How to solve operations with fractions. How to add fractions with different denominators

Multiplication and division of fractions.

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This operation is much nicer than addition-subtraction! Because it's easier. I remind you: 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! Don't need it here...

To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:

For example:

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How to bring this fraction to a decent form? Yes, very easy! Use division through two points:

But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:

then divide-multiply in order, left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:

The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, 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 common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. We reduce all fractions to the stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.

So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. But only Then look at the answers.

Calculate:

Did you decide?

Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.

0; 17/22; 3/4; 2/5; 1; 25.

And now we draw conclusions. If everything worked out - happy for you! Elementary 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. Learning - with interest!)

you can get acquainted with functions and derivatives.

Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.
Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation of the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...

Option 1- you can convert them into ordinary ones and then calculate them.

Option 2- you can separately "work" with the integer and fractional parts.

Examples (2):

And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.

Examples (3):

* Translated into ordinary fractions, calculated the difference, converted the resulting improper fraction into a mixed one.

* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) the unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted into improper ones, then perform the necessary action. After that, if as a result we get an improper fraction, we translate it into a mixed one.

Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.

Consider simple examples:

In these examples, we immediately see how one of the fractions can be converted to get equal denominators.

If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.

That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.

Method SECOND.

Multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.

Example:

*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.

Consider an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THIRD.

Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? This is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each of the numbers into SIMPLE factors

- write out the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Consider examples:

50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number, one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number, two and three are missing

=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144

* The least common multiple of two prime numbers is equal to their product

Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Consider examples:

*51 = 3∙17 119 = 7∙17

in the expansion of a larger number, a triple is missing

=> LCM(51,119) = 3∙7∙17

And now we apply the first method:

* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.

More examples:

* In the second example, it is already clear that the smallest number that is divisible by 40 and 60 is 120.

TOTAL! GENERAL CALCULATION ALGORITHM!
- we bring fractions to ordinary ones, if there is an integer part.
- we bring fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act through the other methods indicated above).
- having received fractions with equal denominators, we perform actions (addition, subtraction).
- if necessary, we reduce the result.
- if necessary, select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

Almost every fifth grader after the first acquaintance with ordinary fractions is in a little shock. Not only do you still need to understand the essence of fractions, but you still have to perform arithmetic operations with them. After that, little students will systematically interrogate their teacher, find out when these fractions will run out.

To avoid such situations, it is enough just to explain this difficult topic to children as simply as possible, and preferably in a playful way.

The essence of the fraction

Before you learn what a fraction is, the child must get acquainted with the concept share . Here the associative method is best suited.

Imagine a whole cake that has been divided into several equal parts, let's say four. Then each piece of the cake can be called a share. If you take one of the four pieces of cake, then it will be one-fourth of a share.

The shares are different, because the whole can be divided into a completely different number of parts. The more shares in general, the smaller they are, and vice versa.

So that the shares could be designated, they came up with such a mathematical concept as common fraction. The fraction will allow us to write down as many shares as needed.

The components of a fraction are the numerator and denominator, which are separated by a fractional bar or a slash. Many children do not understand their meaning, and therefore the essence of the fraction is not clear to them. The fractional bar indicates division, there is nothing complicated here.

It is customary to write the denominator below, under the fractional line or to the right of the overlay line. It shows the number of parts of the whole. The numerator, it is written above the fractional line or to the left of the oblique line, determines how many shares were taken. For example, the fraction 4/7. In this case, 7 is the denominator, shows that there are only 7 shares, and the numerator 4 indicates that four of the seven shares were taken.

The main shares and their record in fractions:

In addition to the ordinary, there is also a decimal fraction.

Actions with fractions Grade 5

In the fifth grade, they learn to perform all arithmetic operations with fractions.

All actions with fractions are performed according to the rules, and it’s not worth hoping that without learning the rule everything will turn out by itself. Therefore, do not neglect the oral part of your math homework.

We have already understood that the decimal and ordinary fractions are different, therefore, arithmetic operations will be performed differently. Actions with ordinary fractions depend on those numbers that are in the denominator, and in decimal, after the decimal point on the right.

For fractions that have the same denominators, the addition and subtraction algorithm is very simple. Actions are performed only with numerators.

For fractions with different denominators, find Least Common Denominator (LCD). This is the number that will be divided without a remainder by all denominators, and will be the smallest of such numbers, if there are several of them.

To add or subtract decimals, you need to write them in a column, comma under comma, and equalize the number of decimal places if necessary.

To multiply ordinary fractions, simply find the product of the numerators and denominators. A very simple rule.

The division is performed according to the following algorithm:

Dividend to write without change
Division turn into multiplication
Flip the divisor (write the reciprocal of the divisor)
Perform multiplication

Addition of fractions, explanation

Let's take a closer look at how to add common and decimal fractions.

As you can see in the image above, the fractions one third and two thirds have a common denominator three. So it is required to add only the numerators one and two, and leave the denominator unchanged. The result is three thirds. Such an answer, when the numerator and denominator of the fraction are equal, can be written as 1, since 3:3 = 1.

It is required to find the sum of fractions two thirds and two ninths. In this case, the denominators are different, 3 and 9. To perform the addition, you need to find a common one. There is a very simple way. We choose the largest denominator, this is 9. We check whether it is divisible by 3. Since 9:3 = 3 without a remainder, therefore 9 is suitable as a common denominator.

The next step is to find additional factors for each numerator. To do this, we divide the common denominator 9 in turn by the denominator of each fraction, the resulting numbers will be added. plural For the first fraction: 9:3 \u003d 3, we add 3 to the numerator of the first fraction. For the second fraction: 9:9 \u003d 1, one can not be added, since when multiplied by it, the same number will be obtained.

Now we multiply the numerators by their complementary factors and add the results. The resulting amount is a fraction of eight ninths.

Adding decimals follows the same rules as adding natural numbers. In a column, the discharge is written below the discharge. The only difference is that in decimal fractions, you need to correctly put a comma in the result. To do this, the fractions are written comma under the comma, and in the sum it is only required to carry the comma down.

Let's find the sum of fractions 38, 251 and 1, 56. To make it more convenient to perform the actions, we leveled the number of decimal places on the right by adding 0.

Adding fractions, ignoring the comma. And in the resulting amount, just drop the comma down. Answer: 39, 811.

Subtraction of fractions, explanation

To find the difference between two-thirds and one-third fractions, you need to calculate the difference between the numerators 2-1 = 1, and leave the denominator unchanged. In the answer we get a difference of one third.

Find the difference between five sixths and seven tenths. We find a common denominator. We use the selection method, out of 6 and 10, the largest is 10. We check: 10: 6 is not divisible without a remainder. We add another 10, it turns out 20:6, it also cannot be divided without a remainder. Again we increase by 10, we got 30:6 = 5. The common denominator is 30. The NOZ can also be found from the multiplication table.

We find additional factors. 30:6 = 5 - for the first fraction. 30:10 = 3 - for the second. We multiply the numerators and their additional multiplier. We get 25/30 reduced and 21/30 subtracted. Next, we subtract the numerators, and leave the denominator unchanged.

The result is a difference of 4/30. The fraction is abbreviated. Divide it by 2. The answer is 2/15.

Division of decimal fractions Grade 5

There are two options for this topic:

Multiplication of decimal fractions Grade 5

Remember how you multiply natural numbers, in exactly the same way you find the product of decimal fractions. First, let's figure out how to multiply a decimal fraction by a natural number. For this:

When multiplying a decimal by a decimal, we act in the same way.

Mixed fractions Grade 5

Five-graders like to call such fractions not mixed, but<<смешные>> probably easier to remember. Mixed fractions are called so because they are obtained by combining a whole natural number and an ordinary fraction.

A mixed fraction consists of an integer part and a fractional part.

When reading such fractions, the whole part is first called, then the fractional part: one whole two thirds, two whole one fifth, three whole two fifths, four point three fourths.

How are they obtained, these mixed fractions? Everything is pretty simple. When we get an improper fraction in the answer (a fraction whose numerator is greater than the denominator), we must always convert it to a mixed one. Just divide the numerator by the denominator. This action is called extracting the integer part:

Converting a mixed fraction back to an improper one is also easy:

Examples with decimals Grade 5 with explanation

Many questions in children are caused by examples of several actions. Let's look at a couple of such examples.

(0.4 8.25 - 2.025) : 0.5 =

The first step is to find the product of the numbers 8.25 and 0.4. We carry out multiplication according to the rule. In the answer, we count from right to left three characters and put a comma.

The second action is in the same place in brackets, this is the difference. Subtract 2.025 from 3.300. We write the action in a column, a comma under a comma.

The third action is division. The resulting difference in the second action is divided by 0.5. The comma is carried over by one character. Result 2.55.

Answer: 2.55.

(0, 93 + 0, 07) : (0, 93 — 0, 805) =

The first action is the sum in brackets. We put it in a column, remember that the comma is under the comma. We get the answer 1.00.

The second action is the difference from the second parenthesis. Since the minuend has fewer decimal places than the subtrahend, we add the missing one. The result of the subtraction is 0.125.

The third step is to divide the sum by the difference. The comma is carried over to three digits. The result was a division of 1000 by 125.

Answer: 8.

Examples with ordinary fractions with different denominators Grade 5 with explanation

In the first example, we find the sum of fractions 5/8 and 3/7. The common denominator will be the number 56. We find additional multipliers, divide 56:8 \u003d 7 and 56:7 \u003d 8. We add them to the first and second fractions, respectively. We multiply the numerators and their factors, we get the sum of fractions 35/56 and 24/56. We got the sum 59/56. The fraction is incorrect, we translate it into a mixed number. The rest of the examples are solved in a similar way.

Examples with fractions grade 5 for training

For convenience, convert mixed fractions to improper and follow the steps.

How to teach a child to easily solve fractions with Lego

With the help of such a constructor, you can not only develop the child’s imagination well, but also explain clearly in a playful way what a share and a fraction are.

The picture below shows that one part with eight circles is a whole. So, taking a puzzle with four circles, you get half, or 1/2. The picture clearly shows how to solve examples with Lego, if you count the circles on the details.

You can build turrets from a certain number of parts and label each of them, as in the picture below. For example, take a turret of seven parts. Each part of the green constructor will be 1/7. If you add two more to one such part, you get 3/7. Visual explanation of the example 1/7+2/7 = 3/7.

To get A's in math, don't forget to learn the rules and practice them.

Let's go to battle with math homework! The enemy is recalcitrant fractions. Grade 5 program. A strategically important task is to explain fractions to the child. Let's change roles with the teacher and try to do it "with little blood", without nerves and in an accessible form. It is much easier to train one soldier than a company...

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How to explain fractions to a child

Don't wait until your child is in 5th grade and encounters fractions on the pages of a math textbook. We recommend looking for the answer to the question “How to explain fractions to a child” in the kitchen! And do it right now! Even if your kid is only 4-5 years old, he is able to understand the meaning of the concept of “fractions” and can even learn the simplest actions with fractions.

We shared an orange.
There are many of us, and he is one
This slice for a hedgehog, this slice for a siskin...
And for a wolf - peel.

Remember the poem? Here is the most illustrative example and the most effective guide to action! It is easiest to explain fractions to a child using food as an example: we cut an apple into halves and quarters, we divide pizza between family members, we cut a loaf of bread before dinner, etc. Most importantly, before you eat the "visual aid" do not forget to voice what part of the whole you are "destroying".

Enter the concept of "share".

Emphasize that a WHOLE orange (apple, chocolate bar, watermelon, etc.) is 1 (denoted by the number 1).

Enter the concept of "fraction".

We divide an orange or a chocolate bar, you can also say “crush” into several parts.

Show your child a well-known object - a ruler. Explain that there are intermediate values between numbers - parts.

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Explain how to write fractions: what the numerator means and what the denominator indicates.

The meaning of the concept of “fractions” and the correct notation can be easily shown using the example of a constructor. In the numerator ABOVE the line we write which part, and in the denominator UNDER the line - into how many such parts the whole was divided.

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Be sure to use a good example to show the difference between fractions with the same numerator but different denominators.

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Using the example of 4 squares of the same size, show how you can divide them into the same / different number of parts. Let the child cut the paper blanks with scissors, and then write down the results using fractions.

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Explain how to write a whole as a fraction.

Remember the square and how we divided it into 4 parts. A square is a whole, we can write it as 1. But how to write it as a fraction: what is in the numerator, what is in the denominator? If we divided the square into 4 parts, then the whole square is 4/4. If we divided the square into 8 parts, then the whole square is 8/8. But it's still a square, i.e. 1. Both 4/4 and 8/8 are a unit, a whole!

How to explain fractions to a child: ask the RIGHT questions

In order for a grade 5 student to understand the topic “Fractions” and learn how to perform calculations with fractions, let's look at the methodology. It is important for us, parents, to understand how the teacher at school explains fractions to children, otherwise we can completely confuse our “soldier”.

A fraction is a number that is part of a whole object. It is always less than one.

Example 1 An apple is a whole, and a half is one second. Is it smaller than a whole apple? Divide the halves in half again. Each slice is one-fourth of a whole apple, and it is less than one-half.

A fraction is the number of parts of a whole.

Example 2 For example, a new product was brought to a clothing store: 30 shirts. Sellers managed to lay out and hang out only one third of all shirts from the new collection. How many shirts did they hang?
The child will easily verbally calculate that a third (one third) is 10 shirts, i.e. 10 were hung up and taken to the trading floor, and another 20 remained in the warehouse.

CONCLUSION: Anything can be measured with fractions, not only slices of pizza, but also liters in barrels, the number of wild animals in the forest, area, etc.

Give a variety of examples from life so that a 5th grade child understands the ESSENCE of fractions: this will help in the future in solving problems and performing calculations with proper and improper fractions, and learning in 5th grade will not be a burden, but a joy.

How to make sure that the child has learned that in the recording of fractions the numbers in the numerator and in the denominator are denoted?

Example 3 Ask what does 5 mean in the fraction 4/5?

- This is how many parts it was divided into.
- What does 4 mean?
- This is how much they took.

Comparing fractions is perhaps the most difficult topic.

Example 4 Invite the child to say which fraction is larger: 3/10 or 3/20? It seems that since 10 is less than 20, then the answer is obvious, but it's not! Remember the squares that we cut into pieces. If two squares of the same size are cut - one into 10, the second into 20 parts - is the answer obvious? So which fraction is bigger?

Actions with fractions

If you see that the child has well mastered the meaning of writing in the form of a fraction, you can proceed to simple arithmetic operations with fractions. On the example of the constructor, you can do this very clearly.

Example 5

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Example 6 Mathematical lotto on the topic "Fractions".

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Dear readers, if you know other effective methods for explaining fractions to a child, share them in the comments. We are happy to replenish our piggy bank of practical school tips.

Task Formulation: Find the value of the expression (actions with fractions).

The task is part of the USE in mathematics at the basic level for grade 11 at number 1 (Actions with fractions).

Let's see how such problems are solved with examples.

Task 1 example:

Find the value of the expression 5/4 + 7/6: 2/3.

Let's calculate the value of the expression. To do this, we define the order of operations: first multiplication and division, then addition and subtraction. And we will perform the necessary actions in the right order:

Answer: 3

Task 2 example:

Find the value of the expression (3.9 - 2.4) ∙ 8.2

Answer: 12.3

Task 3 example:

Find the value of the expression 27 ∙ (1/3 - 4/9 - 5/27).

Let's calculate the value of the expression. To do this, we define the order of operations: first multiplication and division, then addition and subtraction. In this case, the actions in the brackets are executed before the actions outside the brackets. And we will perform the necessary actions in the right order:

Answer: -8

Task 4 example:

Find the value of the expression 2.7 / (1.4 + 0.1)