1. To divide the first fraction by the second, you need to multiply the dividend by the number that is the inverse of the divisor.
For proper and improper fractions, the division rule is as follows:
To divide a common fraction, you must multiply the numerator of the dividend by the denominator of the divisor, and multiply the denominator of the dividend by the numerator of the divisor. We take the first product as the numerator, and the second as the denominator.
Dividing a fraction by a fraction.
To divide the 1st common fraction by the second, do not equal to zero, necessary:
- multiply the numerator of the 1st fraction by the denominator of the 2nd fraction and write the product in the numerator of the resulting fraction;
- multiply the denominator of the 1st fraction by the numerator of the 2nd fraction and write the product in the denominator of the resulting fraction.
In other words, dividing fractions leads to multiplication.
To divide the 1st fraction by the second, you need to multiply the dividend (1st fraction) by the reciprocal fraction of the divisor.
Dividing a fraction by a number.
Schematically dividing a fraction by natural number looks like that:
To divide a fraction by a natural number, use the following method:
We express a natural number as an improper fraction with a numerator that is equal to the number itself and a denominator that is equal to 1.
Multiplying Decimals
The decimal notation allows you to multiply fractions using almost the same rules that you use to multiply natural numbers. The difference is that it is necessary to determine the place of the comma in the resulting product.
Let us explain this with an example; Let's calculate the product 2.5 1.02.
Let's move the comma in the first factor one digit to the right, and in the second factor two digits to the right. Thus, the first factor will increase by 10 times, the second by 10 2 = 100 times, and the product by 10 100 = 1000 times.
Let's define the product of natural numbers 25 and 102:
25 102 = 2550.
This number is 1000 times greater than the required product. Therefore, it is necessary to reduce the number 2550 by 1000 = 10 3 times, that is, move the comma in this number to the left by 3 digits. Thus,
2,5 1,02 = 2,550 = 2,55.
You can think differently:
Thus, in order to multiply two decimal fractions9, it is enough, without paying attention to commas, to multiply them as natural numbers9 and then in the resulting product on the right, separate as many digits with a comma as there were after the commas in both factors together.
For example,
Decimal division
Let's look at an example of dividing a decimal fraction by a natural number.
Example. Calculate 46.8:2.
Solution. Divide 4 tens by 2 - we get the quotient number 2 (2 tens).
We divide 6 units by 2 - we get the quotient number 3 (3 units).
The division of the integer part is complete; we separate the whole part in the quotient with a comma.
We divide 8 tenths by 2 - we get the quotient number 4 (4 tenths). The remainder is 0—the division is complete.
Dividing a decimal by a decimal is reduced to dividing by a natural number by moving the commas in the dividend and divisor so many digits to the right that the divisor becomes a natural number.
Example. Calculate 4.42:0.2.
Solution. Since the divisor has one digit after the decimal point, it is enough to move the commas in the dividend and divisor one digit to the right. Thus, the dividend and divisor increase by 10 times, so the quotient will not change. In this case, the divisor will be a natural number.
You can also reason this way:
But the exact result when dividing is not always obtained decimals. More often you have to be content with an approximate private.
Example. Find the quotient 1.723:0.03.
Solution. Let's get rid of the comma in the divisor: 1.723:0.03= 172.3:3. Let's do the division.
Starting from the hundredths place, the number 3 in the quotient is repeated endlessly, because the remainder, starting from the third stage of the division process, is always equal to the same number 1.
If you leave the first two digits after the decimal point in the quotient, you get an approximate equality: 172.3:3 ≈ 57.43.
6th grade
SUBJECT: “Division of ordinary fractions”, 6th grade.
THE PURPOSE OF THE LESSON: Summarize and systematize theoretical and practical
knowledge, skills and abilities of students. Organize work on
closing gaps in students' knowledge. Improve, expand
and deepen students' knowledge of the topic.
TYPE OF LESSON: Lesson of generalization and systematization of knowledge, skills and abilities.
Equipment: On the board is the topic, purpose, lesson plan.
DURING THE CLASSES.
Each student has a “Check Sheet” on their desk.
1. Homework –
2. review questions –
3. oral counting –
4. class work –
5. independent work –
1. Checking homework:
a) work in pairs on the following questions:
1) Addition, subtraction of ordinary fractions;
2) How to multiply a fraction by a fraction;
3) Multiplication of two fractions;
4) Multiplication of mixed fractions;
5) The rule for dividing fractions;
6) Division of mixed fractions;
7) What is called. reducing fractions.
b) check homework By ready-made solution On the desk:
No. 620 (a), 624, 619 (d).
Purpose: to identify the degree of mastery of homework. Identify typical deficiencies.
Put your grades on the control sheet
Announce the purpose of the lesson: Summarize and systematize knowledge, skills and abilities in
topic: “Dividing ordinary fractions.”
We repeated the theory, let's test our knowledge in practice.
2. Verbal counting.
a) Using cards: 1) Reduce the fraction: ; ; ; ...
2) Convert to an improper fraction: ; ; ...
3) Select the whole part: ; ; ...
b) Number ladder. Whoever gets to the 6th floor faster will find out:
construction of geometry (Euclid)
Option 2 - a person who wanted to be a lawyer, an officer, and a philosopher, but
became a mathematician (Descartes)
l 0.1: ½ 0.4: 0.1 a
and d e l k k a v r e t
Marks on the control sheet, for: 2" - "5", 3" - "4", 4" - "3".
Whoever completed the “ladder” does No. 606 in the notebooks. The first of the students on the wing of the board does No. 606. Then he checks the class.
3.
A) No. 581 (b,d), 587 (with commentary), 591 (l,m,k), 600, 602, 593 (g,k,d,i)
The task is completed in notebooks and on the board.
b) solve the problem: Thousands of rubles were paid for a kg of sweets. How much are
Kg of these sweets?
4.
№ 1 . Follow these steps:
: answers: 1) 2) 3) 4) .
№ 2 . Represent the fraction as common fraction and follow these steps:
0.375: answers: 1) 2) 3) 4)
№ 3 . Solve the equation: answers: 1) 2) 3) 4) 2
№ 4 . On the first day, the tourist walked the entire route, and on the second, the rest. In
how many times more part roads traveled by a tourist on the first day than on
second? Answers: 1) 2) 5 3) 4)
№ 5. Present as a fraction:
: answer: 1) 2) 3) 4)
Check the solution using the template: No. 1 -4; No. 2 – 1; No. 3 – 4; No. 4 – 4; No. 5 – 3.
Put your grades on the control sheet.
Collect control sheets. Summarize. Announce grades for the lesson.
5. Lesson summary:
What basic rules did we repeat today?
6. Homework:
No. 619 (c), 620 (b), 627, individual task No. 617 (a, d, g).
Download:
Preview:
Municipal educational institution "Gymnasium No. 7"
Torzhok, Tver region.
OPEN LESSON ON THE TOPIC:
"DIVISION OF ORDINARY FRACTIONS"
6th grade
Open lesson at the city municipal district of Torzhok
(certification, 2001)
Mathematics teacher: Ufimtseva N.A.
2001
SUBJECT : " Division of ordinary fractions", 6th grade.
THE PURPOSE OF THE LESSON : Summarize and systematize theoretical and practical
Knowledge, abilities, and skills of students. Organize work on
Closing gaps in students' knowledge. Improve, expand
And deepen students' knowledge on the topic.
TYPE OF LESSON : Lesson of generalization and systematization of knowledge, skills and abilities.
Equipment : On the board is the topic, purpose, lesson plan.
DURING THE CLASSES.
Each student has a “Check Sheet” on their desk.
- Homework -
- review questions -
- verbal counting -
- classroom work -
- independent work -
- Checking homework:
A) work in pairs on the following questions:
1) Addition, subtraction of ordinary fractions;
2) How to multiply a fraction by a fraction;
3) Multiplication of two fractions;
4) Multiplication of mixed fractions;
5) The rule for dividing fractions;
6) Division of mixed fractions;
7) What is called. reducing fractions.
B) checking homework using a ready-made solution on the board:
No. 620 (a), 624, 619 (d).
Target : identify the degree of mastery of homework. Identify typical deficiencies.
Put your grades on the control sheet
Announce the purpose of the lesson: Summarize and systematize knowledge, skills and abilities in
Topic: “Dividing ordinary fractions.”
We repeated the theory, let's test our knowledge in practice.
- Verbal counting.
A) Using cards: 1) Reduce the fraction: ; ; ; ...
2) Convert to an improper fraction: ; ; ...
3) Select the whole part: ; ; ...
B) Number ladder. Whoever gets to the 6th floor faster will find out:
Geometry constructions (Euclid)
Option 2 - a person who wanted to be a lawyer, an officer, and a philosopher, but
Became a mathematician (Descartes)
D t
And r
L 0.1: ½ 0.4: 0.1 a
K k
V e
E d
3 2 4 5
I d e l k a v e r t
Marks on the control sheet, for: 2" - "5", 3" - "4", 4" - "3".
Whoever completed the “ladder” does No. 606 in the notebooks. The first of the students on the wing of the board does No. 606. Then he checks the class.
- Repetition and systematization of the main theoretical principles:
A) No. 581 (b,d), 587 (with commentary), 591 (l,m,k), 600, 602, 593 (g,k,d,i)
The task is completed in notebooks and on the board.
B) solve the problem: Thousands of rubles were paid for a kg of sweets. How much are
Kg of these sweets?
- Independent work. Purpose: to check your understanding of this topic.
№ 1 . Follow these steps:
: answers: 1) 2) 3) 4) .
№ 2 . Represent the fraction as a fraction and do the following:
0.375: answers: 1) 2) 3) 4)
№ 3 . Solve the equation: answers: 1) 2) 3) 4) 2
№ 4 . On the first day, the tourist walked the entire route, and on the second, the rest. In
How many times more is the part of the road covered by a tourist on the first day than on
Second? Answers: 1) 2) 5 3) 4)
№ 5. Present as a fraction:
: answer: 1) 2) 3) 4)
Check the solution using the template: No. 1 -4; No. 2 – 1; No. 3 – 4; No. 4 – 4; No. 5 – 3.
Put your grades on the control sheet.
Collect control sheets. Summarize. Announce grades for the lesson.
- Lesson summary:
What basic rules did we repeat today?
- Homework:
No. 619 (c), 620 (b), 627, individual assignment No. 617 (a, e, g)
COURSE WORK
ON ALGEBRA AND PRINCIPLES OF ANALYSIS
ON THIS TOPIC
"TRIGONOMETRIC FUNCTIONS"
Creative group of the Department of Mathematics
“Gymnasium No. 3”, Udomlya.
Lesson No. 3-4 developed by a math teacher
Ufimtseva N.A.
2000
Municipal educational institution "Gymnasium No. 7"
Torzhok, Tver region.
PUBLIC LESSON
IN last time We learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.
Designation:
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If present in fractions whole part, they must be converted into incorrect ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out because there was no pair for it, we take it outside the limits of multiplication. The result is a negative fraction.
Task. Find the meaning of the expression:
We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).
Also note negative numbers: When multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the entire notation more accurate.
Reducing fractions on the fly
Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the meaning of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what remains of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.
There are simply no other reasons for reducing fractions, so the correct solution to the previous problem looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
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 study 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 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 will be 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 denominators, 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 fraction by that number and leave the denominator unchanged.
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:
The number that is being multiplied by the fraction and the denominator of the fraction are resolved if they have common divisor, greater than one.
For example, an expression can be evaluated in two ways.
First way. Multiply the number 4 by the numerator of the fraction, and leave the denominator of the fraction unchanged:
Second way. The four being multiplied and the four in the denominator of the fraction can be reduced. These fours can be reduced by 4, since the greatest common divisor for two fours is the four itself:
We got the same result 3. After reducing the fours, new numbers are formed in their place: two ones. But multiplying one with three, and then dividing by one does not change anything. Therefore, the solution can be written briefly:
The reduction can be performed even when we decided to use the first method, but at the stage of multiplying the number 4 and the numerator 3 we decided to use the reduction:
But for example, the expression can only be calculated in the first way - multiply 7 by the denominator of the fraction, and leave the denominator unchanged:
This is due to the fact that the number 7 and the denominator of the fraction do not have a common divisor greater than one, and accordingly do not cancel.
Some students mistakenly shorten the number being multiplied and the numerator of the fraction. You can't do this. For example, the following entry is not correct:
Reducing a fraction means that both numerator and denominator will be divided by the same number. In the situation with the expression, division is performed only in the numerator, since writing this is the same as writing . We see that division is performed only in the numerator, and no division occurs in the denominator.
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 a pizza looks like, 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 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 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.