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How to convert an improper fraction to a proper fraction. How to turn an improper fraction into a proper fraction

The word “fractions” gives many people goosebumps. Because I remember school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. What if you treated problems involving proper and improper fractions like a puzzle? After all, many adults decide digital and Japanese crosswords. We figured out the rules, and that’s it. It's the same here. One has only to delve into the theory - and everything will fall into place. And the examples will turn into a way to train your brain.

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 in tasks it is more convenient to use correct fraction. 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 person solving the problem.

The mixed number is also compared with the sum of the whole 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 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 = (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 in the form 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 is 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.

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 the 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!

Instructions

Find the numerator of the resulting fraction, which should remain after separating the whole 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 your head, in a column or using a calculator (475-460=15).

Collect the calculated data into one entry in the form of a mixed fraction - first write the whole part (20), then, then write 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). It is ordinary fractions that are proper and improper.

Instructions

Ordinary fraction is called correct if the number in its numerator is less number, standing in the denominator. Reducing fractions is done to work with the smallest numbers.

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 reducing the fraction to the right kind, you need to check whether it can be shortened.
Reducing a fraction is possible if the numerator and denominator have 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 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.

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 shares have their own names, depending on their total number in a 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 two of them were immediately set aside 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 common fraction? These questions are easy to answer:

A mixed digit can always transform to an 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 shorten 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:

The reverse action is also true: 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 a mixed number into a fraction, 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 must separately add the dividend 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?

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

There are also mixed numbers, that is, those that have an integer and a fractional part.

The second type of notation is a decimal fraction. 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 whole 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 forms, 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 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 = (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 is 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 the 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, you do not need to reduce them to a common denominator. It is enough to multiply the numerators and denominators in pairs. The answer will be: 182/55.

The same goes for division. To solve correctly, 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.

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 then 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 a mixed fraction. To put it simply, the fraction is improper because in addition to the fraction it also contains a whole number. 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 turn 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