How to convert an improper fraction to a proper fraction. How to turn an improper fraction into a proper one. How to convert an improper fraction to a mixed fraction. Mixed numbers, converting a mixed number to an improper fraction and vice versa

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 whole part fractions 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 such a denominator, there are no problems. For example, or

If the fraction is, for example . In this case, it is necessary to use the basic property of a fraction and transform 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 in the form 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”. How do we find the extra that remains from the numerator “23” if we remove maximum quantity"3". We leave the denominator unchanged. Everything is done, write down the result

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 integer can be written as an ordinary not proper 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 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 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. 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:

Translation common fraction mixed

An improper fraction can be converted into a mixed fraction by simple division, which results in the integer 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 by corners 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.

A huge block of mathematics is devoted to working with fractions or non-integers. You encounter them very often in life, so knowing how to work with such numbers is important for any person. Mathematics is a science in which the student begins with knowledge of simple things and actions, and then moves on to more complex ones.

Knowledge and ability to work with such numbers will make it easier for him to work with logarithms, rational exponents and integrals in the future. With such numbers you can do everything the same as with ordinary numbers: add fractions, divide, subtract and multiply. In addition, they can be shortened. Working with fractions is simple; the main thing is to know the basic rules and methods for calculating them.

Basic Concepts

In order to understand what kind of meaning this is, it is necessary to imagine a certain whole object. Let's say that there is a cake that has been cut into several identical or equal pieces. Each piece will be called a share.

For example, 10 consists of 5 twos, each two is a part of ten.

The fractions have their own names, depending on their total number in the whole number: 10 can consist of two fives or five twos, in the first case it will be called (one second), and in the second - (one fifth). It should be remembered that it is equal to half a number, (one third) is a third, and (one fourth) is a quarter. They can also be depicted through a dash: ½, 1/3 or 1/5.

A number written on top of a horizontal line or to the left of an inclined line, called the numerator- it shows how many parts were taken from a whole number, and the number under the line or to the right of it - denominator, it shows how many shares were divided into. For example, the cake was divided into 10 pieces and immediately set aside two of them for late guests. It will be 2/10 (two tenths), i.e. took 2 (numerator) pieces from the total 10 (denominator).

What are the fractions, what is an improper fraction, what is a common fraction? These questions are easy to answer:

A mixed digit can always transform V improper fraction and vice versa.

The main property says: when multiplying, as well as dividing the dividend and divisor by the same factor, in general the size of the fraction will not change. This property makes all operations with fractions possible.

How to shorten?

The main rule is that a fractional figure can be reduced by dividing its numerator and denominator by the same divisor(different from 0) so that a new figure is obtained with smaller parameters, but equal to the original in value. Based on this rule, it can be understood that fractions are reducible and irreducible.

An example of reducing fractions: let's reduce 8/24 by dividing its parameters by 2. We get: 8:2=4 and 24:2=12. As a result, the original figure will turn into 4/12. You can repeat the operation by dividing the numbers again: 4:2=2 and 12:2=6. We get 2/6. Let's repeat the operation again: 2:2=1 and 6:2=3. The result is an irreducible figure of 1/3, since its parameters can no longer be divided by the same divisor. Any reducible number can be lead to the irreducible.

You can reduce by multiplying fractional expressions by each other:

*. These numbers themselves are irreducible, but by performing the multiplication operation, you can reduce them diagonally: * = =. You can only abbreviate when multiplying criss-cross: the numerator of the first with the denominator of the second, and vice versa.

You can also shorten a mixed number, i.e. represent the whole part and the proper fraction as an improper fraction. For this should be done some actions:

Fair and reverse action: Make a mixed fraction from an improper fraction. To do this, consider the reverse action with:

It is possible to reduce fractions in any operation using this method. You can reduce the values of its dividend and divisor by multiplying them by the same factor, and turning from mixed number to a share, and vice versa.

Possible actions

All basic types of calculations are available when counting fractions, as with whole numbers: addition, subtraction, and others. Let's look at each action separately with examples:

Addition and subtraction

You can add shares in two ways, depending on their divisor. They are the same and different. Let's consider an example of adding shares with identical divisors.

To solve +, you need to add the dividend separately and leave the divisor alone: 1+1. The result will be the figure, but since it is incorrect, it can be converted into a mixed one by dividing the dividend by the divisor: 2:2= 1. The incorrect fraction should always (!) be given to the correct and irreducible that is, if its dividend and divisor can be divided by the same factor, this must be done in the required order.

In the case of adding shares with different divisors, they must initially be lead to the same. For example, to solve: you need:

Subtraction is carried out in exactly the same way: in the case of identical divisors, we do not touch them, but subtract the numerators sequentially: - = =

. If the denominators are different, then you should proceed as with addition: find the LCM, factors, multiply the shares, and then subtract the shares with the same divisors.

What types of fractions are there?

Let's start with what it is. A fraction is a number that has some part of one. It can be written in two forms. The first one is called ordinary. That is, one that has a horizontal or slanted line. It is equivalent to the division sign.

In such a notation, the number above the line is called the numerator, and the number below it is called the denominator.

Among ordinary fractions, proper and improper fractions are distinguished. For the former, the absolute value of the numerator is always less than the denominator. The wrong ones are called that because they have everything the other way around. The value of a proper fraction is always less than one. While the incorrect 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 recording is decimal. There is a separate conversation about her.

How are improper fractions different from mixed numbers?

In essence, nothing. These are just different recordings of the same number. Improper fractions easily become mixed numbers after simple steps. And vice versa.

It all depends on the specific situation. Sometimes it is more convenient to use an improper fraction in tasks. And sometimes it is necessary to convert 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 skills of the problem solver.

The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second one is always less than one.

How to represent a mixed number as an improper fraction?

If you need to perform any 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 perform the following algorithm:

multiply the denominator by the whole part;
add the numerator value 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 = (17 x 4 + 1) : 4 = 69/4;
39 ½ = (39 x 2 + 1) : 2 = 79/2.

How to write an improper fraction as a mixed number?

The next technique is the opposite of the one discussed above. That is, when all mixed numbers are replaced by improper fractions. The algorithm of actions will be as follows:

divide the numerator by the denominator to obtain the remainder;
write the quotient in place of the whole part of the mixed one;
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 remainder 6; the answer will be 5 whole and 6/14; the fractional part in this example needs to be reduced by 2, resulting in 3/7; the final answer is 5 point 3/7.

108/54; after division, the quotient of 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer will be an integer - 2.

How to turn a whole number into an improper fraction?

There are situations when such action is necessary. To obtain improper fractions with a known denominator, you will need to perform the following algorithm:

multiply an integer by the desired denominator;
write this value above the line;
place the denominator below it.

The simplest option is when the denominator equal to one. Then you don't need to multiply anything. It is enough to simply write the integer given in the example, and place one under the line.

Example: Make 5 an improper fraction with a denominator of 3. Multiplying 5 by 3 gives 15. This number will be the denominator. The answer to the task is a fraction: 15/3.

Two approaches to solving problems with different numbers

The example requires calculating 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 will get the following value: 13/5.

In order to find out the sum, you need to reduce the fractions to same denominator. 13/5 after multiplying by 11 becomes 143/55. And 14/11 after multiplying by 5 will look like: 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, the same numbers are subtracted: 143 - 70 = 73. The answer will be a fraction: 73/55.

When multiplying 13/5 and 14/11 there is no need to lead to common denominator. It is enough to multiply the numerators and denominators in pairs. The answer will be: 182/55.

The same goes for division. For the right decision you need to replace division with multiplication and invert the divisor: 13/5: 14/11 = 13/5 x 11/14 = 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 whole and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 point 48/55. In the first approach the fraction was 213/55. You can check its 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 needs to be converted into a mixed number: 73 is divided by 55 and the quotient is 1 and the remainder is 18.

To find the product and quotient, it is inconvenient to use mixed numbers. It is always recommended to move on to improper fractions here.

How to make a proper fraction from an improper fraction?

The word itself - fraction means that the number is fractional, it is less than a whole (at least one).

Therefore, it is necessary to extract the integer from the numerator. For example, the number 30/4 is an irregular fraction, since 30 is greater than 4. This means that you just need to divide 30 by 4 and we get the number to the decimal point - 7, and we put it in front of the fraction. Multiply 7 by 4 and subtract this number from 30 - you get 2 - it will be in the numerator of the fraction. Total - 7 2/4, reduce - 7 1/2. In your example, the answer is 2 3/4.

For this you need a reader: the denominator.

Write the whole that comes out in the numerator. The denominator is what it was. When you divide, write it down as a whole part.

11:4=2 (3 remainder).

We get the correct fraction: 2 - whole 34

To make an improper fraction into a proper fraction, you need to identify the whole parts and subtract them from the improper fraction. In our case, the improper fraction is 11/4. There will be two (2) whole parts. We subtract them and get the proper fraction: two point three (2 point 3/4).

An improper fraction, in our case 11/4, needs to be converted into a proper fraction, i.e. in this case mixed fraction. To put it simply, the fraction is improper because in addition to the fraction it also contains an integer. It’s like a cake sitting in the refrigerator, unfinished, although cut, and on the table there are a few pieces left from the second one. When we talk about 11/4, we no longer know about two whole cakes, we see only eleven large pieces. 11 divided by 4, we get 2, and the remainder is 11-8 = 3. So, 2 whole 3/4, now the fraction is regular, in it the numerator will be smaller than the denominator, but mixed, since the calculation could not be done without whole units.

To make an improper fraction into a proper one, you need to divide the numerator by the denominator. Place the resulting integer in front of the fraction, and enter the remainder into the numerator. The denominator does not change.

For example: the fraction 11/4 is an improper fraction, where the numerator is 11 and the denominator is 4.

First we divide 11 by 4, we get 2 integers and 3 remainder. We put 2 in front of the fraction, and write the remainder 3 in the numerator 3/4. Thus, the fraction becomes correct - 2 whole and 3/4.

An improper fraction has a denominator that is smaller than the numerator, which indicates that this fraction has integer parts that can be separated to form a proper fraction with an integer.

The easiest way to divide the numerator by the denominator. We put the resulting integer to the left of the fraction, and write the remainder in the numerator, the denominator remains the same.

For example 11/4. Divide 11 by 4 and get 2 and the remainder 3. Two is the number that we put next to the fraction, and we write three in the numerator of the fraction. Comes out 2 and 3/4.

To answer this simple question, you can solve the same simple problem:

Petya and Valya came to the company of their peers. All together there were 11 of them. Valya had apples with him (but not many) and in order to treat everyone, Petya cut each one into four parts and distributed them. There was enough for everyone and there were even five pieces left.

How many apples did Petya give away and how many apples are left? How many were there in total?

Can we write this down mathematically?

11 pieces of apple in our case is 11/4 - we got an improper fraction, since the numerator is greater than the denominator.

To select a whole part (convert improper fraction into a proper fraction), you need numerator divided by denominator, write the incomplete quotient (in our case 2) on the left, leave the remainder (3) in the numerator and do not touch the denominator.

As a result we get 11/4 = 11:4 = 2 3/4 Petya gave away the apples.

Likewise, 5/4 = 1 1/4 apples left.

(11+5)/4 = 16/4 = Valya brought 4 apples

Every person, when solving problems in mathematics, often comes across problems involving fractions. There are a lot of them, so we'll look at different options solving the main such 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 an integer 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 some 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.

We'll do everything necessary steps 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 by either one equal to it 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 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