With fractions, you can perform all actions, including division. This article shows the division of ordinary fractions. Definitions will be given, examples will be considered. Let us dwell on the division of fractions by natural numbers and vice versa. The division of an ordinary fraction by a mixed number will be considered.
Division of ordinary fractions
Division is the inverse of multiplication. When dividing, the unknown factor is at the known product and another factor, where its given meaning is preserved with ordinary fractions.
If it is necessary to divide the ordinary fraction a b by c d, then to determine such a number, you need to multiply by the divisor c d, this will eventually give the dividend a b. Let's get a number and write it a b · d c , where d c is the reciprocal of c d number. Equalities can be written using the properties of multiplication, namely: a b d c c d = a b d c c d = a b 1 = a b , where the expression a b d c is the quotient of dividing a b by c d .
From here we obtain and formulate the rule for dividing ordinary fractions:
Definition 1
To divide an ordinary fraction a b by c d, it is necessary to multiply the dividend by the reciprocal of the divisor.
Let's write the rule as an expression: a b: c d = a b d c
The rules of division are reduced to multiplication. To stick to it, you need to be well versed in performing multiplication of ordinary fractions.
Let's move on to the division of ordinary fractions.
Example 1
Perform division 9 7 by 5 3 . Write the result as a fraction.
Decision
The number 5 3 is the reciprocal of 3 5 . You must use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 \u003d 9 7 3 5 \u003d 9 3 7 5 \u003d 27 35.
Answer: 9 7: 5 3 = 27 35 .
When reducing fractions, you should highlight the whole part if the numerator is greater than the denominator.
Example 2
Divide 8 15: 24 65 . Write the answer as a fraction.
Decision
The solution is to switch from division to multiplication. We write it in this form: 8 15: 24 65 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
It is necessary to make a reduction, and this is done as follows: 8 65 15 24 \u003d 2 2 2 5 13 3 5 2 2 2 3 \u003d 13 3 3 \u003d 13 9
We select the integer part and get 13 9 = 1 4 9 .
Answer: 8 15: 24 65 = 1 4 9 .
Division of an extraordinary fraction by a natural number
We use the rule of dividing a fraction by a natural number: to divide a b by a natural number n, you need to multiply only the denominator by n. From here we get the expression: a b: n = a b · n .
The division rule is a consequence of the multiplication rule. Therefore, representing a natural number as a fraction will give an equality of this type: a b: n \u003d a b: n 1 \u003d a b 1 n \u003d a b n.
Consider this division of a fraction by a number.
Example 3
Divide the fraction 1645 by the number 12.
Decision
Apply the rule for dividing a fraction by a number. We get an expression like 16 45: 12 = 16 45 12 .
Let's reduce the fraction. We get 16 45 12 = 2 2 2 2 (3 3 5) (2 2 3) = 2 2 3 3 3 5 = 4 135 .
Answer: 16 45: 12 = 4 135 .
Division of a natural number by a common fraction
The division rule is similar about the rule of dividing a natural number by an ordinary fraction: to divide a natural number n by an ordinary a b , it is necessary to multiply the number n by the reciprocal of the fraction a b .
Based on the rule, we have n: a b \u003d n b a, and thanks to the rule of multiplying a natural number by an ordinary fraction, we get our expression in the form n: a b \u003d n b a. It is necessary to consider this division with an example.
Example 4
Divide 25 by 15 28 .
Decision
We need to move from division to multiplication. We write in the form of an expression 25: 15 28 = 25 28 15 = 25 28 15 . Let's reduce the fraction and get the result in the form of a fraction 46 2 3 .
Answer: 25: 15 28 = 46 2 3 .
Division of a common fraction by a mixed number
When dividing an ordinary fraction by a mixed number, you can easily shine to dividing ordinary fractions. You need to convert a mixed number to an improper fraction.
Example 5
Divide the fraction 35 16 by 3 1 8 .
Decision
Since 3 1 8 is a mixed number, let's represent it as an improper fraction. Then we get 3 1 8 = 3 8 + 1 8 = 25 8 . Now let's divide the fractions. We get 35 16: 3 1 8 = 35 16: 25 8 = 35 16 8 25 = 35 8 16 25 = 5 7 2 2 2 2 2 2 2 (5 5) = 7 10
Answer: 35 16: 3 1 8 = 7 10 .
Dividing a mixed number is done in the same way as ordinary numbers.
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Lesson contentAdding fractions with the same denominators
Adding fractions is of two types:
- Adding fractions with the same denominators
- Adding fractions with different denominators
Let's start with adding fractions with the same denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators, and leave the denominator unchanged. For example, let's add the fractions and . We add the numerators, and leave the denominator unchanged:
This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:
Example 2 Add fractions and .
The answer is an improper fraction. If the end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the integer part is allocated easily - two divided by two is equal to one:
This example can be easily understood if we think of a pizza that is divided into two parts. If you add more pizzas to the pizza, you get one whole pizza:
Example 3. Add fractions and .
Again, add the numerators, and leave the denominator unchanged:
This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:
Example 4 Find the value of an expression 
This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:
Let's try to depict our solution using a picture. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.
As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:
- To add fractions with the same denominator, you need to add their numerators, and leave the denominator unchanged;
Adding fractions with different denominators
Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those fractions must be the same. But they are not always the same.
For example, fractions can be added because they have the same denominators.
But fractions cannot be added at once, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
There are several ways to reduce fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem complicated for a beginner.
The essence of this method lies in the fact that first (LCM) of the denominators of both fractions is sought. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and the second additional factor is obtained.
Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.
Example 1. Add fractions and
First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6
LCM (2 and 3) = 6
Now back to fractions and . First, we divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.
The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor above it:
We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.
The resulting number 3 is the second additional factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:
Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:
Look closely at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's complete this example to the end:
Thus the example ends. To add it turns out.
Let's try to depict our solution using a picture. If you add pizzas to a pizza, you get one whole pizza and another sixth of a pizza:
Reduction of fractions to the same (common) denominator can also be depicted using a picture. Bringing the fractions and to a common denominator, we get the fractions and . These two fractions will be represented by the same slices of pizzas. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).
The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).
Note that we have painted this example in too much detail. In educational institutions it is not customary to write in such a detailed manner. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the additional factors found by your numerators and denominators. While at school, we would have to write this example as follows:
But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.
To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:
- Find the LCM of the denominators of fractions;
- Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
- Multiply the numerators and denominators of fractions by their additional factors;
- Add fractions that have the same denominators;
- If the answer turned out to be an improper fraction, then select its whole part;
Example 2 Find the value of an expression 
.
Let's use the instructions above.
Step 1. Find the LCM of the denominators of fractions
Find the LCM of the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4
Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction
Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:
Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:
Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:
Step 3. Multiply the numerators and denominators of fractions by your additional factors
We multiply the numerators and denominators by our additional factors:
Step 4. Add fractions that have the same denominators
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add up:
The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.
Step 5. If the answer turned out to be an improper fraction, then select the whole part in it
Our answer is an improper fraction. We must single out the whole part of it. We highlight:
Got an answer
Subtraction of fractions with the same denominators
There are two types of fraction subtraction:
- Subtraction of fractions with the same denominators
- Subtraction of fractions with different denominators
First, let's learn how to subtract fractions with the same denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.
For example, let's find the value of the expression . To solve this example, it is necessary to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. Let's do this:
This example can be easily understood if we think of a pizza that is divided into four parts. If you cut pizzas from a pizza, you get pizzas:
Example 2 Find the value of the expression .
Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:
This example can be easily understood if we think of a pizza that is divided into three parts. If you cut pizzas from a pizza, you get pizzas:
Example 3 Find the value of an expression 
This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:
As you can see, there is nothing complicated in subtracting fractions with the same denominators. It is enough to understand the following rules:
- To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
- If the answer turned out to be an improper fraction, then you need to select the whole part in it.
Subtraction of fractions with different denominators
For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.
The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.
Example 1 Find the value of an expression:
These fractions have different denominators, so you need to bring them to the same (common) denominator.
First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12
LCM (3 and 4) = 12
Now back to fractions and
Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:
We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a triple over the second fraction:
Now we are all set for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's complete this example to the end:
Got an answer
Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.
This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:
Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):
The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.
Example 2 Find the value of an expression 
These fractions have different denominators, so you first need to bring them to the same (common) denominator.
Find the LCM of the denominators of these fractions.
The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30
LCM(10, 3, 5) = 30
Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.
Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:
Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:
Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:
Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.
The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:
The answer turned out to be a correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it easier. What can be done? You can reduce this fraction.
To reduce a fraction, you need to divide its numerator and denominator by (gcd) the numbers 20 and 30.
So, we find the GCD of the numbers 20 and 30:
Now we return to our example and divide the numerator and denominator of the fraction by the found GCD, that is, by 10
Got an answer
Multiplying a fraction by a number
To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator the same.
Example 1. Multiply the fraction by the number 1.
Multiply the numerator of the fraction by the number 1
The entry can be understood as taking half 1 time. For example, if you take pizza 1 time, you get pizza
From the laws of multiplication, we know that if the multiplicand and the multiplier are interchanged, then the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying an integer and a fraction works:
This entry can be understood as taking half of the unit. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:
Example 2. Find the value of an expression
Multiply the numerator of the fraction by 4
The answer is an improper fraction. Let's take a whole part of it:
The expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.
And if we swap the multiplicand and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:
Multiplication of fractions
To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.
Example 1 Find the value of the expression .
Got an answer. It is desirable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:
The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:
How to take two-thirds from this half? First you need to divide this half into three equal parts:
And take two from these three pieces:
We'll get pizza. Remember what a pizza looks like divided into three parts:
One slice from this pizza and the two slices we took will have the same dimensions:
In other words, we are talking about the same pizza size. Therefore, the value of the expression is
Example 2. Find the value of an expression
Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:
The answer is an improper fraction. Let's take a whole part of it:
Example 3 Find the value of an expression
Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:
The answer turned out to be a correct fraction, but it will be good if it is reduced. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.
So, let's find the GCD of the numbers 105 and 450:
Now we divide the numerator and denominator of our answer to the GCD that we have now found, that is, by 15
Representing an integer as a fraction
Any whole number can be represented as a fraction. For example, the number 5 can be represented as . From this, the five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:
Reverse numbers
Now we will get acquainted with a very interesting topic in mathematics. It's called "reverse numbers".
Definition. Reverse to numbera is the number that, when multiplied bya gives a unit.
Let's substitute in this definition instead of a variable a number 5 and try to read the definition:
Reverse to number 5 is the number that, when multiplied by 5 gives a unit.
Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent five as a fraction:
Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let's multiply the fraction by itself, only inverted:
What will be the result of this? If we continue to solve this example, we get one:
This means that the inverse of the number 5 is the number, since when 5 is multiplied by one, one is obtained.
The reciprocal can also be found for any other integer.
You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.
Division of a fraction by a number
Let's say we have half a pizza:
Let's divide it equally between two. How many pizzas will each get?
It can be seen that after splitting half of the pizza, two equal pieces were obtained, each of which makes up a pizza. So everyone gets a pizza.
Division of fractions is done using reciprocals. Reciprocals allow you to replace division with multiplication.
To divide a fraction by a number, you need to multiply this fraction by the reciprocal of the divisor.
Using this rule, we will write down the division of our half of the pizza into two parts.
So, you need to divide the fraction by the number 2. Here the dividend is a fraction and the divisor is 2.
To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is a fraction. So you need to multiply by
Last time we learned how to add and subtract fractions (see the lesson "Addition and subtraction of fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.
To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the "inverted" second.
Designation:
From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what exactly will not happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.
By definition we have:
Multiplication of fractions with an integer part and negative fractions
If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:
- We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.
A task. Find the value of the expression:
We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:
Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
A task. Find the value of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what is left of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.
There is simply no other reason to reduce fractions, so the correct solution to the previous problem looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
Sooner or later, all children in school begin to learn fractions: their addition, division, multiplication, and all the possible actions that it is only possible to perform with fractions. In order to provide proper assistance to the child, parents themselves should not forget how whole numbers are divided into fractions, otherwise, you will not be able to help him in any way, but only confuse him. If you need to remember this action, but you can’t bring all the information in your head into a single rule, then this article will help you: you will learn how to divide a number by a fraction and see illustrative examples.
How to divide a number into a fraction
Write down your example on a draft so that you can take notes and blots. Remember that an integer is written between cells, right at their intersection, and fractional numbers - each in its own cell.
- In this method, you need to turn the fraction upside down, that is, write the denominator to the numerator, and the numerator to the denominator.
- The sign of division must be changed to multiplication.
- Now you just have to perform the multiplication according to the rules already studied: the numerator is multiplied by an integer, and the denominator is not touched.
Of course, as a result of such an action, you will get a very large number in the numerator. It is impossible to leave a fraction in this state - the teacher simply will not accept this answer. Reduce the fraction by dividing the numerator by the denominator. Write the resulting integer to the left of the fraction in the middle of the cells, and the remainder will be the new numerator. The denominator remains unchanged.
This algorithm is quite simple, even for a child. After completing it five or six times, the baby will remember the procedure and will be able to apply it to any fractions.
How to divide a number by a decimal
There are other types of fractions - decimals. The division into them occurs according to a completely different algorithm. If you are faced with such an example, then follow the instructions:
- First, convert both numbers to decimals. This is easy to do: your divisor is already represented as a fraction, and you separate the divisible natural number with a comma, getting a decimal fraction. That is, if the dividend was the number 5, you get a fraction of 5.0. You need to separate the number by as many digits as it stands after the decimal point and the divisor.
- After that, you must make both decimal fractions natural numbers. It may seem a little confusing at first, but it is the fastest way to divide and will take you seconds after a few practice sessions. A fraction of 5.0 will become the number 50, a fraction of 6.23 will be 623.
- Do the division. If the numbers turned out to be large, or the division will occur with a remainder, perform it in a column. So you will clearly see all the actions of this example. You do not need to specifically put a comma, as it will appear itself in the process of dividing into a column.
This kind of division initially seems too confusing, since you need to turn the dividend and divisor into a fraction, and then back into natural numbers. But after a short training, you will immediately begin to see those numbers that you just need to divide by each other.
Remember that the ability to correctly divide fractions and integers into them can be useful more than once in life, therefore, the child needs to know these rules and simple principles perfectly so that in older grades they do not become a stumbling block because of which the child cannot decide more complex tasks.
To solve various tasks from the course of mathematics, physics has to divide fractions. This is very easy to do if you know certain rules for performing this mathematical operation.
Before moving on to formulate a rule on how to divide fractions, let's recall some mathematical terms:
- The top of a fraction is called the numerator and the bottom is called the denominator.
- When dividing, numbers are called like this: dividend: divisor \u003d quotient
How to divide fractions: simple fractions
To divide two simple fractions, multiply the dividend by the reciprocal of the divisor. This fraction is also called inverted in another way, because it is obtained as a result of swapping the numerator and denominator. For example:
3/77: 1/11 = 3 /77 * 11 /1 = 3/7
How to divide fractions: mixed fractions
If we have to divide mixed fractions, then everything is also quite simple and clear here. First, convert the mixed fraction to an ordinary improper fraction. To do this, we multiply the denominator of such a fraction by an integer and add the numerator to the resulting product. As a result, we got a new numerator of the mixed fraction, and its denominator will remain unchanged. Further division of fractions will be carried out in the same way as the division of simple fractions. For example:
10 2/3: 4/15 = 32/3: 4/15 = 32/3 * 15 /4 = 40/1 = 40
How to divide a fraction by a number
In order to divide a simple fraction by a number, the latter should be written as a fraction (improper). This is very easy to do: this number is written in place of the numerator, and the denominator of such a fraction is equal to one. Further division is carried out in the usual way. Let's look at this with an example:
5/11: 7 = 5/11: 7/1 = 5/11 * 1/7 = 5/77
How to divide decimals
Often, an adult has difficulty, if necessary, without the help of a calculator, to divide an integer or a decimal fraction into a decimal fraction.
So, in order to divide decimal fractions, you just need to cross out the comma in the divisor and stop paying attention to it. In the divisible, the comma must be moved to the right exactly as many characters as it was in the fractional part of the divisor, adding zeros if necessary. And then produce the usual division by an integer. To make this clearer, let's take the following example.
