You can do everything with fractions, including division. This article shows the division of ordinary fractions. Definitions will be given and examples will be discussed. Let us dwell in detail on dividing fractions by natural numbers and vice versa. Dividing a common fraction by a mixed number will be discussed.
Dividing fractions
Division is the inverse of multiplication. When dividing unknown multiplier is located at famous work and another factor, where its given meaning is preserved with ordinary fractions.
If it is necessary to divide a common fraction a b by c d, then to determine such a number you need to multiply by the divisor c d, this will ultimately give the dividend a b. Let's get a number and write it a b · d c , where d c is the inverse of the 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 a common fraction a b by c d, you need to multiply the dividend by the reciprocal of the divisor.
Let's write the rule in the form of an expression: a b: c d = a b · d c
The rules of division come down to multiplication. To stick with it, you need to have a good understanding of multiplying fractions.
Let's move on to considering the division of ordinary fractions.
Example 1
Divide 9 7 by 5 3. Write the result as a fraction.
Solution
The number 5 3 is the reciprocal fraction 3 5. It is necessary to use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 = 9 7 · 3 5 = 9 · 3 7 · 5 = 27 35.
Answer: 9 7: 5 3 = 27 35 .
When reducing fractions, separate out the whole part if the numerator is greater than the denominator.
Example 2
Divide 8 15: 24 65. Write the answer as a fraction.
Solution
To solve, you need to move from division to multiplication. Let's 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 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9
Select the whole part and get 13 9 = 1 4 9.
Answer: 8 15: 24 65 = 1 4 9 .
Dividing an extraordinary fraction by a natural number
We use the rule for dividing a fraction by a natural number: to divide a b by a natural number n, you only need to multiply 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 = a b: n 1 = a b · 1 n = a b · n.
Consider this division of a fraction by a number.
Example 3
Divide the fraction 16 45 by the number 12.
Solution
Let's apply the rule for dividing a fraction by a number. We obtain an expression of the form 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 .
Dividing a natural number by a fraction
The division rule is similar O the rule for dividing a natural number by an ordinary fraction: in order to divide a natural number n by an ordinary fraction 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 = 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 = n · b a. It is necessary to consider this division with an example.
Example 4
Divide 25 by 15 28.
Solution
We need to move from division to multiplication. Let's write it in the form of the expression 25: 15 28 = 25 28 15 = 25 28 15. Let's reduce the fraction and get the result in the form of the fraction 46 2 3.
Answer: 25: 15 28 = 46 2 3 .
Dividing a fraction by a mixed number
When dividing a common fraction by a mixed number, you can easily begin to divide common fractions. Need to make a transfer mixed number V improper fraction.
Example 5
Divide the fraction 35 16 by 3 1 8.
Solution
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 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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A fraction is one or more parts of a whole, usually taken to be one (1). As with natural numbers, you can perform all basic arithmetic operations (addition, subtraction, division, multiplication) with fractions; to do this, you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fraction has its own specifics, but once you thoroughly understand how to handle them, you will be able to solve any examples with fractions, since you will know the basic principles of performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by a whole number using different types of fractions.
How to split simple fraction to a natural number?Ordinary or simple fractions are fractions that are written in the form of a ratio of numbers in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated at the bottom. How to divide such a fraction by a whole number? Let's look at an example! Let's say we need to divide 8/12 by 2.
To do this we must perform a number of actions:
Thus, if we are faced with the task of dividing a fraction by a whole number, the solution diagram will look something like this: 
In a similar way, you can divide any ordinary (simple) fraction by an integer.
How to divide a decimal by a whole number?
A decimal is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic operations with decimals are quite simple.
Let's look at an example of how to divide a fraction by a whole number. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.
To summarize, let us dwell on two main points that are important when performing the division operation decimals to an integer: - to divide a decimal fraction by a natural number, long division is used;
- A comma is placed in a quotient when the division of the whole part of the dividend is completed.
Adding fractions with like denominators
There are two types of addition of fractions:
- Adding fractions with like denominators
- Adding fractions with different denominators
First, let's learn the addition of fractions with like 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 . Add the numerators and leave the denominator unchanged:
This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:
Example 2. Add fractions and .
The answer was not proper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:
This example can be easily understood if we remember about a pizza that is divided into two parts. If you add more pizza to the pizza, you get one whole pizza:
Example 3. Add fractions and .
Again, we add up the numerators and leave the denominator unchanged:
This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:
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 drawing. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.
As you can see, there is nothing complicated about adding fractions with the same denominators. 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 let's learn how to add fractions with different denominators. When adding fractions, the denominators of the 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 right away, since 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 look at only one of them, since the other methods may seem complicated for a beginner.
The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.
The numerators and denominators of the fractions are then 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. Let's add the 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 let's return to fractions and . First, 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 multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found 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 multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:
Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:
Look carefully 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 take this example to the end:
This completes the example. It turns out to add .
Let's try to depict our solution using a drawing. If you add pizza to a pizza, you get one whole pizza and another sixth of a pizza:
Reducing fractions to the same (common) denominator can also be depicted using a picture. Reducing the fractions and to a common denominator, we got the fractions and . These two fractions will be represented by the same pieces of pizza. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).
The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).
Please note that we have described this example in too much detail. IN educational institutions It’s not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:
But there is also another side to the coin. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “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 obtain an additional factor for each fraction;
- Multiply the numerators and denominators of fractions by their additional factors;
- Add fractions that have the same denominators;
- If the answer turns out to be an improper fraction, then select its whole part;
Example 2. Find the value of an expression 
.
Let's use the instructions given above.
Step 1. Find the LCM of the denominators of the fractions
Find the LCM of the denominators of both fractions. The denominators of fractions are the numbers 2, 3 and 4
Step 2. Divide the LCM by the denominator of each fraction and get an additional factor 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 above 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 get the second additional factor 4. We write it above 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 get the third additional factor 3. We write it above the third fraction:
Step 3. Multiply the numerators and denominators of the fractions by their additional factors
We multiply the numerators and denominators by their additional factors:
Step 4. Add fractions with the same denominators
We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it 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 moved 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 the 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 turns out to be an improper fraction, then select the whole part of it
Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:
We received an answer
Subtracting fractions with like denominators
There are two types of subtraction of fractions:
- Subtracting fractions with like denominators
- Subtracting fractions with different denominators
First, let's learn how to subtract fractions with like 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, but leave the denominator the same.
For example, let's find the value of the expression . To solve this example, you need 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 remember the pizza, which 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 remember the pizza, which 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 about 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 turns out to be an improper fraction, then you need to highlight the whole part of it.
Subtracting fractions with different denominators
For example, you can subtract a fraction from a fraction because the fractions have the same denominators. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
The common denominator is found using 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 above 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 above the second fraction.
The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators are converted into fractions that have the same denominators. And we already know how to subtract such fractions.
Example 1. Find the meaning of the expression:
These fractions have different denominators, so you need to reduce 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 let's return to fractions and
Let's find an additional factor for the first fraction. To do this, 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. Write a four above the first fraction:
We do the same with the second fraction. 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 three over the second fraction:
Now we are 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 had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:
We received an answer
Let's try to depict our solution using a drawing. If you cut pizza from a pizza, you get pizza
This is the detailed version of the solution. If we were at school, we would have to solve this example shorter. Such a solution would look like this:
Reducing fractions to a common denominator can also be depicted using a picture. Reducing these fractions to a common denominator, we got the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into equal shares (reduced to the same denominator):
The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting 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 first you need to reduce them to the same (common) denominator.
Let's 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, 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 above 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 above 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 above 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 had 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 regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler. What can be done? You can shorten this fraction.
To reduce a fraction, you need to divide its numerator and denominator by (GCD) of the numbers 20 and 30.
So, we find the gcd of 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
We received 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 that number and leave the denominator the same.
Example 1. Multiply a fraction by the number 1.
Multiply the numerator of the fraction by the number 1
The recording can be understood as taking half 1 time. For example, if you take pizza once, you get pizza
From the laws of multiplication we know that if the multiplicand and the factor are swapped, the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying a whole number and a fraction works:
This notation can be understood as taking half of one. 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 was an improper fraction. Let's highlight the whole part of it:
The expression can be understood as taking two quarters 4 times. For example, if you take 4 pizzas, you will get two whole pizzas
And if we swap the multiplicand and the multiplier, we get the expression . It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:
Multiplying fractions
To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.
Example 1. Find the value of the expression.
We received an answer. It is advisable 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 make pizza. Remember what pizza looks like when divided into three parts:
One piece of this pizza and the two pieces we took will have the same dimensions:
In other words, we are talking about the same size pizza. 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 was an improper fraction. Let's highlight the 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 regular fraction, but it would be good if it was shortened. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the largest common divisor(GCD) numbers 105 and 450.
So, let’s find the gcd of numbers 105 and 450:
Now we divide the numerator and denominator of our answer by the gcd that we have now found, that is, by 15
Representing a whole number as a fraction
Any whole number can be represented as a fraction. For example, the number 5 can be represented as . This will not change the meaning of five, since the expression means “the number five divided by one,” and this, as we know, is equal to five:
Reciprocal numbers
Now we will get acquainted with very interesting topic in mathematics. It's called "reverse numbers".
Definition. Reverse to numbera is a number that, when multiplied bya gives one.
Let's substitute in this definition instead of the variable a number 5 and try to read the definition:
Reverse to number 5 is a number that, when multiplied by 5 gives one.
Is it possible to find a number that, when multiplied by 5, gives one? It turns out it is possible. Let's imagine 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 upside down:
What will happen as a result of this? If we continue solving this example, we get one:
This means that the inverse of the number 5 is the number , since when you multiply 5 by you get one.
The reciprocal of a number can also be found for any other integer.
You can also find the reciprocal of any other fraction. To do this, just turn it over.
Dividing a fraction by a number
Let's say we have half a pizza:
Let's divide it equally between two. How much pizza will each person get?
It can be seen that after dividing half the pizza, two equal pieces were obtained, each of which constitutes a pizza. So everyone gets a pizza.
Division of fractions is done using reciprocals. Reciprocal numbers allow you to replace division with multiplication.
To divide a fraction by a number, you need to multiply the fraction by the inverse 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 the fraction and the divisor is the number 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 the fraction. So you need to multiply by
To solve various problems from mathematics and physics courses, you have to divide fractions. This is very easy to do if you know certain rules for performing this mathematical operation.
Before we move on to formulating the rule for dividing fractions, let's remember some mathematical terms:
- The top part of the fraction is called the numerator, and the bottom part is called the denominator.
- When dividing, numbers are called as follows: dividend: divisor = 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 because it is obtained by 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 here is also quite simple and clear. First, we convert the mixed fraction to a regular improper fraction. To do this, multiply the denominator of such a fraction by an integer and add the numerator to the resulting product. As a result, we received a new numerator of the mixed fraction, but its denominator will remain unchanged. Further, the division of fractions will be carried out in exactly 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 (irregular). 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 performed 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 dividing a whole number or a decimal fraction by a decimal fraction without the help of a calculator.
So, to divide decimals, you just need to cross out the comma in the divisor and stop paying attention to it. In the dividend, the comma must be moved to the right exactly as many places as it was in the fractional part of the divisor, adding zeros if necessary. And then they perform the usual division by an integer. To make this more clear, consider the following example.
Multiplying and dividing fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:
For example:
Everything is extremely simple. And please don't look common denominator! There is no need for him here...
To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:
For example:
If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:
In high school, you often have to deal with three-story (or even four-story!) fractions. For example:
How can I make this fraction look decent? Yes, very simple! Use two-point division:
But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:
then divide and multiply in order, from left to right!
And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:
The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.
That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer of them (errors)!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! Is not common words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.
2. In examples with different types fractions - go to ordinary fractions.
3. We reduce all fractions until they stop.
4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.
So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.
Calculate:
Have you decided?
We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.
0; 17/22; 3/4; 2/5; 1; 25.
Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.
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