While studying the queen of all sciences - mathematics, at some point everyone comes across fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not at all complicated, it must be treated carefully, because in real life It will be very useful outside of school. So, let's refresh our knowledge about fractions: what they are, what they are for, what types they are and how to do different things with them arithmetic operations.
Her Majesty fraction: what is it
Fractions in mathematics are numbers, each of which consists of one or more parts of a unit. Such fractions are also called ordinary or simple. As a rule, they are written in the form of two numbers that are separated by a horizontal or slash line, it is called a “fractional” line. For example: ½, ¾.
The upper, or first, of these numbers is the numerator (shows how many parts are taken from the number), and the lower, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fraction bar actually functions as a division sign. For example, 7:9=7/9
Traditionally, common fractions are less than one. While decimals can be larger than it.
What are fractions for? Yes for everything, because in real world Not all numbers are integers. For example, two schoolgirls in the cafeteria bought one delicious chocolate bar together. When they were about to share dessert, they met a friend and decided to treat her too. However, now it is necessary to correctly divide the chocolate bar, considering that it consists of 12 squares.
At first, the girls wanted to divide everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their friend, not 1/3, but 1/4 of the chocolate. And since the schoolgirls did not study fractions well, they did not take into account that in such a situation they would end up with 9 pieces, which are very difficult to divide into two. This fairly simple example shows how important it is to be able to correctly find a part of a number. But in life similar cases much more.
Types of fractions: ordinary and decimal
All mathematical fractions are divided into two large categories: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it’s worth paying attention to the second.
Decimal is a positional notation of a fraction of a number, which is written in writing separated by a comma, without a dash or slash. For example: 0.75, 0.5.
Actually decimal is identical to ordinary, however, its denominator is always one followed by zeros - this is where its name comes from.
The number preceding the decimal point is whole part, and everything after it is fractional. I love it simple fraction can be converted to decimal. Thus, the decimal fractions indicated in the previous example can be written as usual: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be either positive or negative. If they are preceded by a "-" sign, given fraction negative, if "+" - then positive.
Subtypes of ordinary fractions
There are these types of simple fractions.
Subtypes of decimal fraction
Unlike a simple fraction, a decimal fraction is divided into only 2 types.
- Final - received this name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25.
- An infinite fraction is a number with an infinite number of digits after the decimal point. For example, when dividing 10 by 3, the result is infinite fraction 3,333…
Adding Fractions
Carrying out various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you understand the basic rules, solving any example with them will not be difficult.
For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect value because the numerator is greater than the denominator. It can and should be transformed into a correct mixed one by dividing 17:12 = 1 and 5/12.
When mixed fractions are added, operations are performed first with whole numbers, and then with fractions.
If the example contains a decimal fraction and an ordinary fraction, it is necessary to make both simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as mixed fraction 3 and 1/10 or as incorrect - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator of 1/2 by 5 alternately, you get 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper reducible fraction, we reduce it to normal look, reducing by 5: 7/2 = 3 and 1/2, or decimal - 3.5.
When adding 2 decimal fractions, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add required quantity zeros, because in decimal fractions this can be done painlessly. For example, 3.5+3.005. To solve this problem, you need to add 2 zeros to the first number and then add one by one: 3.500+3.005=3.505.
Subtracting Fractions
When subtracting fractions, you should do the same as when adding: reduce to common denominator, subtract one numerator from the other, and, if necessary, convert the result to a mixed fraction.
For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator by multiplying both its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both sides by 2 and the result is 3/10.
Multiplying fractions
Dividing and multiplying fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.
To multiply fractions, you simply need to multiply both numerators one by one, and then both denominators. Reduce the resulting result if the fraction is a reducible quantity.
For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. This fraction can be reduced by 4, so the final answer in the example is 5/18.
How to divide fractions
Dividing fractions is also a simple operation; in fact, it still comes down to multiplying them. To divide one fraction by another, you need to reverse the second and multiply by the first.
For example, dividing the fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.
If you need to divide a fraction by a prime number, the technique is slightly different. Initially, you should write this number as an improper fraction, and then divide according to the same scheme. For example, 2/13:5 should be written as 2/13: 5/1. Now you need to turn over 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to treat them as you would with whole numbers: turn them into improper fractions, reverse the divisor and multiply everything. For example, 8 ½: 3. Turn everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you should convert the improper fraction to the correct one - 2 whole and 5/6.
So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it correctly.
We come across fractions in life much earlier than we begin studying them at school. If we cut a whole apple in half, we get ½ of the fruit. Let's cut it again - it will be ¼. These are fractions. And everything seemed simple. For an adult. For the child (and this topic start studying at the end junior school) abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must clearly explain what a proper fraction and an improper fraction, an ordinary and a decimal are, what operations can be performed with them and, most importantly, what all this is needed for.
What types of fractions are there?
Getting to know new topic at school it starts with ordinary fractions. They are easily recognized by the horizontal line separating the two numbers - above and below. The top one is called the numerator, the bottom one is the denominator. There is also a lowercase option for writing improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to use a “two-story” entry form. Why? Yes, because it is more convenient. We'll see this a little later.
In addition to ordinary fractions, there are also decimal fractions. It is very simple to distinguish them: if in one case a horizontal or slash is used, in the other a comma is used to separate sequences of numbers. Let's look at an example: 2.9; 163.34; 1.953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two point nine.”
New concepts
Let's return to ordinary fractions. They come in two types.
The definition of a proper fraction is as follows: it is a fraction whose numerator less than the denominator. Why is this important? We'll see now!
You have several apples, halved. Total - 5 parts. How would you say: do you have “two and a half” or “five and a half” apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if we need to calculate how many fruits each person will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from a mathematical point of view, this will be more clear.
So, for naming proper and improper fractions, the rule is this: if a whole part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is irregular. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.
The main property of a fraction
If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value does not change. Imagine: they cut the cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Does it really matter? After all, ¼ and 2/8 are the same thing!
Reduction
Authors of problems and examples in mathematics textbooks often seek to confuse students by offering fractions that are cumbersome to write but can actually be abbreviated. Here is an example of a proper fraction: 167/334, which, it would seem, looks very “scary”. But we can actually write it as ½. The number 334 is divisible by 167 without a remainder - after performing this operation, we get 2.
Mixed numbers
An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.
To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division on top, above the line, and the whole part - before the expression. Thus, we get two structural parts: whole units + proper fraction.
It is possible to carry out reverse operation- to do this, you need to multiply the whole part by the denominator and add the resulting value to the numerator. Nothing complicated.
Multiplication and division
Oddly enough, multiplying fractions is easier than adding. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5.
With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7*15 / 8*14 = 15/16.
Adding Fractions
What to do if you need to perform addition or their denominator is different numbers? It will not work to do the same as with multiplication - here you should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the lower part of both fractions must have the same numbers.
To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½.
How to choose which denominator to reduce the terms to? This must be the minimum number that is a multiple of both numbers in the denominators of the fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because smaller value, divisible by 2 and 7 without a remainder, does not exist.
Usage
What are improper fractions used for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and be done with it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use irregular ones.
Let's take next example: (2 + 3/17) / (37 / 68).
It would seem that there is nothing to cut at all. But what if we write the addition result in the first parentheses as an improper fraction? Look: (37/17) / (37/68)
Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37*68) / (17*37).
Let's cancel 37 in the numerator and denominator and finally divide the top and bottom by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number as long as we do it for the numerator and denominator at the same time.
So, we get the answer: 4. The example looked complicated, but the answer contains only one number. This happens often in mathematics. The main thing is not to be afraid and follow simple rules.
Common Mistakes
When implementing, a student can easily make one of the common mistakes. Usually they occur due to inattention, and sometimes due to the fact that the material studied has not yet been properly stored in the head.
Often the sum of numbers in the numerator makes you want to reduce its individual components. Let’s say in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, due to inexperience, cross out 13 above and below. But this should not be done under any circumstances, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.
Guys also often make mistakes when dividing fractions. Let's take two correct ones irreducible fractions and divide by each other: (5/6) / (25/33). The student can mix it up and write the resulting expression as (5*25) / (6*33). But this would happen with multiplication, but in our case everything will be somewhat different: (5*33) / (6*25). We reduce what is possible, and the answer will be 11/10. We write the resulting improper fraction as a decimal - 1.1.
Parentheses
Remember that in any mathematical expressions the order of operations is determined by the priority of the operation signs and the presence of parentheses. All other things being equal, the order of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.
After all, this is the result of dividing one number by another. If they are not evenly divided, it becomes a fraction - that's all.
How to write a fraction on a computer
Since standard tools do not always allow creating a fraction consisting of two “tiers,” students sometimes resort to various tricks. For example, they copy the numerators and denominators into the Paint graphic editor and glue them together, drawing between them horizontal line. Of course, there is a simpler option, which, by the way, provides a lot of additional features, which will be useful to you in the future.
Open Microsoft Word. One of the panels at the top of the screen is called “Insert” - click it. On the right, on the side where the close and minimize window icons are located, there is a “Formula” button. This is exactly what we need!
If you use this function, a rectangular area will appear on the screen in which you can use any mathematical signs, missing on the keyboard, and also write fractions in the classic form. That is, dividing the numerator and denominator with a horizontal line. You might even be surprised that such a proper fraction is so easy to write.
Learn math
If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, you cannot do without fractions. Soon you will learn to calculate everything in your mind, without even writing down the expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with curriculum, do your homework on time and you will succeed.
Common fractions are divided into \textit (proper) and \textit (improper) fractions. This division is based on a comparison of the numerator and denominator.
Proper Fractions
Proper fraction called common fraction$\frac(m)(n)$, whose numerator is less than the denominator, i.e. $m
Example 1
For example, the fractions $\frac(1)(3)$, $\frac(9)(123)$, $\frac(77)(78)$, $\frac(378567)(456298)$ are correct, so how in each of them the numerator is less than the denominator, which meets the definition of a proper fraction.
There is a definition of a proper fraction, which is based on comparing the fraction with one.
correct, if it is less than one:
Example 2
For example, the common fraction $\frac(6)(13)$ is proper because condition $\frac(6)(13) is satisfied
Improper fractions
Improper fraction is an ordinary fraction $\frac(m)(n)$ whose numerator is greater than or equal to the denominator, i.e. $m\ge n$.
Example 3
For example, the fractions $\frac(5)(5)$, $\frac(24)(3)$, $\frac(567)(113)$, $\frac(100001)(100000)$ are irregular, so how in each of them the numerator is greater than or equal to the denominator, which meets the definition of an improper fraction.
Let's give a definition of an improper fraction, which is based on its comparison with one.
The common fraction $\frac(m)(n)$ is wrong, if it is equal to or greater than one:
\[\frac(m)(n)\ge 1\]
Example 4
For example, the common fraction $\frac(21)(4)$ is improper because the condition $\frac(21)(4) >1$ is satisfied;
the common fraction $\frac(8)(8)$ is improper because the condition $\frac(8)(8)=1$ is satisfied.
Let's take a closer look at the concept of an improper fraction.
Let's take the improper fraction $\frac(7)(7)$ as an example. The meaning of this fraction is to take seven shares of an object, which is divided into seven equal parts. Thus, from the seven shares that are available, the entire object can be composed. Those. the improper fraction $\frac(7)(7)$ describes whole subject and $\frac(7)(7)=1$. So, improper fractions, in which the numerator is equal to the denominator, describe one whole object and such a fraction can be replaced by the natural number $1$.
$\frac(5)(2)$ - it is quite obvious that from these five second parts you can make up $2$ whole objects (one whole object will be made up of $2$ parts, and to compose two whole objects you need $2+2=4$ shares) and one second share remains. That is, the improper fraction $\frac(5)(2)$ describes $2$ of an object and $\frac(1)(2)$ the share of this object.
$\frac(21)(7)$ -- from twenty-one-sevenths parts you can make $3$ whole objects ($3$ objects with $7$ shares in each). Those. the fraction $\frac(21)(7)$ describes $3$ whole objects.
From the examples considered, one can conclude next output: An improper fraction can be replaced by a natural number if the numerator is divisible by the denominator (for example, $\frac(7)(7)=1$ and $\frac(21)(7)=3$), or the sum natural number and a proper fraction if the numerator is not completely divisible by the denominator (for example, $\ \frac(5)(2)=2+\frac(1)(2)$). That's why such fractions are called wrong.
Definition 1
The process of representing an improper fraction as the sum of a natural number and a proper fraction (for example, $\frac(5)(2)=2+\frac(1)(2)$) is called separating the whole part from an improper fraction.
When working with improper fractions, one can observe close connection between them and mixed numbers.
An improper fraction is often written as a mixed number - a number that consists of an integer part and a fraction part.
To write an improper fraction as a mixed number, you must divide the numerator by the denominator with a remainder. The quotient will be the integer part of the mixed number, the remainder will be the numerator of the fractional part, and the divisor will be the denominator of the fractional part.
Example 5
Write the improper fraction $\frac(37)(12)$ as a mixed number.
Solution.
Divide the numerator by the denominator with a remainder:
\[\frac(37)(12)=37:12=3\ (remainder\ 1)\] \[\frac(37)(12)=3\frac(1)(12)\]
Answer.$\frac(37)(12)=3\frac(1)(12)$.
To write a mixed number as an improper fraction, you need to multiply the denominator by the whole part of the number, add the numerator of the fractional part to the resulting product, and write the resulting amount into the numerator of the fraction. The denominator of the improper fraction will be equal to the denominator of the fractional part of the mixed number.
Example 6
Write the mixed number $5\frac(3)(7)$ as an improper fraction.
Solution.
Answer.$5\frac(3)(7)=\frac(38)(7)$.
Adding mixed numbers and proper fractions
Mixed Number Addition$a\frac(b)(c)$ and proper fraction$\frac(d)(e)$ is performed by adding to a given fraction the fractional part of a given mixed number:
Example 7
Add the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$.
Solution.
Let's use the formula for adding a mixed number and a proper fraction:
\[\frac(4)(15)+3\frac(2)(5)=3+\left(\frac(2)(5)+\frac(4)(15)\right)=3+\ left(\frac(2\cdot 3)(5\cdot 3)+\frac(4)(15)\right)=3+\frac(6+4)(15)=3+\frac(10)( 15)\]
By dividing by the number \textit(5) we can determine that the fraction $\frac(10)(15)$ is reducible. Let's perform the reduction and find the result of the addition:
So, the result of adding the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$ is $3\frac(2)(3)$.
Answer:$3\frac(2)(3)$
Adding mixed numbers and improper fractions
Adding improper fractions and mixed numbers reduces to the addition of two mixed numbers, for which it is enough to isolate the whole part from the improper fraction.
Example 8
Calculate the sum of the mixed number $6\frac(2)(15)$ and the improper fraction $\frac(13)(5)$.
Solution.
First, let's extract the integer part from the improper fraction $\frac(13)(5)$:
Answer:$8\frac(11)(15)$.
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 solve 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 this 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 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 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 person solving the problem.
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; fractional part in this example you need to reduce by 2, you get 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, 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. 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.
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– 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. Most close number which 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 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.
