Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions with different denominators are multiplied.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.
When multiplying simple fractions with different denominators for two or more factors the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional line will be a product of different numbers and, naturally, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way of representing a mixed fraction as an improper fraction, and can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
) and denominator by denominator (we get the denominator of the product).
Formula for multiplying fractions:
For example:
Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.
Dividing a common fraction by a fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper fractions;
- multiplying the numerators and denominators of fractions;
- reduce the fraction;
- If you get an improper fraction, then we convert the improper fraction into a mixed fraction.
Note! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It may be more convenient to use the second method of multiplying a common fraction by a number.
Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multistory fractions.
In high school, three-story (or more) fractions are often encountered. Example:
To bring such a fraction to its usual form, use division through 2 points:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.
2. In tasks with different types of fractions, go to the type of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.
5. Divide a unit by a fraction in your head, simply turning the fraction over.
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 pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.
As you can see, there is nothing complicated about adding fractions with the same denominators. It is enough to understand the following rules:
- To add fractions with the same denominator, you need to add their numerators and leave the denominator unchanged;
Adding fractions with different denominators
Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the fractions must be the same. But they are not always the same.
For example, fractions can be added because they have the same denominators.
But fractions cannot be added right away, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
There are several ways to reduce fractions to the same denominator. Today we will look at only one of them, since the other methods may seem complicated for a beginner.
The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.
The numerators and denominators of the fractions are then multiplied by their additional factors. As a result of these actions, fractions that had different denominators are converted 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. While 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
We got an improper fraction in our answer. 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. All that remains is 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. All that remains is 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:
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 being multiplied by the fraction and the denominator of the fraction are resolved if they have a common factor 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 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:
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 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, there were two equal pieces, each of which constituted a pizza. So everyone gets a pizza.
SUBJECT: Dividing fractions.
- Learning the rules for dividing fractions; Formation of basic skills in dividing fractions;
- development of basic skills to divide fractions using the basic algorithm; Development of attention, logical thinking;
- nurturing interest in studying the subject and the ability to work in groups.
LESSON PLAN:
1. Organizational moment.
2. Oral work leading to a new rule.
3. Introduction of the definition.
4. Work with cards for assimilation.
5. Physical exercises.
6. Oral work “find the mistake.”
7. Pinning: chain calculations.
8. Summing up the lesson.
DURING THE CLASSES
1) Today in class, guys, we have to do some serious work. You will need perseverance, desire, attention, consistency and correctness in completing tasks.
Oral work: write the inverse of this number:
2) How can you check whether the multiplication operation is performed correctly? (By the action of division).
We don't know how fractions are divided. It's time to get acquainted with this new action.
Dividing and dividing can sometimes be difficult, so the operation of dividing fractions itself requires special attention.
Let's remember what division is as a mathematical operation? (action inverse to multiplication; action when one of the factors and the product is used to find another factor).
Now together we will try to see a rule for dividing fractions that is new to us while considering the next problem.
Now our solutions will diverge.
What suggestions do you have to solve this equation?
Firstly, we know how to solve such equations using the concept of reciprocal numbers (it is enough to multiply both sides of the equation by the inverse of the coefficient of the variable X).
Secondly, we know the standard rule for finding an unknown factor (the product must be divided by a known factor).
Let's consider both of these cases:
Look carefully at the two resulting expressions for finding the value of X. These are answers to the same problem, which means the answers must be the same. In one case we multiply by 7/6, and in the other we divide by 6/7.
We find that when divided by 6/7, the same answer should be obtained if multiplied by 7/6. This means that the meaning of dividing fractions comes down to multiplying by the reciprocal of the divisor. This is not a random feature we noticed.
Introduce the new rule on page 100 of the textbook, repeat several times, ask several students from memory.
3) Using the learned rule, consider its application in various examples .
Children receive special cards, which they fill out together with the teacher, with comments from the place. You should consider dividing a fraction by a fraction, dividing a natural number by a fraction and fractions by a natural number, and dividing mixed numbers. When filling out, the children say the rule again. Pay special attention to three stages when performing division: the dividend remains unchanged; division is replaced by multiplication; multiply by the inverse of the divisor.
Division |
Application |
Rule |
Conversion |
||
| 5/7: 3/4 = | 5/7 * 4/3= | (5*4) / (7*3) = | 20/21 | 20/21 | |
| 5: 2/5 = | 5 * | ||||
| 7/8: 2 = | 7/8: 2/1= | 7/8 * | |||
| 4 1/2: 1 1/2= | 9/2: 3/2 = | 9/2 * |
On the back of the card there are three tasks that children solve after filling out the card on the spot, then check the solutions and results obtained.
DECIDE FOR YOURSELF |
| 1. 4/6: 3 = |
| 2. 8: 4/5 = |
| 3 . 1 2/3: 1 1/10 = |
4) Conducting physical exercises.
5) Stage of mastering the definition.
Let's check how you have learned today's rule and find out how attentive you are: “FIND THE ERROR”
6) Solving problems from the textbook: No. 619 (a, b, d).
7) Work in groups. Children take turns going to the board and writing down the solution to the example.
8) Well done. Well done. Let's summarize:
What new did you learn in class today?
How are fractions divided?
What are reciprocal numbers?
At home: Rule No. 617.
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 fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the 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 out of 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 pay attention to 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 due to the fact that when adding the numerator of a fraction, the sum appears, and not the product of numbers. Therefore, 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.
