Simple mathematical rules and tricks, if they are not used constantly, are forgotten the fastest. Terms are fading out of memory even faster.
One of these simple actions- converting an improper fraction into a proper one or, in other words, a mixed one.
Improper fraction
An improper fraction is a fraction in which the numerator (the number above the fractional bar) is greater than or equal to the denominator (the number below the bar). Such a fraction is obtained by adding fractions or multiplying a fraction by an integer. According to the rules of mathematics, such a fraction must be turned into a regular one.
Proper fraction
It is logical to assume that all other fractions are called correct. Strict definition - a correct fraction is called, in which the numerator is less than the denominator. A fraction that has an integer part is sometimes called a mixed fraction.
Converting an Improper Fraction to a Proper Fraction
- First case: numerator and denominator are equal to each other. As a result of the transformation of any such fraction, one will be obtained. It doesn't matter if it's three-thirds or one hundred and twenty-five one hundred and twenty-fifths. In fact, such a fraction denotes the action of dividing a number by itself.
- Second case: the numerator is greater than the denominator. Here you need to remember the method of dividing numbers with a remainder.
To do this, you need to find the number closest to the value of the numerator, which is divisible by the denominator without a remainder. For example, you have a fraction of nineteen thirds. The closest number that can be divided by three is eighteen. Get six. Now subtract the resulting number from the numerator. We get a unit. This is the remainder. Write down the result of the transformation: six integers and one third.
But before reducing the fraction to correct form, we need to check whether it can be reduced.
A fraction can be reduced if the numerator and denominator have a common divisor. That is, a number by which both are divisible without a remainder. If there are several such divisors, you need to find the largest one.
For example, all even numbers have a common divisor - two. And the fraction of sixteenth twelfths has another common divisor - four. This is the largest divisor. Divide the numerator and denominator by four. Reduction result: four-thirds. Now, as a practice, convert this fraction to a proper one.
In this material, we will analyze such a thing as mixed numbers. We start, as always, with a definition and small examples, then we will explain the connection between mixed numbers and improper fractions. After that, we will learn how to correctly extract the integer part from a fraction and get an integer as a result.
The concept of a mixed number
If we take the sum n + a b , where the value of n can be any natural number, and a b is a proper ordinary fraction, then we can write the same thing without using a plus: n a b . Let's take specific numbers for clarity: so, 28 + 5 7 is the same as 28 5 7 . Writing a fraction next to an integer is called a mixed number.
Definition 1
mixed number is a number that is equal to the sum of a natural number n with a proper ordinary fraction a b . In this case, n is the integer part of the number, and a b is its fractional part.
It follows from the definition that any mixed number is equal to what will result from the addition of its integer and fractional parts. Thus, the equality n a b = n + a b will hold.
It can also be written as n + a b = n a b .
What are some examples of mixed numbers? So, 5 1 8 belongs to them, while five is its whole part, and one eighth is fractional. More examples: 1 1 2 , 234 34 53 , 34000 6 25 .
Above we wrote that in the fractional part mixed number must be a proper fraction. Sometimes you can find entries like 5 22 3 , 75 7 2 . They are not mixed numbers, because their fractional part is wrong. They need to be understood as the sum of an integer and a fractional part. Such numbers can be standard view writing mixed numbers by extracting the integer part from the improper fraction and adding it to 5 and 75 in these examples, respectively.
Numbers of the form 0 3 14 are also not mixed. The first part of the condition is not met here: the whole part must be represented only natural number, and zero is not.
How are improper fractions and mixed numbers related?
This connection is easiest to trace on a concrete example.
Example 1
Let's take a whole cake and another three quarters of the same. According to the addition rules, we have 1 + 3 4 cakes on the table. This sum can be represented as a mixed number as 1 3 4 cakes. If we take a whole cake and also cut it into four equal parts, then we will have 7 4 cakes on the table. Obviously, the amount did not increase from cutting, and 1 3 4 = 7 4 .
Our example proves that any number can be represented as a mixed number. improper fraction.
Let's go back to our 7 4 cakes left on the table. Let's put one cake back from its pieces (1 + 3 4). We will again have 1 3 4 .
Answer: 7 4 = 1 3 4 .
We figured out how to convert an improper fraction to a mixed number. If the numerator of an improper fraction contains a number that can be divided by the denominator without a remainder, then you can do this, and then our improper fraction will become a natural number.
Example 2
For example,
8 4 = 2 since 8: 4 = 2 .
How to convert a mixed number to an improper fraction
To successfully solve problems, it is useful to be able to perform the reverse action, that is, to make improper fractions from mixed numbers. In this paragraph, we will analyze how to do it correctly.
To do this, you need to reproduce the following sequence of actions:
1. To begin with, we present the available mixed number n a b as the sum of the integer and fractional parts. It turns out n + a b
3. After that, we perform an already familiar action - we add two ordinary fractions n 1 and a b. The resulting improper fraction will be equal to the mixed number given in the condition.
Let's analyze this action on a specific example.
Example 3
Write 5 3 7 as an improper fraction.
Decision
We perform the steps of the above algorithm in sequence. Our number 5 3 7 is the sum of the integer and fractional parts, that is, 5 + 3 7. Now let's write the five as 5 1 . We got the sum 5 1 + 3 7 .
The last step is to add fractions with different denominators:
5 1 + 3 7 = 35 7 + 3 7 = 38 7
All solution to short form can be written as 5 3 7 = 5 + 3 7 = 5 1 + 3 7 = 35 7 + 3 7 = 38 7 .
Answer: 5 3 7 = 38 7 .
Thus, with the help of the above chain of actions, we can convert any mixed number n a b into an improper fraction. We have obtained the formula n a b = n b + a b , which we will take to solve further problems.
Example 4
Write 15 2 5 as an improper fraction.
Decision
Take this formula and substitute the desired values into it. We have n = 15 , a = 2 , b = 5 , therefore 15 2 5 = 15 5 + 2 5 = 77 5 .
Answer: 15 2 5 = 77 5 .
We usually don't list the improper fraction as the final answer. It is customary to bring the calculations to the end and replace it with either a natural number (dividing the numerator by the denominator) or a mixed number. As a rule, the first method is used when it is possible to divide the numerator by the denominator without a remainder, and the second - if such an action is impossible.
When we extract the whole part from an improper fraction, we simply replace it with an equal mixed number.
Let's see how exactly this is done.
Definition 2
We present a proof of this assertion.
We need to explain why q r b = a b . To do this, the mixed number q r b must be represented as an improper fraction by following all the steps of the algorithm from the previous paragraph. Since is an incomplete quotient, and r is the remainder of dividing a by b, then the equality a = b · q + r must hold.
So q b + r b = a b so q r b = a b . This is the proof of our assertion. To summarize:
Definition 3
The selection of the integer part from the improper fraction a b is carried out as follows:
1) we divide a by b with a remainder and write the incomplete quotient q and the remainder r separately.
2) Write the results as q r b . This is our mixed number, equal to the original improper fraction.
Example 5
Express 1074 as a mixed number.
Decision
We divide 104 by 7 in a column:
Dividing the numerator a = 118 by the denominator b = 7 gives us the incomplete quotient q = 16 and the remainder r = 6.
As a result, we get that the improper fraction 118 7 is equal to the mixed number q r b = 16 6 7 .
Answer: 118 7 = 16 6 7 .
It remains for us to see how to replace an improper fraction with a natural number (provided that its numerator is divisible by the denominator without a remainder).
To do this, remember what relationship exists between ordinary fractions and division. From this we can derive the equalities: a b = a: b = c . It turns out that the improper fraction a b can be replaced by a natural number c.
Example 6
For example, if the answer turned out to be an improper fraction 27 3, then we can write 9 instead, since 27 3 \u003d 27: 3 \u003d 9.
Answer: 27 3 = 9 .
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A fraction is a number that consists of one or more fractions of a unit. There are three types of fractions in mathematics: common, mixed, and decimal.
Common fractions
An ordinary fraction is written as a ratio in which the numerator reflects how many parts of the number are taken, and the denominator shows how many parts the unit is divided into. If the numerator is less than the denominator, then we have a proper fraction. For example: ½, 3/5, 8/9.
If the numerator is equal to or greater than the denominator, then we are dealing with an improper fraction. For example: 5/5, 9/4, 5/2 Dividing the numerator can result in a finite number. For example, 40/8 \u003d 5. Therefore, any integer can be written as an ordinary improper fraction or a series of such fractions. Consider writing the same number as a series of different .
- mixed fractions
AT general view A mixed fraction can be represented by the formula: 
Thus, a mixed fraction is written as an integer and an ordinary proper fraction, and such a record is understood as the sum of a whole and its fractional part.
- Decimals
The decimal is special variety a fraction whose denominator can be represented as a power of 10. There are infinite and finite decimal fractions. When writing this type of fraction, the integer part is first indicated, then the fractional part is fixed through the separator (dot or comma).
The record of the fractional part is always determined by its dimension. The decimal entry looks like this: 
Translation rules between different types of fractions
- Translation mixed fraction into ordinary
A mixed fraction can only be converted to an improper fraction. For translation, it is necessary to bring the whole part to the same denominator as the fractional part. In general, it will look like this: 
Consider the use of this rule on specific examples:
- Translation common fraction into mixed
An improper common fraction can be converted to a mixed one by simple division, which results in the integer part and the remainder (fractional part).
For example, let's translate the fraction 439/31 into a mixed one: 
- Translation of an ordinary fraction
In some cases, converting a fraction to a decimal is quite simple. In this case, the basic property of a fraction is applied, the numerator and denominator are multiplied by the same number, in order to bring the divisor to the power of 10.
For example:
In some cases, you may need to find the quotient by dividing by a corner or using a calculator. And some fractions cannot be reduced to a final decimal fraction. For example, the fraction 1/3 will never give the final result when divided.
An improper fraction is one of the formats for writing an ordinary fraction. Like any ordinary fraction, it has a number above the line (numerator) and below it - the denominator. If the numerator is greater than the denominator, it is hallmark fraction irregularities. In this form, you can convert a mixed ordinary fraction. The decimal can also be represented in the wrong ordinary form records, but only if the separating comma is preceded by a non-zero number.
Instruction
In mixed fraction format, the numerator and denominator are separated from the integer part by a space. To convert such an entry to , first multiply its integer part (the number before the space) by the denominator of the fractional part. Add the resulting value to the numerator. The value calculated in this way will be the numerator of an improper fraction, and put the denominator of the mixed fraction in its denominator without any changes. For example, 5 7/11 in regular irregular format can be written like this: (5*11+7)/11 = 62/11.
To convert a decimal fraction to an incorrect ordinary notation, determine the number of digits after the decimal point separating the integer part from the fractional - it is equal to the number of digits to the right of this comma. Use the resulting number as an indicator of the power to which you need to raise ten in order to calculate the denominator of an improper fraction. The numerator is obtained without any calculations - just remove the comma from the decimal fraction. For example, if the original decimal is 12.585, the numerator of the corresponding wrong number should be 10³ = 1000, and the denominator should be 12585: 12.585 = 12585/1000.
Like any ordinary fraction, it can and should be reduced. To do this, after obtaining the result in the ways described in the previous two steps, try to find the greatest common divisor for the numerator and denominator. If you can do this, divide by what you found on both sides of the solid bar. For the example from the second step, this divisor will be the number 5, so the improper fraction can be reduced: 12.585 = 12585/1000 = 2517/200. And for the example from the first step common divisor no, so there is no need to reduce the resulting improper fraction.
Related videos
Decimal fractions are more convenient for automated calculations than natural ones. Any natural fraction can be converted into natural numbers either without loss of accuracy, or with an accuracy of up to a given number of decimal places, depending on the ratio between the numerator and denominator.
Instruction
If necessary, round the result to the required number of decimal places. The rounding rules are as follows: if the highest of the deleted digits contains a digit from 0 to 4, then the next highest digit (which is not deleted) does not change, and if the digit is from 5 to 9, it increases by one. If the last of these operations is subjected to a digit with the number 9, the unit is transferred to another, even more senior digit, like a column. Please note that rounding up to the available number of character spaces does not always perform this operation. Sometimes there are hidden digits in his memory that are not displayed on the indicator. Logarithmic, having low accuracy (up to two decimal places), often at the same time copes with rounding in the right direction better.
If you find that a certain sequence of digits is repeated after the decimal point, place this sequence in brackets. They say about her that she is "", because she repeats periodically. For example, number 53.7854785478547854... can be written as 53,(7854).
A proper fraction, the value of which is greater than one, consists of two parts: a whole and a fraction. First, divide the numerator of the fractional part by its denominator. Then add the result of the division to the integer part. Then, if necessary, round the result to required amount decimal places or find the frequency and highlight it with brackets.
Decimals are easy to handle. They are recognized by calculators and many computer programs. But sometimes it is necessary, for example, to draw up a proportion. To do this, you have to translate decimal into a common fraction. It won't be difficult if you do small digression into the school curriculum.
Instruction
Reduce the fractional part of the resulting . To do this, the numerator and denominator of the fraction must be divided by the same divisor. In this case, it is the number "5". So "5/10" is converted to "1/2".
Choose a number so that the result of its multiplication by the denominator is 10. Reasoning from the reverse: is it possible to turn the number 4 into 10? Answer: no, because 10 is not divisible by 4. Then 100? Yes, 100 is divisible by 4 without a remainder, the result is 25. Multiply the numerator and denominator by 25 and write the answer in decimal form:
¼ = 25/100 = 0.25.
It is not always possible to use the selection method, there are two more ways. Their principle is almost the same, only the recording differs. One of them is the gradual allocation of decimal places. Example: translate the fraction 1/8.
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 the extraction of the integer part from an 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 of a 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 particular 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. An improper fraction is preliminarily replaced either with a natural number equal to it (when the numerator is divided entirely by the denominator), or the so-called selection of the whole 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
