these numbers is in the numerator of the fraction, and the second, accordingly, is the denominator of this fraction. The types of such ordinary fractions are determined by comparing the values of their denominator and numerator.
Proper fraction
In the case when the denominator of a fraction is a natural number, which in its value is greater than its numerator, also a natural number, then the fraction is called proper. Examples of these could be: 8/19; 9/14; 31/162; 5/37 and so on.
If the denominator of a fraction is less than or equal to its numerator, then such a fraction is already called improper. For example, these are: 7/4; 19/6; 15/3; 231/83 and the like.
Why convert an improper fraction to a proper fraction?
Such mathematical manipulation is necessary if an operation is performed with several fractions, for example, they are added.
Advice
If there is a mixed fraction, then you should first convert it to an improper fraction, then perform other mathematical operations.
Converting to an improper fraction
To turn any mixed fraction into an improper one, you first need to multiply its whole part by the denominator of its fractional part, and then add the numerator to this product. Next, the sum is taken as the numerator, but with the same denominator as before. To convert an improper fraction to a proper fraction, you will need to divide the numerator of such an improper fraction by its denominator. Further, the integer obtained in this way should be taken as the whole part of the fraction, while the remainder, if there is one, of course, should be made the numerator of the fractional part of the proper fraction. The denominator is written the same as it was. To convert any improper fraction to a decimal, you must first find out whether there is such a factor at all that allows you to reduce the denominator of its fractional part in the irregular format to a number that is equal to ten or ten raised to any power. That is, 10, 100, 1000 and so on. If there is such a factor, then you should multiply both the numerator and the denominator of the improper fraction by this factor, thereby, as it were, checking it. And then the multiplied numerator will need to be added, separated by a comma, to the integer part of the improper fraction.
Cannot be converted by rounding to tenths
In the case where such a factor does not exist as such, this means that such an improper fraction does not have a clear equivalent in decimal form. Simply put, not every improper fraction can be converted into a decimal. In this case, you will need to find the approximate, maximum corresponding value of the fraction. It all depends on the degree of accuracy required in the conditions of a particular task. The easiest way to calculate this fraction is on a calculator, but you can also do it in your head or simply in a column. For example, "41/7 = 5(6/7) = 5.9", this is rounded to the nearest tenth, or "= 5.86" when rounded to the hundredth is required, and also "= 5.857" when rounded to the nearest thousandths Many of the fractions cannot be clearly converted into decimals, so it is easier to count them not in your head or in a column, but using a calculator.
Conclusion:
Without manipulating fractions, not a single school mathematics course is possible. And in everyday life you rarely have to deal only with whole numbers, and therefore everyone needs to be able to convert regular fractions into improper ones, or convert them into such mixed fractions. This is very simple and therefore you can remember how to do it literally after a couple of practical examples, solved on paper, and then generally in your mind. With decimal fractions the situation is somewhat different and not everything can be accurately converted into decimal form.
Mathematical 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 point five hundredths, we get a fraction. Three point seventeen thousandths, we get the fraction. The numbers before the decimal point are the whole part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a single-digit number after the decimal point, the denominator will be 10, if there is a two-digit number - 100, a three-digit number - 1000, etc. Some resulting fractions can be reduced. In our examples 
Converting a fraction to a decimal
This is the reverse of the previous transformation. What is the characteristic of a decimal fraction? Its denominator is always 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, or
If the fraction is, for example . In this case, it is necessary 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 a decimal number 0.12.
Some fractions are easier to divide than to convert the denominator. For example,
Some fractions cannot be converted to decimals!
For example,
Converting a mixed fraction to an improper fraction
A mixed fraction, for example, can be easily converted to an improper fraction. To do this, you need to multiply the whole part by the denominator (bottom) and add it with the numerator (top), leaving the denominator (bottom) unchanged. That is
When converting a mixed fraction to an improper fraction, you can remember that you can use fraction addition
Converting an improper fraction to a mixed fraction (highlighting the whole part)
An improper fraction can be converted to a mixed fraction by highlighting the whole part. Let's look at an example. We determine how many integer times “3” fits into “23”. Or divide 23 by 3 on a calculator, the whole number 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”. 
It’s as if we find the extra that remains from the numerator “23” if we remove the maximum amount of “3”. We leave the denominator unchanged. Everything is done, write down the result
Every person, when solving problems in mathematics, often comes across problems involving fractions. There are a lot of them, so we will look at different options for solving these basic problems.
What are fractions
The top number of any fraction is called the numerator, and the bottom number is the denominator. An ordinary fraction is the quotient of two numbers, one of these numbers is in the numerator of the fraction, the second is in the denominator of the fraction. The types of these common fractions will be determined by comparing the denominator and numerator of the fraction.
If the denominator of a fraction (natural number) is greater than the numerator of the fraction (natural number), then the fraction is called proper. Here are some examples: 7/19; 9/13; 31/152; 5/17.
If the denominator of a fraction (natural number) is less than or equal to the numerator of the fraction (natural number), then the fraction is called improper. Here are some examples: 7/5; 19/3; 15/9; 231/63.
How to convert improper fraction
To convert a mixed fraction to an improper fraction, you need to multiply the whole part of the fraction by the denominator in the fractional part and add the numerator to this product. Then take the amount as the numerator, writing the same denominator as before. Here are some examples:
- 4(3/11) = (4x11+3)/11 = (44+3)/11 = 47/11.
- 11(5/9) = (11x9+5)/9 = (99+5)/9 = 104/9.
To convert an improper fraction to a proper fraction, you must divide the numerator of the improper fraction by its denominator. Take the resulting integer as the whole part of the fraction, and take the remainder (of course, if there is one) as the numerator of the fractional part of the proper fraction, writing the same denominator as before. Here are some examples:
- 150/13 = (143/13)+(7/13) = 11(7/13).
- 156/12 = (13x12)/12 = 13.
To convert an improper fraction to a decimal, it is necessary to find out whether there is such a factor that will allow the denominator of the fractional part of the improper fraction to be reduced to a number that is equal to ten (or a ten that is raised to any power (10, 100, 1000 and more). If such a factor is, then you need to multiply the numerator and denominator of the improper fraction by this factor to check it. Now the multiplied numerator must be added, separated by a comma, to the integer part of the improper fraction. Here are examples:
- Multiplier “5” - 8/20 = (8x5)/(20x5) = 40/100 = 0.4.
- Multiplier "4" - 14/25 = (14x4)/(25x4) = 56/100 = 0.56.
- Multiplier "25" - 3/40 = (3x25)/(40x25) = 75/1000 = 0.075.
If such a factor does not exist, this means that this improper fraction in decimal form does not have a clear equivalent. That is, not every improper fraction can be converted to a decimal. In this case, you need to find the approximate value of the fraction with the degree of accuracy you require. You can calculate such a fraction on a calculator, in your head, or in a column. Here are examples: 41/7 = 5(6/7) = 5.9 (rounded to tenths), = 5.86 (rounded to hundredths), = 5.857 (rounded to thousandths); 3/7, 7/6, 1/3 and others. They are also not clearly translated and are calculated on a calculator, in the head or in a column.
Now you know how to convert an improper fraction to a proper or decimal fraction!