The rule for multiplying ordinary fractions of mixed numbers. Rule for multiplying fractions by whole numbers

Multiplication ordinary fractions Let's look at several possible options.

Multiplying a fraction by a fraction

This is the simplest case, in which you need to use the following fraction multiplication rules.

To multiply a fraction by a fraction, necessary:

• multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;
• multiply the denominator of the first fraction by the denominator of the second fraction and write their product into the denominator of the new fraction;

Before multiplying numerators and denominators, check if the fractions can be reduced. Reducing fractions in calculations will greatly facilitate your calculations.



Multiplying a fraction by a natural number

To fraction multiply by natural number you need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.

If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, select the whole part.



Multiplication of mixed numbers

To multiply mixed numbers, you must first convert them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.



Another way to multiply a fraction by a natural number

Sometimes in calculations it is more convenient to use a different method of multiplying an ordinary fraction by a number.

To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.



As can be seen from the example, it is more convenient to use this version of the rule if the denominator of the fraction is divisible without a remainder by a natural number.

Actions with fractions

Adding fractions with the same denominators

Adding fractions is of two types:

• Adding fractions with the same denominators
• Adding fractions with different denominators

Let's start with adding fractions with the same 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 . We add the numerators, and leave the denominator unchanged:



This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2 Add fractions and .

Again, add the numerators, and leave the denominator unchanged:



The answer is an improper fraction. If the end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the integer part is allocated easily - two divided by two is equal to one:



This example can be easily understood if we think of a pizza that is divided into two parts. If you add more pizzas to the pizza, you get one whole pizza:

Example 3. Add fractions and .



This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:

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 picture. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:

1. To add fractions with the same denominator, you need to add their numerators, and leave the denominator the same;
2. If the answer turned out to be an improper fraction, then you need to select the whole part in it.

Adding fractions with different denominators

Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those 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 at once, because 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 consider only one of them, since the rest of the methods may seem complicated for a beginner.

The essence of this method is that the least common multiple (LCM) of the denominators of both fractions is first searched for. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the NOC is divided by the denominator of the second fraction and the second additional factor is obtained.

Then the numerators and denominators of the fractions are 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. Add fractions and

These fractions have different denominators, so you need to bring them to the same (common) denominator.

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 back to fractions and . First, we 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 factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor 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 factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:

Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:



Look closely 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 complete this example to the end:



Thus the example ends. To add it turns out.

Let's try to depict our solution using a picture. If you add pizzas to a pizza, you get one whole pizza and another sixth of a pizza:

Reduction of fractions to the same (common) denominator can also be depicted using a picture. Bringing the fractions and to a common denominator, we get the fractions and . These two fractions will be represented by the same slices of pizzas. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).



The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).

Note that we have painted given example too detailed. AT educational institutions it is not customary to write in such a detailed manner. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the additional factors found by your numerators and denominators. While at school, we would have to write this example as follows:



But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “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:

3. Find the LCM of the denominators of fractions;
4. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
5. Multiply the numerators and denominators of fractions by their additional factors;
6. Add fractions that have the same denominators;
7. If the answer turned out to be an improper fraction, then select its whole part;

Example 2 Find the value of an expression .

Let's use the diagram above.

Step 1. Find the LCM for the denominators of fractions

We find the LCM for the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4. You need to find the LCM for these numbers:

Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier 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 over 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 got the second additional factor 4. We write it over 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 got the third additional factor 3. We write it over the third fraction:

Step 3. Multiply the numerators and denominators of fractions by your additional factors

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions that have the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add 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 carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning 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 turned out to be an improper fraction, then select its integer part

Our answer is an improper fraction. We must single out the whole part of it. We highlight:



Got an answer

Subtraction of fractions with the same denominators

There are two types of fraction subtraction:

8. Subtraction of fractions with the same denominators
9. Subtraction of fractions with different denominators

First, let's learn how to subtract fractions with the same 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, and leave the denominator the same.

For example, let's find the value of the expression . To solve this example, it is necessary to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same. Let's do this:

This example can be easily understood if we think of a pizza that 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 the same:

This example can be easily understood if we think of a pizza that 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:

The answer is an improper fraction. If the example is complete, then it is customary to get rid of the improper fraction. Let's get rid of the wrong fraction in the answer. To do this, select its whole part:

As you can see, there is nothing complicated in subtracting fractions with the same denominators. It is enough to understand the following rules:

Subtraction of fractions with different denominators

For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found according to 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 over 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 over the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1 Find the value of an expression:

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 back to fractions and

Let's find an additional factor for the first fraction. To do this, we 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. We write the four over the first fraction:

We do the same with the second fraction. 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 4. Divide 12 by 4, we get 3. We write the triple over the second fraction:

Now we are all set 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 complete this example to the end:

Got an answer

Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.

This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:

Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):

The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off 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 you first need to bring them to the same (common) denominator.

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, we 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 over 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 over 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 over 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 have 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 correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler and more aesthetically pleasing. What can be done? You can reduce this fraction. Recall that the reduction of a fraction is the division of the numerator and denominator by the largest common divisor numerator and denominator.

To correctly reduce a fraction, you need to divide its numerator and denominator by the greatest common divisor (GCD) of the numbers 20 and 30.

Do not confuse GCD with NOC. The most common mistake many beginners make. GCD is the greatest common divisor. We find it for fraction reduction.

And LCM is the least common multiple. We find it in order to bring fractions to the same (common) denominator.

Now we will find the greatest common divisor (gcd) of the numbers 20 and 30.

So, we find the GCD for the numbers 20 and 30:

GCD (20 and 30) = 10

Now we return to our example and divide the numerator and denominator of the fraction by 10:

Got a nice answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator the same.

Example 1. Multiply the fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The entry can be understood as taking half 1 time. For example, if you take pizza 1 time, you get pizza

From the laws of multiplication, we know that if the multiplicand and the multiplier are interchanged, then the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying an integer and a fraction works:

This entry can be understood as taking half of the unit. 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 expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.

And if we swap the multiplicand and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.

Example 1 Find the value of the expression .

Got an answer. It is desirable to reduce given 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 get pizza. Remember what a pizza looks like divided into three parts:

One slice from this pizza and the two slices we took will have the same dimensions:

In other words, we are talking about the same pizza size. 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 is an improper fraction. Let's take a whole part of it:

Example 3 Find the value of an expression

The answer turned out to be a correct fraction, but it will be good if it is reduced. To reduce this fraction, it must be divided by the gcd of the numerator and denominator. So, let's find the GCD of the numbers 105 and 450:

GCD for (105 and 150) is 15

Now we divide the numerator and denominator of our answer to the GCD:

Representing an integer as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . From this, the five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:

Reverse numbers

Now we will get acquainted with interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to number a is the number that, when multiplied by a gives a unit.

Let's substitute in this definition instead of a variable a number 5 and try to read the definition:

Reverse to number 5 is the number that, when multiplied by 5 gives a unit.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, multiply the fraction by itself, only inverted:

What will be the 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 5 is multiplied by one, one is obtained.

The reciprocal can also be found for any other integer.

• the reciprocal of 3 is a fraction
• the reciprocal of 4 is a fraction

You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.

Multiplying a whole number by a fraction is a simple task. But there are subtleties that you probably understood at school, but have since forgotten.

How to multiply an integer by a fraction - a few terms

If you remember what the numerator and denominator are and how a proper fraction differs from an improper one, skip this paragraph. It is for those who have completely forgotten the theory.

The numerator is top part fractions are what we divide. The denominator is the bottom one. This is what we share.
A proper fraction is one whose numerator is less than the denominator. An improper fraction is a fraction whose numerator is greater than or equal to the denominator.

How to multiply a whole number by a fraction

The rule for multiplying an integer by a fraction is very simple - we multiply the numerator by the integer, and do not touch the denominator. For example: two multiplied by one fifth - we get two fifths. Four times three sixteenths is twelve sixteenths.

Reduction

In the second example, the resulting fraction can be reduced.
What does it mean? Note that both the numerator and denominator of this fraction are divisible by four. Dividing both numbers by a common divisor is called reducing the fraction. We get three quarters.

Improper fractions

But suppose we multiply four times two fifths. Got eight fifths. This is the wrong fraction.
It must be brought to correct form. To do this, you need to select a whole part from it.
Here you need to use division with a remainder. We get one and three in the remainder.
One whole and three fifths is our proper fraction.

Correcting thirty-five eighths is a bit more difficult. The closest number to thirty-seven that is divisible by eight is thirty-two. When divided, we get four. We subtract thirty-two from thirty-five - we get three. Outcome: four whole and three eighths.

Equality of the numerator and denominator. And here everything is very simple and beautiful. When the numerator and denominator are equal, the result is just one.

In this article, we will analyze multiplication of mixed numbers. First, let's voice the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next, we will talk about the multiplication of a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and an ordinary fraction.

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Multiplication of mixed numbers.

Multiplication of mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers into improper fractions.

Let's write down multiplication rule for mixed numbers:

• First, the mixed numbers to be multiplied must be replaced by improper fractions;
• Secondly, you need to use the rule of multiplying a fraction by a fraction.

Consider examples of applying this rule when multiplying a mixed number by a mixed number.

Perform mixed number multiplication and .

First, we represent the multiplied mixed numbers as improper fractions: and . Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: . Applying the rule of multiplication of fractions, we get . The resulting fraction is irreducible (see reducible and irreducible fractions), but it is incorrect (see regular and improper fractions), therefore, to get the final answer, it remains to extract the integer part from the improper fraction: .

Let's write the whole solution in one line: .

.

To consolidate the skills of multiplying mixed numbers, consider the solution of another example.

Do the multiplication.

Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then . At this stage, it's time to remember about fraction reduction: let's replace all the numbers in the fraction with their expansions into prime factors, and perform the reduction of the same factors.

Multiplication of a mixed number and a natural number

After replacing the mixed number, proper fraction, multiplying a mixed number and a natural number is reduced to the multiplication of an ordinary fraction and a natural number.

Multiply the mixed number and the natural number 45 .

A mixed number is a fraction, then . Let's replace the numbers in the resulting fraction with their expansions into prime factors, make a reduction, after which we select the integer part: .

.

Multiplication of a mixed number and a natural number is sometimes conveniently done using the distributive property of multiplication with respect to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, .

Compute the product.

We replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .

Multiplying a mixed number and a common fraction it is most convenient to reduce to the multiplication of ordinary fractions, representing the multiplied mixed number as an improper fraction.

Multiply the mixed number by the common fraction 4/15.

Replacing the mixed number with a fraction, we get .

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Multiplication of fractional numbers

§ 140. Definitions. 1) The multiplication of a fractional number by an integer is defined in the same way as the multiplication of integers, namely: to multiply some number (multiplier) by an integer (multiplier) means to make a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

So multiplying by 5 means finding the sum:
2) To multiply some number (multiplier) by a fraction (multiplier) means to find this fraction of the multiplicand.

Thus, finding a fraction of a given number, which we considered before, we will now call multiplication by a fraction.

3) To multiply some number (multiplier) by a mixed number (factor) means to multiply the multiplier first by the integer of the factor, then by the fraction of the factor, and add the results of these two multiplications together.

For example:

The number obtained after multiplication is in all these cases called work, i.e., in the same way as when multiplying integers.

From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.

§ 141. Expediency of these definitions. To understand the expediency of introducing the last two definitions of multiplication into arithmetic, let us take the following problem:

A task. The train, moving evenly, travels 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?

If we remained with the same definition of multiplication, which is indicated in the arithmetic of integers (addition of equal terms), then our problem would have three different solutions, namely:

If the given number of hours is an integer (for example, 5 hours), then to solve the problem, 40 km must be multiplied by this number of hours.

If a given number of hours is expressed as a fraction (for example, hours), then you will have to find the value of this fraction from 40 km.

Finally, if the given number of hours is mixed (for example, hours), then it will be necessary to multiply 40 km by an integer contained in the mixed number, and add to the result such a fraction from 40 km as is in the mixed number.

The definitions we have given allow us to give one general answer to all these possible cases:

40 km must be multiplied by the given number of hours, whatever it may be.

Thus, if the task is presented in general view So:

A train moving uniformly travels v km per hour. How many kilometers will the train cover in t hours?

then, whatever the numbers v and t, we can express one answer: the desired number is expressed by the formula v · t.

Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, to find 5% (i.e. five hundredths) of a given number means the same as multiplying the given number by or by; finding 125% of a given number is the same as multiplying that number by or by , etc.

§ 142. A note about when a number increases and when it decreases from multiplication.

From multiplication by a proper fraction, the number decreases, and from multiplication by improper fraction the number increases if this improper fraction is greater than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so,.

§ 143. Derivation of multiplication rules.

1) Multiplying a fraction by an integer. Let the fraction be multiplied by 5. This means to increase by 5 times. To increase a fraction by 5, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).

That's why:
Rule 1. To multiply a fraction by an integer, you must multiply the numerator by this integer, and leave the denominator the same; instead, you can also divide the denominator of the fraction by the given integer (if possible), and leave the numerator the same.

Comment. The product of a fraction and its denominator is equal to its numerator.

So:
Rule 2. To multiply an integer by a fraction, you need to multiply the integer by the numerator of the fraction and make this product the numerator, and sign the denominator of the given fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator and make the first product the numerator and the second the denominator of the product.

Comment. This rule can also be applied to the multiplication of a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:

Thus, the three rules now stated are contained in one, which can be expressed in general terms as follows:
4) Multiplication of mixed numbers.

Rule 4. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For example:
§ 144. Reduction in multiplication. When multiplying fractions, if possible, a preliminary reduction should be done, as can be seen from the following examples:

Such a reduction can be done because the value of the fraction will not change if the numerator and denominator are reduced in the same number once.

§ 145. Change of product with change of factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .

So, if in the example:
in order to multiply several fractions, it is necessary to multiply their numerators among themselves and the denominators among themselves and make the first product the numerator and the second the denominator of the product.

Comment. This rule can also be applied to such products in which some factors of the number are integer or mixed, if only we consider the whole number as a fraction whose denominator is one, and we turn mixed numbers into improper fractions. For example:
§ 147. Basic properties of multiplication. Those properties of multiplication that we have indicated for integers (§ 56, 57, 59) also belong to the multiplication of fractional numbers. Let's specify these properties.

1) The product does not change from changing the places of the factors.

For example:

Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their members differ only in the order of the integer factors, and the product of integers does not change when the factors change places.

2) The product will not change if any group of factors is replaced by their product.

For example:

The results are the same.

From this property of multiplication, we can deduce the following conclusion:

to multiply a number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, and so on.

For example:
3) The distributive law of multiplication (with respect to addition). To multiply the sum by some number, you can multiply each term by this number separately and add the results.

This law has been explained by us (§ 59) as applied to whole numbers. It remains true without any changes for fractional numbers.

Let us show, in fact, that the equality

(a + b + c + .)m = am + bm + cm + .

(the distributive law of multiplication with respect to addition) remains true even when the letters mean fractional numbers. Let's consider three cases.

1) Suppose first that the factor m is an integer, for example m = 3 (a, b, c are any numbers). According to the definition of multiplication by an integer, one can write (limited for simplicity to three terms):

(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).

On the basis of the associative law of addition, we can omit all brackets on the right side; applying the commutative law of addition, and then again the combination law, we can obviously rewrite the right-hand side as follows:

(a + a + a) + (b + b + b) + (c + c + c).

(a + b + c) * 3 = a * 3 + b * 3 + c * 3.

Hence, the distributive law in this case is confirmed.

Multiplication and division of fractions

Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in 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 easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.

To multiply two fractions, you need to 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.

From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.

As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what definitely won’t happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.

By definition we have:

Multiplication of fractions with an integer part and negative fractions

If there is an integer part in the fractions, 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 limits of multiplication or removed altogether according to the following rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:

1. We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
2. If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.

A task. Find the value of the expression:

We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:

Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).

Also pay attention to negative numbers: When multiplied, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.

Reducing fractions on the fly

Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more 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:

A task. Find the value of the expression:

By definition we have:

In all examples, the numbers that have been reduced and what is left of them are marked in red.

Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, in no case do not 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 a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.

There is simply no other reason to reduce fractions, so the right decision the previous task looks like this:

As you can see, the correct answer turned out to be not so beautiful. In general, be careful.

Multiplication of fractions.

To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.

Multiplying a fraction by a fraction.

To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.

Consider an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.

Multiplying a fraction by a number.

Let's start with the rule any number can be represented as a fraction \(\bf n = \frac \) .

Let's use this rule for multiplication.

The improper fraction \(\frac = \frac = \frac + \frac = 2 + \frac = 2\frac \\\) was converted to mixed fraction.

In other words, When multiplying a number by a fraction, multiply the number by the numerator and leave the denominator unchanged. Example:

Multiplication of mixed fractions.

To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied with the denominator.

Multiplication of reciprocal fractions and numbers.

Related questions:
How to multiply a fraction by a fraction?
Answer: the product of ordinary fractions is the multiplication of the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.

How to multiply fractions with different denominators?
Answer: it doesn’t matter if the denominators of fractions are the same or different, multiplication occurs according to the rule for finding the product of the numerator with the numerator, the denominator with the denominator.

How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction to an improper fraction and then find the product according to the rules of multiplication.

How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and leave the denominator the same.

Example #1:
Calculate the product: a) \(\frac \times \frac \) b) \(\frac \times \frac \)

Example #2:
Calculate the product of a number and a fraction: a) \(3 \times \frac \) b) \(\frac \times 11\)

Example #3:
Write the reciprocal of the fraction \(\frac \)?
Answer: \(\frac = 3\)

Example #4:
Calculate the product of two reciprocals: a) \(\frac \times \frac \)

Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?

Solution:
a) Let's use an example to answer the first question. The fraction \(\frac \) is correct, its reciprocal will be equal to \(\frac \) - an improper fraction. Answer: no.

b) in almost all enumerations of fractions, this condition is not met, but there are some numbers that fulfill the condition of being an improper fraction at the same time. For example, the improper fraction is \(\frac \) , its reciprocal is \(\frac \). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.

c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac \), then its reciprocal will be \(\frac \). The fraction \(\frac \) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac = \frac = 1\). The number 1 is a natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.

Example #6:
Perform the product of mixed fractions: a) \(4 \times 2\frac \) b) \(1\frac \times 3\frac \)

Solution:
a) \(4 \times 2\frac = \frac \times \frac = \frac = 11\frac \\\\ \)
b) \(1\frac \times 3\frac = \frac \times \frac = \frac = 4\frac \)

Example #7:
Can two mutually reciprocals be simultaneously mixed numbers?

Let's look at an example. Take a mixed fraction \(1\frac \), find for it reciprocal, for this we translate it into an improper fraction \(1\frac = \frac \) . Its reciprocal will be equal to \(\frac \) . The fraction \(\frac \) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.

Multiplying a decimal by a natural number

Presentation for the lesson

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.

• In a fun way, introduce students to the rule of multiplying a decimal fraction by a natural number, by a bit unit and the rule of expressing a decimal fraction as a percentage. Develop the ability to apply the acquired knowledge in solving examples and problems.
• Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their work and the work of each other.
• To cultivate interest in mathematics, activity, mobility, ability to communicate.

Equipment: interactive board, a poster with a cyphergram, posters with mathematicians' statements.

1. Organizing time.
2. Oral counting is a generalization of previously studied material, preparation for the study of new material.
3. Explanation of new material.
4. Homework assignment.
5. Mathematical physical education.
6. Generalization and systematization of the acquired knowledge in a playful way with the help of a computer.
7. Grading.

2. Guys, today our lesson will be somewhat unusual, because I will not spend it alone, but with my friend. And my friend is also unusual, now you will see him. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, friend? Komposha replies: "My name is Komposha." Are you ready to help me today? YES! Well then, let's start the lesson.

Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is posted on the board with oral count for addition and subtraction of decimal fractions, as a result of which the guys get the following code 523914687. )

Komposha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is the keyword of the topic of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”

Guys, we know how the multiplication of natural numbers is performed. Today we are going to look at multiplication. decimal numbers to a natural number. The multiplication of a decimal fraction by a natural number can be considered as the sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 3 = 5.21 + 5, 21 + 5.21 = 15.63 So 5.21 3 = 15.63. Representing 5.21 as an ordinary fraction of a natural number, we get

And in this case, we got the same result of 15.63. Now, ignoring the comma, let's take the number 521 instead of the number 5.21 and multiply by the given natural number. Here we must remember that in one of the factors the comma is moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now, in this example, we will move the comma to the left by two digits. Thus, by how many times one of the factors was increased, the product was reduced by so many times. Based on the similar points of these methods, we draw a conclusion.

To multiply decimal to a natural number, you need:
1) ignoring the comma, perform the multiplication of natural numbers;
2) in the resulting product, separate with a comma on the right as many characters as there are in a decimal fraction.

The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 3 = 15.63 and 7.624 15 = 114.34. After I show the multiplication by round number 12.6 50 = 630. Next, I turn to the multiplication of a decimal fraction by a bit unit. I show the following examples: 7.423 100 \u003d 742.3 and 5.2 1000 \u003d 5200. So, I introduce the rule for multiplying a decimal fraction by a bit unit:

To multiply a decimal fraction by bit units 10, 100, 1000, etc., it is necessary to move the comma to the right in this fraction by as many digits as there are zeros in the bit unit record.

I end the explanation with the expression of a decimal fraction as a percentage. I enter the rule:

To express a decimal as a percentage, multiply it by 100 and add the % sign.

I give an example on a computer 0.5 100 = 50 or 0.5 = 50%.

4. At the end of the explanation, I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.

5. In order for the guys to rest a little, to consolidate the topic, we do a mathematical physical education session together with Komposha. Everyone stands up, shows the class the solved examples and they must answer whether the example is correct or incorrect. If the example is solved correctly, then they raise their hands above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and knead their fingers.

6. And now you have a little rest, you can solve the tasks. Open your textbook to page 205, № 1029. in this task it is necessary to calculate the value of expressions:

Tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when fully assembled, sails away.

Solving this task on a computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the Kazakh land from the Baikonur cosmodrome take off to the stars spaceships. Near Baikonur, Kazakhstan is building its own new spaceport Baiterek.

How far will a car travel in 4 hours if the speed is passenger car 74.8 km/h.

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) and the denominator by the denominator (we get the denominator of the product).

Fraction multiplication formula:

For example:

Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:

Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper fraction to a mixed one.

Note! To multiply a mixed fraction by another mixed fraction, you first need to bring 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 is more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the above example, 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.

Multilevel fractions.

In high school, three-story (or more) fractions are often found. Example:

To bring such a fraction to its usual form, division through 2 points is used:

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 in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

2. In tasks with different types fractions - go to the form of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.

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

Multiplication of ordinary fractions

Consider an example.

Let there be $\frac(1)(3)$ part of an apple on the plate. We need to find the $\frac(1)(2)$ part of it. The required part is the result of multiplying the fractions $\frac(1)(3)$ and $\frac(1)(2)$. The result of multiplying two common fractions is a common fraction.

Multiplying two common fractions

Rule for multiplying ordinary fractions:

The result of multiplying a fraction by a fraction is a fraction whose numerator is equal to the product of the numerators of the multiplied fractions, and the denominator is equal to the product of the denominators:

Example 1

Multiply ordinary fractions $\frac(3)(7)$ and $\frac(5)(11)$.

Solution.

Let's use the rule of multiplication of ordinary fractions:

\[\frac(3)(7)\cdot \frac(5)(11)=\frac(3\cdot 5)(7\cdot 11)=\frac(15)(77)\]

Answer:$\frac(15)(77)$

If as a result of multiplying fractions a cancellable or improper fraction is obtained, then it is necessary to simplify it.

Example 2

Multiply fractions $\frac(3)(8)$ and $\frac(1)(9)$.

Solution.

We use the rule for multiplying ordinary fractions:

\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)\]

As a result, we got a reducible fraction (on the basis of division by $3$. Divide the numerator and denominator of the fraction by $3$, we get:

\[\frac(3)(72)=\frac(3:3)(72:3)=\frac(1)(24)\]

Short solution:

\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)=\frac(1) (24)\]

Answer:$\frac(1)(24).$

When multiplying fractions, you can reduce the numerators and denominators to find their product. In this case, the numerator and denominator of the fraction is decomposed into simple factors, after which the repeating factors are reduced and the result is found.

Example 3

Calculate the product of fractions $\frac(6)(75)$ and $\frac(15)(24)$.

Solution.

Let's use the formula for multiplying ordinary fractions:

\[\frac(6)(75)\cdot \frac(15)(24)=\frac(6\cdot 15)(75\cdot 24)\]

Obviously, the numerator and denominator contain numbers that can be reduced in pairs by the numbers $2$, $3$, and $5$. We decompose the numerator and denominator into simple factors and make the reduction:

\[\frac(6\cdot 15)(75\cdot 24)=\frac(2\cdot 3\cdot 3\cdot 5)(3\cdot 5\cdot 5\cdot 2\cdot 2\cdot 2\cdot 3)=\frac(1)(5\cdot 2\cdot 2)=\frac(1)(20)\]

Answer:$\frac(1)(20).$

When multiplying fractions, the commutative law can be applied:

Multiplying a fraction by a natural number

The rule for multiplying an ordinary fraction by a natural number:

The result of multiplying a fraction by a natural number is a fraction in which the numerator is equal to the product of the numerator of the multiplied fraction by the natural number, and the denominator is equal to the denominator of the multiplied fraction:

where $\frac(a)(b)$ is a common fraction, $n$ is a natural number.

Example 4

Multiply the fraction $\frac(3)(17)$ by $4$.

Solution.

Let's use the rule of multiplying an ordinary fraction by a natural number:

\[\frac(3)(17)\cdot 4=\frac(3\cdot 4)(17)=\frac(12)(17)\]

Answer:$\frac(12)(17).$

Do not forget about checking the result of multiplication for the contractibility of a fraction or for an improper fraction.

Example 5

Multiply the fraction $\frac(7)(15)$ by $3$.

Solution.

Let's use the formula for multiplying a fraction by a natural number:

\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)\]

By the criterion of division by the number $3$), it can be determined that the resulting fraction can be reduced:

\[\frac(21)(15)=\frac(21:3)(15:3)=\frac(7)(5)\]

The result is an improper fraction. Let's take the whole part:

\[\frac(7)(5)=1\frac(2)(5)\]

Short solution:

\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)=\frac(7)(5)=1\frac(2) (5)\]

It was also possible to reduce fractions by replacing the numbers in the numerator and denominator with their expansions into prime factors. In this case, the solution could be written as follows:

\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(7\cdot 3)(3\cdot 5)=\frac(7)(5)= 1\frac(2)(5)\]

Answer:$1\frac(2)(5).$

When multiplying a fraction by a natural number, you can use the commutative law:

Division of ordinary fractions

The division operation is the inverse of multiplication and its result is a fraction by which you need to multiply a known fraction to get a known product of two fractions.

Division of two common fractions

The rule for dividing ordinary fractions: Obviously, the numerator and denominator of the resulting fraction can be decomposed into simple factors and reduce:

\[\frac(8\cdot 35)(15\cdot 12)=\frac(2\cdot 2\cdot 2\cdot 5\cdot 7)(3\cdot 5\cdot 2\cdot 2\cdot 3)= \frac(2\cdot 7)(3\cdot 3)=\frac(14)(9)\]

As a result, we got an improper fraction, from which we select the integer part:

\[\frac(14)(9)=1\frac(5)(9)\]

Answer:$1\frac(5)(9).$