Examples with fractions are one of the basic elements of mathematics. There are many different types of equations with fractions. Below are detailed instructions for solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, be it addition, subtraction, multiplication or division, you need to know the basic rules:
- In order to add fractional expressions with the same denominator (the denominator is the number located at the bottom of the fraction, the numerator is at the top), you need to add their numerators, and leave the denominator the same.
- In order to subtract a second fractional expression (with the same denominator) from one fraction, you need to subtract their numerators and leave the denominator the same.
- To add or subtract fractions with different denominators, you need to find the lowest common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, and, if possible, reduce.
- To divide a fraction by a fraction, you multiply the first fraction by the second fraction reversed.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if two (or more) fractions have the same denominator, you simply add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will equal 1.
Answer: 3/4 + 1/4 = 4/4 = 1.
Rule 2, example 1:
Calculate: 3/4 – 1/4
Using rule number 2, to solve this equation you need to subtract 1 from 3 and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.
Answer: 3/4 – 1/4 = 2/4 = 1/2.
Rule 3, Example 1
Calculate: 3/4 + 1/6
Solution: Using the 3rd rule, we find the lowest common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions in the example. Thus, we need to find the minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. Divide 12 by the denominator of the first fraction, we get 3, multiply by 3, write 3 in the numerator *3 and + sign. Divide 12 by the denominator of the second fraction, we get 2, multiply 2 by 1, write 2*1 in the numerator. So, we get a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.
Answer: 11/12
Rule 3, Example 2:
Calculate 3/4 – 1/6. This example is very similar to the previous one. We do all the same steps, but in the numerator instead of the + sign, we write a minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.
Answer: 7/12
Rule 4, Example 1:
Calculate: 3/4 * 1/4
Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.
Answer: 3/16
Rule 4, Example 2:
Calculate 2/5 * 10/4.
This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are canceled.
2 cancels from 4. 10 cancels from 5. We get 1 * 2/2 = 1*1 = 1.
Answer: 2/5 * 10/4 = 1
Rule 5, Example 1:
Calculate: 3/4: 5/6
Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.
Answer: 9/10.
How to solve examples with fractions - fractional equations
Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.
Let's look at an example:
Solve the equation 15/3x+5 = 3
Let us remember that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. For this purpose, there is an OA (permissible value range).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
At x = 5/3 the equation simply has no solution.
Having specified the ODZ, the best way to solve this equation is to get rid of the fractions. To do this, we first present all non-fractional values as a fraction, in this case the number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions you need to multiply each of them by the lowest common denominator. In this case it will be (3x+5)*1. Sequence of actions:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Open the brackets: 15*(3x+5) = 45x + 75.
- We do the same with the right side of the equation: 3*(3x+5) = 9x + 15.
- Equate the left and right sides: 45x + 75 = 9x +15
- Move the X's to the left, numbers to the right: 36x = – 50
- Find x: x = -50/36.
- We reduce: -50/36 = -25/18
Answer: ODZ x ≠ 5/3. x = -25/18.
How to solve examples with fractions - fractional inequalities
Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the number axis. Let's look at this example.
Sequence of actions:
- We equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
2. 2-x=0 => x=2 - We draw a number axis, writing the resulting values on it.
- Draw a circle under the value. There are two types of circles - filled and empty. A filled circle means that the given value is within the solution range. An empty circle indicates that this value is not included in the solution area.
- Since the denominator cannot be equal to zero, there will be an empty circle under the 2nd.
- To determine the signs, we substitute any number greater than two into the equation, for example 3. (3*3-5)/(2-3)= -4. the value is negative, which means we write a minus above the area after the two. Then substitute for X any value of the interval from 5/3 to 2, for example 1. The value is again negative. We write a minus. We repeat the same with the area located up to 5/3. We substitute any number less than 5/3, for example 1. Again, minus.
- Since we are interested in the values of x at which the expression will be greater than or equal to 0, and there are no such values (there are minuses everywhere), this inequality has no solution, that is, x = Ø (an empty set).
Answer: x = Ø
Fraction- a form of representing numbers in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.
Correct A fraction whose numerator is greater than its denominator is called a fraction. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.
The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:
The main property of a fraction
If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Reducing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- Multiply the numerator of the first fraction by the denominator of the second
- Multiply the numerator of the second fraction by the denominator of the first
- Replace the denominators of both fractions with their product
Operations with fractions
Addition. To add two fractions you need
- Add the new numerators of both fractions and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Reduce fractions to a common denominator
- Subtract the numerator of the second from the numerator of the first fraction and leave the denominator unchanged
Example:
Multiplication. To multiply one fraction by another, multiply their numerators and denominators:
Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:
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 for a 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. Take practical advice into account, and there will be fewer of them (mistakes)!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general 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 of fractions, we move on 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 definitely need to solve. 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. Right 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. And 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.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Lesson contentAdding 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 turned out to be an improper 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 pizza 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 is 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 obtain 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 highlight 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. The 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. The 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 greatest common divisor (GCD) of the 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 a 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 to solve 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 number for 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
I myself was faced with the fact that fractions turned out to be a rather difficult topic for my children.
There is a very good game Nikitin's Fractions, it is intended for preschoolers, but also at school it will perfectly help the child figure out what they are - fractions, their relationship to each other..., and all in an accessible, visual and exciting form.
It consists of twelve multi-colored circles. One circle is whole, and all the rest are divided into equal parts - two, three.... (up to twelve).
The child is asked to complete simple game tasks, for example:
What are the parts of the circles called? or
Which part is bigger? (put the smaller one on top of the larger one.)
This technique helped me. In general, I really regret that all these Nikitin developments did not catch my eye when the children were still babies.
You can make the game yourself or buy a ready-made one, and find out more about everything -.
Solving fractions can also be explained using Lego bricks. It develops not only imagination, but also creative and logical thinking, which means it can also be used as a teaching aid.
Alicia Zimmerman came up with the idea of using the blocks of the famous designer to teach children the basics of mathematics.
And here's how to explain fractions using Lego.
Practice shows that the most difficulties arise when adding (subtracting) fractions with different denominators and when dividing fractions.
Difficulties arise due to incorrect instructions in the textbook, such as dividing a fraction by a fraction.
To divide a fraction by a fraction, you multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction.
Can a child in 4th grade understand this and not get confused? NO!
And the teacher explained it to us in an elementary way: we need to turn the second fraction over and then multiply it!
Same thing with addition.
To add two fractions, you need to multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction, add the resulting numbers and write them in the numerator. And in the denominator you need to write the product of the denominators of the fractions. After this, the resulting fraction can (or should) be reduced.
And it’s simpler: Reduce the fractions to a common denominator, which is equal to the LCM of the denominators, and then add the numerators.
Show them with a clear example. For example, cut an apple into 4 parts, put it into 8 parts, add 12 parts into a whole, add several parts, subtract. At the same time, explain on paper using rules. Rules for addition and subtraction. dividing fractions, as well as how to isolate a whole from an improper fraction - learn all this while manipulating with an apple. Do not rush the children; let them carefully sort out the slices with your help.
Teaching children to solve fractions, in particular, is quite common and will not create much trouble. The simplest thing you can do is take something whole, for example a tangerine, or any other fruit, divide it into parts, and use an example to show subtraction, addition and other operations with pieces of this fruit, which will be fractions from the whole. Everything needs to be explained and shown, and the final factor will be to explain and solve problems together using mathematical examples until the child learns to do these tasks himself.
The figure clearly shows what corresponds to what and how the fraction looks on a real object, this is exactly how it needs to be explained.
You need to approach this issue thoroughly, since solving fractions will come in handy in life. It is necessary in this matter, as they say, to be on an equal footing with children, and to explain the theory in a language they understand, for example, in the language of cake or tangerine. You need to divide the cake into do and give it to friends, after which the child will begin to understand the essence of solving fractions. Don't start with heavy fractions, start with the concepts of 1/2, 1/3, 1/10. First, subtract and add, and then move on to more complex concepts like multiplication and division.
There are different types of problems with fractions. One child cannot understand that one second and five tenths are the same thing, others are perplexed by bringing different fractions to the same denominator, and still others are confused by the division of fractions. Therefore, there is no one rule for all occasions.
The main thing in problems involving fractions is not to miss the moment when what is understandable ceases to be so. Return to the stove and repeat everything all over again, even if it seems wretchedly primitive. For example, go back to what is one second.
The child must understand that mathematical concepts are abstract, that the same phenomenon can be described in different words and expressed in different numbers.
I like the answer given by Mefody66. I will add from many years of personal practice: teaching how to solve problems with fractions (and not solving fractions; solving fractions is impossible, just as it is impossible to solve numbers) is quite simple, you just need to be close to the child when he first starts solving such problems, and correct his solution in time , so that mistakes, which are inevitable in any learning, do not have time to take hold in the child’s mind. Relearning is more difficult than learning something new. And solve such problems as much as possible. Bringing the solution of such tasks to automaticity would be a good thing to do. The ability to solve problems with ordinary fractions is as important in a school mathematics course as knowledge of the multiplication table. So you need to take the time to watch how your child solves such problems.
And don’t rely too much on the textbook: teachers in schools explain exactly as Mefody66 wrote in his answer. It is better to talk with the teacher, find out in what words the teacher explained this topic. And use the same words and phrases if possible (so as not to confuse the child too much)
Also: I advise you to use visual examples only at the initial stage of explanation, then quickly abstract and move on to the solution algorithm. Otherwise, clarity may be detrimental when solving more complex problems. For example, if you need to add fractions with denominators 29 and 121, what kind of visual aid will help? It will only confuse.
Fractions are one of those blessed mathematical topics where there are no abstractions that are not applicable. Products should be used (on cakes, like Juanita Solis in Desperate Housewives - a really cool method of explanation). All these numerator-denominators come later. Then it is necessary for the child to understand that dividing by a fraction is no longer a decrease at all, and multiplication is not an increase. Here it is better to show how to divide by a fraction in the form of multiplication by inversion. Present the abbreviation in a playful way; if they are divided by one number, then divide, it almost turns out to be Sudoku, if you are interested. The main thing is to notice misunderstandings in time, because further on there will be more interesting topics that are not easy to understand. Therefore, have more practice solving fractions and everything will get better quickly. To me, the purest humanist, far from the slightest degree of abstraction, fractions have always been clearer than other topics.
