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 assigned, 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!
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 valid, 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 when a is divided by b. That is, the integer part is equal to the partial 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
Simple mathematical rules and techniques, if they are not used constantly, are forgotten most quickly. Terms disappear from memory even faster.
One of these simple actions is converting an improper fraction into a proper or, in other words, a mixed fraction.
Improper fraction
An improper fraction is one in which the numerator (the number above the line) is greater than or equal to the denominator (the number below the line). This fraction is obtained by adding fractions or multiplying a fraction by a whole number. According to the rules of mathematics, such a fraction must be converted into a proper one.
Proper fraction
It is logical to assume that all other fractions are called proper. A strict definition is that a fraction whose numerator is less than its denominator is called proper. A fraction that has an integer part is sometimes called a mixed fraction.
Converting an improper fraction to a proper fraction
- First case: the numerator and denominator are equal to each other. The result of converting any such fraction is one. It doesn't matter if it's three-thirds or one hundred and twenty-five one hundred and twenty-fifths. Essentially, 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 numerator value, which is divisible by the denominator without a remainder. For example, you have the fraction nineteen thirds. The closest number that can be divided by three is eighteen. That's six. Now subtract the resulting number from the numerator. We get one. This is the remainder. Write down the result of the conversion: six whole and one third.
But before you can reduce a fraction to its correct form, you need to check whether it can be reduced.
You can reduce a fraction if the numerator and denominator have a common factor. 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 such a common divisor - two. And the fraction sixteen-twelfths has one more common divisor - four. This is the greatest divisor. Divide the numerator and denominator by four. Result of reduction: four thirds. Now, as a practice, convert this fraction to a proper fraction.
You can convert an improper fraction to a proper fraction by dividing the numerator of such a fraction by the denominator - this way we get a proper fraction. Alternatively, an improper fraction can be written as a simple decimal number.
An improper fraction is a fraction in which the numerator is greater than the denominator. A proper fraction is one whose numerator is smaller than its denominator. There is no way to turn an improper fraction into a proper fraction, but it can be represented as a mixed number consisting of two parts (one part will be an integer, and the other will be a proper fraction).
for example 5/2=2+1/2 (only the fraction is usually written immediately after the whole number without the plus sign)
Here you need to divide the numerator of the improper fraction by the denominator. We write down the integer part of the division (in our case 2). then we write the remainder of the division (that is, 1) as the numerator of the fraction, which we write next to the two.
We know from the school mathematics course. that 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 decimal fraction.
An improper fraction, for example: 9/5, let’s select the whole part of it, it will be: 1 4/5 now it looks a little like the correct one only with the whole part being one.
You can turn it into a decimal fraction in our case it will be 1.8
To solve the problem, you first need to clearly understand for yourself what a proper fraction is and what an improper fraction is.
Let's start with the fact that the statement
This is not true for all numbers on the number line.
numerator is (-10), denominator is (-4)
similar statement
not always true either
numerator is 2, denominator is (-3)
An improper fraction can be written using the sum of a whole number 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, take the minus outside the fraction - (15/2), divide 15 by 2, put the integer 7 in the whole part of the fraction, write the remainder of the division 1 in the numerator, and leave the denominator 2 without changes.
In order to convert an improper fraction into a proper fraction, you first need to say:
An improper fraction has a numerator (the top number in the fraction) greater than or equal to the denominator;
For a proper fraction the opposite is true.
Let us analyze the conversion process using the example of the fraction 260/7:
1) First, divide 260 by 7, we get the number 37.14..
2) The number 37 will appear in front of the fraction as a whole number
3) Now 37 * 7 = 259
4) From the numerator we subtract the resulting number 260 - 259 = 1 - this number will be in the numerator of our proper fraction.
5) When writing a new fraction, the denominator remains unchanged. In this case it is 7. The proper fraction would look like this:
Checking the converted fraction:
We multiply the integer by the denominator and add the numerator 37 * 7 + 1 = 260.
A proper fraction is a fraction in which the denominator is greater than the numerator. This suggests that this fraction shows some part of the whole. For example, the fraction 1/2 means that we have half of a watermelon, for example, and the fraction 7/9 means that we have seven pieces of watermelon left, cut into 9 parts. Someone ate two parts.
If the fraction is improper, that is, the numerator is greater than the denominator, then it is completely unclear what part of the whole, but cut watermelon we have and how many more whole watermelons are available. Therefore, we have to convert an improper fraction into a proper one. in this case we will get some kind of integer and the remainder - exactly a proper fraction.
To convert, divide the numerator by the denominator in a column. Example: 7/4. Seven times 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 call a fraction such that numerator is greater than denominator. This means that a proper fraction is one whose numerator is less than its denominator. To turn an improper fraction into a proper fraction, 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 proper fraction, you need to divide the numerator of the improper fraction by its denominator, the result will be a number with a remainder.
For example, 4 whole and three elevenths, we multiply 4 by 11 and +3, then we divide by 11, we get 44 +3 and divide by 11, and we get the fraction 47/11. An improper fraction is when there is an integer, for example 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 reduce it by two, we get six thirds, we reduce six thirds by three, we get the first two, we divide two by one we get two. That is, 6.6 = 2.
A fraction is a number that is made up of one or more units. 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 are taken from the number, 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 = 5. Therefore, any whole number can be written as an ordinary improper fraction or a series of such fractions. Let's consider the entries of the same number in the form of a number of different ones.
- Mixed fractions
In general, 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 notation is understood as the sum of the whole and its fractional part.
- Decimals
A decimal is a special type of fraction in which the denominator can be represented as a power of 10. There are infinite and finite decimals. When writing this type of fraction, the whole part is first indicated, then the fractional part is recorded through a separator (period or comma).
The notation of a fractional part is always determined by its dimension. The decimal notation looks like this: 
Rules for converting between different types of fractions
- Converting a mixed fraction to a common fraction
A mixed fraction can only be converted to an improper fraction. To translate, it is necessary to bring the whole part to the same denominator as the fractional part. In general it will look like this: 
Let's look at the use of this rule using specific examples:
- Converting a common fraction to a mixed fraction
An improper fraction can be converted into a mixed fraction by simple division, resulting in the whole part and the remainder (fractional part).
For example, let's convert the fraction 439/31 to mixed: 
- Converting fractions
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 a power of 10.
For example:
In some cases, you may need to find the quotient by dividing with a corner or using a calculator. And some fractions cannot be reduced to a final decimal. For example, the fraction 1/3 when divided will never give the final result.
