Multiply an improper fraction. Multiplying a fraction by a number

Multiplication and division of fractions.

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For those who strongly "not very..."
And for those who "very much...")

This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). I.e:

For example:

Everything is extremely simple. And please don't look for a common denominator! Don't need it here...

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

For example:

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

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

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

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

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

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

then divide-multiply in order, left to right!

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

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

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, and there will be fewer of them (mistakes)!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

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

3. We reduce all fractions to the stop.

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

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

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

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

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

Calculate:

Did you decide?

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

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

And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

) 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 of fractions - go to the type 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.

We continue to study actions with ordinary fractions. Now in the spotlight multiplication of common fractions. In this article, we will give a rule for multiplying ordinary fractions, consider the application of this rule when solving examples. We will also focus on multiplying an ordinary fraction by a natural number. In conclusion, consider how the multiplication of three or more fractions is carried out.

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Multiplying a common fraction by a common fraction

Let's start with the wording rules for multiplying common fractions: multiplying a fraction by a fraction gives a fraction whose numerator is equal to the product of the numerators of the multiplied fractions, and whose denominator is equal to the product of the denominators.

That is, the formula corresponds to the multiplication of ordinary fractions a / b and c / d.

Let us give an example illustrating the rule of multiplication of ordinary fractions. Consider a square with a side of 1 unit. , while its area is 1 unit 2 . Divide this square into equal rectangles with sides 1/4 units. and 1/8 units. , while the original square will consist of 4 8 = 32 rectangles, therefore, the area of each rectangle is 1/32 of the area of the original square, that is, it is equal to 1/32 units 2. Now let's paint over part of the original square. All our actions are reflected in the figure below.

The sides of the filled rectangle are 5/8 units. and 3/4 units. , which means that its area is equal to the product of fractions 5/8 and 3/4, that is, units 2. But the filled rectangle consists of 15 "small" rectangles, so its area is 15/32 units 2 . Hence, . Since 5 3=15 and 8 4=32 , the last equality can be rewritten as , which confirms the formula for multiplying ordinary fractions of the form .

Note that with the help of the voiced multiplication rule, you can multiply both regular and improper fractions, and fractions with the same denominators, and fractions with different denominators.

Consider examples of multiplying common fractions.

Multiply the common fraction 7/11 by the common fraction 9/8.

The product of the numerators of the multiplied fractions 7 and 9 is 63, and the product of the denominators of 11 and 8 is 88. Thus, multiplying the common fractions 7/11 and 9/8 gives the fraction 63/88.

Here is a summary of the solution: .

We should not forget about the reduction of the resulting fraction, if as a result of multiplication a reducible fraction is obtained, and about the selection of the whole part from an improper fraction.

Multiply fractions 4/15 and 55/6.

Let's apply the rule of multiplication of ordinary fractions: .

Obviously, the resulting fraction is reducible (the sign of divisibility by 10 allows us to assert that the numerator and denominator of the fraction 220/90 have a common factor of 10). Let's reduce the fraction 220/90: GCD(220, 90)=10 and . It remains to select the integer part from the resulting improper fraction: .

Note that fraction reduction can be carried out before calculating the products of the numerators and the products of the denominators of the multiplied fractions, that is, when the fraction has the form . For this number, a, b, c, and d are replaced by their prime factorizations, after which the same factors of the numerator and denominator are cancelled.

To clarify, let's go back to the previous example.

Calculate the product of fractions of the form .

By the formula for multiplying ordinary fractions, we have .

Since 4=2 2 , 55=5 11 , 15=3 5 and 6=2 3 , then . Now we cancel the common prime factors: .

It remains only to calculate the products in the numerator and denominator, and then select the integer part from the improper fraction: .

It should be noted that the multiplication of fractions is characterized by a commutative property, that is, the multiplied fractions can be interchanged: .

Multiplying a fraction by a natural number

Let's start with the wording rules for multiplying a common fraction by a natural number: multiplying a fraction by a natural number gives a fraction whose 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.

With the help of letters, the rule for multiplying a fraction a / b by a natural number n has the form .

The formula follows from the formula for multiplying two ordinary fractions of the form . Indeed, representing a natural number as a fraction with a denominator of 1, we obtain .

Consider examples of multiplying a fraction by a natural number.

Multiply the fraction 2/27 by 5.

Multiplying the numerator 2 by the number 5 gives 10, therefore, by virtue of the rule of multiplying a fraction by a natural number, the product of 2/27 by 5 is equal to the fraction 10/27.

The whole solution can be conveniently written as follows: .

When multiplying a fraction by a natural number, the resulting fraction often has to be reduced, and if it is also incorrect, then represent it as a mixed number.

Multiply the fraction 5/12 by the number 8.

According to the formula for multiplying a fraction by a natural number, we have . Obviously, the resulting fraction is reducible (the sign of divisibility by 2 indicates a common divisor 2 of the numerator and denominator). Let's reduce the fraction 40/12: since LCM(40, 12)=4, then . It remains to select the whole part: .

Here is the whole solution: .

Note that the reduction could be done by replacing the numbers in the numerator and denominator by their expansions into prime factors. In this case, the solution would look like this:

In conclusion of this paragraph, we note that the multiplication of a fraction by a natural number has the commutative property, that is, the product of a fraction by a natural number is equal to the product of this natural number by a fraction: .

Multiply three or more fractions

The way we have defined ordinary fractions and the action of multiplication with them allows us to assert that all the properties of multiplication of natural numbers apply to the multiplication of fractions.

The commutative and associative properties of multiplication make it possible to uniquely determine multiplying three or more fractions and natural numbers. In this case, everything happens by analogy with the multiplication of three or more natural numbers. In particular, fractions and natural numbers in the product can be rearranged for convenience of calculation, and in the absence of brackets indicating the order in which actions are performed, we can arrange the brackets ourselves in any of the allowed ways.

Consider examples of multiplication of several fractions and natural numbers.

Multiply three common fractions 1/20, 12/5, 3/7 and 5/8.

Let's write the product that we need to calculate . By virtue of the rule for multiplying fractions, the written product is equal to a fraction whose numerator is equal to the product of the numerators of all fractions, and the denominator is the product of the denominators: .

Before calculating the products in the numerator and denominator, it is advisable to replace all factors by their expansions into prime factors and reduce (of course, you can reduce the fraction after multiplication, but in many cases this requires a lot of computational effort): .

Multiply five numbers .

In this product, it is convenient to group the fraction 7/8 with the number 8, and the number 12 with the fraction 5/36, this will simplify the calculations, since with such a grouping the reduction is obvious. We have
.

Multiplication of fractions

We will consider the multiplication of ordinary fractions in several possible ways.

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 a 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.

Multiplication of mixed numbers: rules, examples, solutions.

In this article, we will analyze multiplication of mixed numbers. First, we will 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 to 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 the fraction reduction: we will replace all the numbers in the fraction with their expansions into prime factors, and we will perform the reduction of the same factors.

Multiplication of a mixed number and a natural number

After replacing the mixed number with an improper 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 .

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 (factor) 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:

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 problem is presented in general form as follows:

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 an 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).

So:
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 a fraction will not change if the numerator and denominator are reduced by the same number of times.

§ 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, one can deduce the following conclusion:

to multiply some 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.

Division of a fraction by a natural number

Sections: Mathematics

T class type: ONZ (discovery of new knowledge - according to the technology of the activity method of teaching).

Deduce methods of dividing a fraction by a natural number;
To form the ability to perform the division of a fraction by a natural number;
Repeat and consolidate the division of fractions;
Train the ability to reduce fractions, analyze and solve problems.

Equipment demo material:

1. Tasks for updating knowledge:

2. Trial (individual) task.

1. Perform division:

2. Perform the division without performing the entire chain of calculations: .

When dividing a fraction by a natural number, you can multiply the denominator by this number, and leave the numerator the same.

If the numerator is divisible by a natural number, then when dividing a fraction by this number, you can divide the numerator by the number, and leave the denominator the same.

I. Motivation (self-determination) for learning activities.

Organize the actualization of the requirements for the student on the part of educational activities (“must”);
Organize the activities of students to establish a thematic framework (“I can”);
To create conditions for the student to have an internal need for inclusion in educational activities (“I want”).

Organization of the educational process at stage I.

Hello! I'm glad to see you all in math class. I hope it's mutual.

Guys, what new knowledge did you acquire in the last lesson? (Divide fractions).

Right. What helps you divide fractions? (Rule, properties).

Where do we need this knowledge? (In examples, equations, tasks).

Well done! You did well in the last lesson. Would you like to discover new knowledge yourself today? (Yes).

Then - go! And the motto of the lesson is the statement “Mathematics cannot be learned by watching how your neighbor does it!”.

II. Actualization of knowledge and fixation of an individual difficulty in a trial action.

To organize the actualization of the studied methods of action, sufficient to build new knowledge. Fix these methods verbally (in speech) and symbolically (standard) and generalize them;
Organize the actualization of mental operations and cognitive processes sufficient to build new knowledge;
Motivate for a trial action and its independent implementation and justification;
Present an individual task for a trial action and analyze it in order to identify new educational content;
Organize the fixation of the educational goal and the topic of the lesson;
Organize the implementation of a trial action and fixing the difficulty;
Organize an analysis of the responses received and record individual difficulties in performing a trial action or justifying it.

Organization of the educational process at stage II.

Frontally, using tablets (individual boards).

1. Compare expressions:

(These expressions are equal)

What interesting things did you notice? (The numerator and denominator of the dividend, the numerator and denominator of the divisor in each expression increased by the same number of times. Thus, the dividends and divisors in the expressions are represented by fractions that are equal to each other).

Find the meaning of the expression and write it down on the tablet. (2)

How to write this number as a fraction?

How did you perform the division action? (Children pronounce the rule, the teacher hangs letters on the board)

2. Calculate and record only the results:

3. Add up your results and write down your answer. (2)

What is the name of the number obtained in task 3? (Natural)

Do you think you can divide a fraction by a natural number? (Yes, we will try)

Try this.

4. Individual (trial) task.

Do the division: (example a only)

What rule did you use to divide? (According to the rule of dividing a fraction by a fraction)

And now divide the fraction by a natural number in a simpler way, without performing the entire chain of calculations: (example b). I give you 3 seconds for this.

Who failed to complete the task in 3 seconds?

Who made it? (There are no such)

Why? (We don't know the way)

What did you get? (Difficulty)

What do you think we will do in class? (Divide fractions by natural numbers)

That's right, open your notebooks and write down the topic of the lesson "Dividing a fraction by a natural number."

Why does this topic sound new when you already know how to divide fractions? (Need a new way)

Right. Today we will establish a technique that simplifies the division of a fraction by a natural number.

III. Identification of the location and cause of the difficulty.

Organize the restoration of completed operations and fix (verbal and symbolic) place - step, operation, where the difficulty arose;
To organize the correlation of students' actions with the method (algorithm) used and the fixation in external speech of the cause of the difficulty - those specific knowledge, skills or abilities that are not enough to solve the initial problem of this type.

Organization of the educational process at stage III.

What task did you have to complete? (Divide a fraction by a natural number without doing the whole chain of calculations)

What caused you difficulty? (Could not solve in a short time in a fast way)

What is the purpose of our lesson? (Find a quick way to divide a fraction by a natural number)

What will help you? (Already known rule for dividing fractions)

IV. Construction of the project of an exit from difficulty.

Clarification of the purpose of the project;
Choice of method (clarification);
Definition of funds (algorithm);
Building a plan to achieve the goal.

Organization of the educational process at stage IV.

Let's go back to the test case. Did you say that you divided by the rule of dividing fractions? (Yes)

To do this, replace a natural number with a fraction? (Yes)

What step(s) do you think you can skip?

(The solution chain is open on the board:

Analyze and draw a conclusion. (Step 1)

If there is no answer, then we summarize through the questions:

Where did the natural divisor go? (to the denominator)

Has the numerator changed? (Not)

So what step can be "omitted"? (Step 1)

Multiply the denominator of a fraction by a natural number.
The numerator does not change.
We get a new fraction.

V. Implementation of the constructed project.

Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
Organize the fixation of the constructed method of action in speech and signs (with the help of a standard);
Organize the solution of the original problem and record the overcoming of the difficulty;
Organize a clarification of the general nature of the new knowledge.

Organization of the educational process at stage V.

Now run the test case in the new way quickly.

Are you able to complete the task quickly now? (Yes)

Explain how you did it? (Children speak)

This means that we have received new knowledge: the rule for dividing a fraction by a natural number.

Well done! Say it in pairs.

Then one student speaks to the class. We fix the rule-algorithm verbally and in the form of a standard on the board.

Now enter the letter designations and write down the formula for our rule.

The student writes on the board, pronouncing the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, and leave the numerator the same.

(Everyone writes the formula in notebooks).

And now once again analyze the chain of solving the trial task, paying special attention to the answer. What did they do? (The numerator of the fraction 15 was divided (reduced) by the number 3)

What is this number? (Natural, divisor)

So how else can you divide a fraction by a natural number? (Check: if the numerator of a fraction is divisible by this natural number, then you can divide the numerator by this number, write the result into the numerator of the new fraction, and leave the denominator the same)

Write this method in the form of a formula. (The student writes down the rule on the board. Everyone writes down the formula in notebooks.)

Let's go back to the first method. Can it be used if a:n? (Yes, this is the general way)

And when is the second method convenient to use? (When the numerator of a fraction is divisible by a natural number without a remainder)

VI. Primary consolidation with pronunciation in external speech.

To organize the assimilation by children of a new method of action when solving typical problems with their pronunciation in external speech (frontally, in pairs or groups).

Organization of the educational process at stage VI.

Calculate in a new way:

No. 363 (a; d) - perform at the blackboard, pronouncing the rule.
No. 363 (d; f) - in pairs with a check on the sample.

VII. Independent work with self-test according to the standard.

To organize the students' independent fulfillment of tasks for a new mode of action;
Organize self-test based on comparison with the standard;
Based on the results of independent work, organize a reflection on the assimilation of a new mode of action.

Organization of the educational process at stage VII.

Calculate in a new way:

Students check the standard, note the correctness of the performance. The causes of errors are analyzed and errors are corrected.

The teacher asks those students who made mistakes, what is the reason?

At this stage, it is important that each student independently check their work.

Before solving task 8) consider an example from the textbook:

IX. Reflection of learning activities in the classroom.

Organize the fixation of new content studied in the lesson;
Organize a reflective analysis of educational activities in terms of fulfilling the requirements known to students;
Organize students' assessment of their own activities in the lesson;
Organize the fixation of unresolved difficulties in the lesson as a direction for future learning activities;
Organize discussion and recording of homework.

Organization of the educational process at stage IX.

Guys, what new knowledge did you discover today? (We learned to divide a fraction by a natural number in a simple way)

Formulate a general way. (They say)

In what way, and in what cases can you still use it? (They say)

What is the advantage of the new method?

Have we reached our goal of the lesson? (Yes)

What knowledge did you use to achieve the goal? (They say)

Have you succeeded?

What were the difficulties?

§ 87. Addition of fractions.

Adding fractions has many similarities to adding integers. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), which contains all units and fractions of units of terms.

We will consider three cases in turn:

1. Addition of fractions with the same denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.

1. Addition of fractions with the same denominators.

Consider an example: 1 / 5 + 2 / 5 .

Take the segment AB (Fig. 17), take it as a unit and divide it into 5 equal parts, then the part AC of this segment will be equal to 1/5 of the segment AB, and the part of the same segment CD will be equal to 2/5 AB.

It can be seen from the drawing that if we take the segment AD, then it will be equal to 3/5 AB; but segment AD is precisely the sum of segments AC and CD. So, we can write:

1 / 5 + 2 / 5 = 3 / 5

Considering these terms and the resulting amount, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.

From this we get the following rule: To add fractions with the same denominators, you must add their numerators and leave the same denominator.

Consider an example:

2. Addition of fractions with different denominators.

Let's add fractions: 3/4 + 3/8 First they need to be reduced to the lowest common denominator:

The intermediate link 6/8 + 3/8 could not have been written; we have written it here for greater clarity.

Thus, to add fractions with different denominators, you must first bring them to the lowest common denominator, add their numerators and sign the common denominator.

Consider an example (we will write additional factors over the corresponding fractions):

3. Addition of mixed numbers.

Let's add the numbers: 2 3 / 8 + 3 5 / 6.

Let us first bring the fractional parts of our numbers to a common denominator and rewrite them again:

Now add the integer and fractional parts in sequence:

§ 88. Subtraction of fractions.

Subtraction of fractions is defined in the same way as subtraction of whole numbers. This is an action by which, given the sum of two terms and one of them, another term is found. Let's consider three cases in turn:

1. Subtraction of fractions with the same denominators.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.

1. Subtraction of fractions with the same denominators.

Consider an example:

13 / 15 - 4 / 15

Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then the AC part of this segment will be 1/15 of AB, and the AD part of the same segment will correspond to 13/15 AB. Let's set aside another segment ED, equal to 4/15 AB.

We need to subtract 4/15 from 13/15. In the drawing, this means that the segment ED must be subtracted from the segment AD. As a result, segment AE will remain, which is 9/15 of segment AB. So we can write:

The example we made shows that the numerator of the difference was obtained by subtracting the numerators, and the denominator remained the same.

Therefore, in order to subtract fractions with the same denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.

2. Subtraction of fractions with different denominators.

Example. 3/4 - 5/8

First, let's reduce these fractions to the smallest common denominator:

The intermediate link 6 / 8 - 5 / 8 is written here for clarity, but it can be skipped in the future.

Thus, in order to subtract a fraction from a fraction, you must first bring them to the smallest common denominator, then subtract the numerator of the subtrahend from the numerator of the minuend and sign the common denominator under their difference.

Consider an example:

3. Subtraction of mixed numbers.

Example. 10 3 / 4 - 7 2 / 3 .

Let's bring the fractional parts of the minuend and the subtrahend to the lowest common denominator:

We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the integer part of the reduced, split it into those parts in which the fractional part is expressed, and add to the fractional part of the reduced. And then the subtraction will be performed in the same way as in the previous example:

§ 89. Multiplication of fractions.

When studying the multiplication of fractions, we will consider the following questions:

1. Multiplying a fraction by an integer.
2. Finding a fraction of a given number.
3. Multiplication of a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding percentages of a given number. Let's consider them sequentially.

1. Multiplying a fraction by an integer.

Multiplying a fraction by an integer has the same meaning as multiplying an integer by an integer. Multiplying a fraction (multiplicand) by an integer (multiplier) means composing the sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

So, if you need to multiply 1/9 by 7, then this can be done like this:

We easily got the result, since the action was reduced to adding fractions with the same denominators. Hence,

Consideration of this action shows that multiplying a fraction by an integer is equivalent to increasing this fraction as many times as there are units in the integer. And since the increase in the fraction is achieved either by increasing its numerator

or by decreasing its denominator , then we can either multiply the numerator by the integer, or divide the denominator by it, if such a division is possible.

From here we get the rule:

To multiply a fraction by an integer, you need to multiply the numerator by this integer and leave the same denominator or, if possible, divide the denominator by this number, leaving the numerator unchanged.

When multiplying, abbreviations are possible, for example:

2. Finding a fraction of a given number. There are many problems in which you have to find, or calculate, a part of a given number. The difference between these tasks and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce the method of solving them.

Task 1. I had 60 rubles; 1 / 3 of this money I spent on the purchase of books. How much did the books cost?

Task 2. The train must cover the distance between cities A and B, equal to 300 km. He has already covered 2/3 of that distance. How many kilometers is this?

Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there?

Here are some of the many problems that we have to deal with to find a fraction of a given number. They are usually called problems for finding a fraction of a given number.

Solution of problem 1. From 60 rubles. I spent 1 / 3 on books; So, to find the cost of books, you need to divide the number 60 by 3:

Problem 2 solution. The meaning of the problem is that you need to find 2 / 3 of 300 km. Calculate first 1/3 of 300; this is achieved by dividing 300 km by 3:

300: 3 = 100 (that's 1/3 of 300).

To find two-thirds of 300, you need to double the resulting quotient, that is, multiply by 2:

100 x 2 = 200 (that's 2/3 of 300).

Solution of problem 3. Here you need to determine the number of brick houses, which are 3/4 of 400. Let's first find 1/4 of 400,

400: 4 = 100 (that's 1/4 of 400).

To calculate three quarters of 400, the resulting quotient must be tripled, that is, multiplied by 3:

100 x 3 = 300 (that's 3/4 of 400).

Based on the solution of these problems, we can derive the following rule:

To find the value of a fraction of a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.

3. Multiplication of a whole number by a fraction.

Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 \u003d 5 + 5 + 5 + 5 \u003d 20). In this paragraph (paragraph 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.

In both cases, the multiplication consisted in finding the sum of identical terms.

Now we move on to multiplying a whole number by a fraction. Here we will meet with such, for example, multiplication: 9 2 / 3. It is quite obvious that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.

Because of this, we will have to give a new definition of multiplication, i.e., in other words, to answer the question of what should be understood by multiplication by a fraction, how this action should be understood.

The meaning of multiplying an integer by a fraction is clear from the following definition: to multiply an integer (multiplier) by a fraction (multiplier) means to find this fraction of the multiplier.

Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it's easy to figure out that we end up with 6.

But now an interesting and important question arises: why such seemingly different actions as finding the sum of equal numbers and finding the fraction of a number are called the same word “multiplication” in arithmetic?

This happens because the previous action (repeating the number with terms several times) and the new action (finding the fraction of a number) give an answer to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by one and the same action.

To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?

This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).

Let's take the same problem, but in it the amount of cloth will be expressed as a fractional number: “1 m of cloth costs 50 rubles. How much will 3/4 m of such cloth cost?

This problem also needs to be solved by multiplying the number of rubles (50) by the number of meters (3/4).

You can also change the numbers in it several times without changing the meaning of the problem, for example, take 9/10 m or 2 3/10 m, etc.

Since these problems have the same content and differ only in numbers, we call the actions used in solving them the same word - multiplication.

How is a whole number multiplied by a fraction?

Let's take the numbers encountered in the last problem:

According to the definition, we must find 3 / 4 of 50. First we find 1 / 4 of 50, and then 3 / 4.

1/4 of 50 is 50/4;

3/4 of 50 is .

Hence.

Consider another example: 12 5 / 8 = ?

1/8 of 12 is 12/8,

5/8 of the number 12 is .

Hence,

From here we get the rule:

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.

We write this rule using letters:

To make this rule perfectly clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for multiplying a number by a quotient, which was set out in § 38

It must be remembered that before performing multiplication, you should do (if possible) cuts, For example:

4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying an integer by a fraction, that is, when multiplying a fraction by a fraction, you need to find the fraction in the multiplier from the first fraction (multiplier).

Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.

How do you multiply a fraction by a fraction?

Let's take an example: 3/4 times 5/7. This means that you need to find 5 / 7 from 3 / 4 . Find first 1/7 of 3/4 and then 5/7

1/7 of 3/4 would be expressed like this:

5 / 7 numbers 3 / 4 will be expressed as follows:

Thus,

Another example: 5/8 times 4/9.

1/9 of 5/8 is ,

4/9 numbers 5/8 are .

Thus,

From these examples, the following rule can be deduced:

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 product the denominator of the product.

This rule can be written in general as follows:

When multiplying, it is necessary to make (if possible) reductions. Consider examples:

5. Multiplication of mixed numbers. Since mixed numbers can easily be replaced by improper fractions, this circumstance is usually used when multiplying mixed numbers. This means that in those cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, then they are replaced by improper fractions. Multiply, for example, mixed numbers: 2 1/2 and 3 1/5. We turn each of them into an improper fraction and then we will multiply the resulting fractions according to the rule of multiplying a fraction by a fraction:

Rule. To multiply mixed numbers, you must first convert them to improper fractions and then multiply according to the rule of multiplying a fraction by a fraction.

Note. If one of the factors is an integer, then the multiplication can be performed based on the distribution law as follows:

6. The concept of interest. When solving problems and when performing various practical calculations, we use all kinds of fractions. But one must keep in mind that many quantities admit not any, but natural subdivisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a penny, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of the ruble, it will be "10 kopecks, or a dime. You can take a quarter of the ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take, for example , 2/7 rubles because the ruble is not divided into sevenths.

The unit of measurement for weight, i.e., the kilogram, allows, first of all, decimal subdivisions, for example, 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are uncommon.

In general our (metric) measures are decimal and allow decimal subdivisions.

However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (uniform) method of subdividing quantities. Many years of experience have shown that such a well-justified division is the "hundredths" division. Let's consider a few examples related to the most diverse areas of human practice.

1. The price of books has decreased by 12/100 of the previous price.

Example. The previous price of the book is 10 rubles. She went down by 1 ruble. 20 kop.

2. Savings banks pay out during the year to depositors 2/100 of the amount that is put into savings.

Example. 500 rubles are put into the cash desk, the income from this amount for the year is 10 rubles.

3. The number of graduates of one school was 5/100 of the total number of students.

EXAMPLE Only 1,200 students studied at the school, 60 of them graduated from school.

The hundredth of a number is called a percentage..

The word "percent" is borrowed from the Latin language and its root "cent" means one hundred. Together with the preposition (pro centum), this word means "for a hundred." The meaning of this expression follows from the fact that initially in ancient Rome interest was the money that the debtor paid to the lender “for every hundred”. The word "cent" is heard in such familiar words: centner (one hundred kilograms), centimeter (they say centimeter).

For example, instead of saying that the plant produced 1/100 of all the products produced by it during the past month, we will say this: the plant produced one percent of the rejects during the past month. Instead of saying: the plant produced 4/100 more products than the established plan, we will say: the plant exceeded the plan by 4 percent.

The above examples can be expressed differently:

1. The price of books has decreased by 12 percent of the previous price.

2. Savings banks pay depositors 2 percent per year of the amount put into savings.

3. The number of graduates of one school was 5 percent of the number of all students in the school.

To shorten the letter, it is customary to write the% sign instead of the word "percentage".

However, it must be remembered that the % sign is usually not written in calculations, it can be written in the problem statement and in the final result. When performing calculations, you need to write a fraction with a denominator of 100 instead of an integer with this icon.

You need to be able to replace an integer with the specified icon with a fraction with a denominator of 100:

Conversely, you need to get used to writing an integer with the indicated icon instead of a fraction with a denominator of 100:

7. Finding percentages of a given number.

Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch wood was there?

The meaning of this problem is that birch firewood was only a part of the firewood that was delivered to the school, and this part is expressed as a fraction of 30 / 100. So, we are faced with the task of finding a fraction of a number. To solve it, we must multiply 200 by 30 / 100 (tasks for finding the fraction of a number are solved by multiplying a number by a fraction.).

So 30% of 200 equals 60.

The fraction 30 / 100 encountered in this problem can be reduced by 10. It would be possible to perform this reduction from the very beginning; the solution to the problem would not change.

Task 2. There were 300 children of various ages in the camp. Children aged 11 were 21%, children aged 12 were 61% and finally 13 year olds were 18%. How many children of each age were in the camp?

In this problem, you need to perform three calculations, that is, successively find the number of children 11 years old, then 12 years old, and finally 13 years old.

So, here it will be necessary to find a fraction of a number three times. Let's do it:

1) How many children were 11 years old?

2) How many children were 12 years old?

3) How many children were 13 years old?

After solving the problem, it is useful to add the numbers found; their sum should be 300:

63 + 183 + 54 = 300

You should also pay attention to the fact that the sum of the percentages given in the condition of the problem is 100:

21% + 61% + 18% = 100%

This suggests that the total number of children in the camp was taken as 100%.

3 a da cha 3. The worker received 1,200 rubles per month. Of these, he spent 65% on food, 6% on an apartment and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% he saved. How much money was spent on the needs indicated in the task?

To solve this problem, you need to find a fraction of the number 1,200 5 times. Let's do it.

1) How much money is spent on food? The task says that this expense is 65% of all earnings, i.e. 65/100 of the number 1,200. Let's do the calculation:

2) How much money was paid for an apartment with heating? Arguing like the previous one, we arrive at the following calculation:

3) How much money did you pay for gas, electricity and radio?

4) How much money is spent on cultural needs?

5) How much money did the worker save?

For verification, it is useful to add the numbers found in these 5 questions. The amount should be 1,200 rubles. All earnings are taken as 100%, which is easy to check by adding up the percentages given in the problem statement.

We have solved three problems. Despite the fact that these tasks were about different things (delivery of firewood for the school, the number of children of different ages, the expenses of the worker), they were solved in the same way. This happened because in all tasks it was necessary to find a few percent of the given numbers.

§ 90. Division of fractions.

When studying the division of fractions, we will consider the following questions:

1. Divide an integer by an integer.
2. Division of a fraction by an integer
3. Division of an integer by a fraction.
4. Division of a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number given its fraction.
7. Finding a number by its percentage.

Let's consider them sequentially.

1. Divide an integer by an integer.

As was indicated in the section on integers, division is the action consisting in the fact that, given the product of two factors (the dividend) and one of these factors (the divisor), another factor is found.

The division of an integer by an integer we considered in the department of integers. We met there two cases of division: division without a remainder, or "entirely" (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 in the remainder). We can therefore say that in the realm of integers, exact division is not always possible, because the dividend is not always the product of the divisor and the integer. After the introduction of multiplication by a fraction, we can consider any case of division of integers as possible (only division by zero is excluded).

For example, dividing 7 by 12 means finding a number whose product times 12 would be 7. This number is the fraction 7/12 because 7/12 12 = 7. Another example: 14: 25 = 14/25 because 14/25 25 = 14.

Thus, to divide an integer by an integer, you need to make a fraction, the numerator of which is equal to the dividend, and the denominator is the divisor.

2. Division of a fraction by an integer.

Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find such a second factor that, when multiplied by 3, would give the given product 6 / 7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6 / 7 by 3 times.

We already know that the reduction of a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore, you can write:

In this case, the numerator 6 is divisible by 3, so the numerator should be reduced by 3 times.

Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:

Based on this, we can state the rule: To divide a fraction by an integer, you need to divide the numerator of the fraction by that integer(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.

3. Division of an integer by a fraction.

Let it be required to divide 5 by 1 / 2, i.e. find a number that, after multiplying by 1 / 2, will give the product 5. Obviously, this number must be greater than 5, since 1 / 2 is a proper fraction, and when multiplying a number by a proper fraction, the product must be less than the multiplicand. To make it clearer, let's write our actions as follows: 5: 1 / 2 = X , so x 1 / 2 \u003d 5.

We must find such a number X , which, when multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is 5, and the whole number X twice as much, i.e. 5 2 \u003d 10.

So 5: 1 / 2 = 5 2 = 10

Let's check:

Let's consider one more example. Let it be required to divide 6 by 2 / 3 . Let's first try to find the desired result using the drawing (Fig. 19).

Fig.19

Draw a segment AB, equal to 6 of some units, and divide each unit into 3 equal parts. In each unit, three-thirds (3 / 3) in the entire segment AB is 6 times larger, i.e. e. 18/3. We connect with the help of small brackets 18 obtained segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in b units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 integer units. Hence,

How to get this result without a drawing using only calculations? We will argue as follows: it is required to divide 6 by 2 / 3, i.e., it is required to answer the question, how many times 2 / 3 is contained in 6. Let's find out first: how many times is 1 / 3 contained in 6? In a whole unit - 3 thirds, and in 6 units - 6 times more, i.e. 18 thirds; to find this number, we must multiply 6 by 3. Hence, 1/3 is contained in b units 18 times, and 2/3 is contained in b units not 18 times, but half as many times, i.e. 18: 2 = 9. Therefore , when dividing 6 by 2 / 3 we did the following:

From here we get the rule for dividing an integer by a fraction. To divide an integer by a fraction, you need to multiply this integer by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.

We write the rule using letters:

To make this rule perfectly clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for dividing a number by a quotient, which was set out in § 38. Note that the same formula was obtained there.

When dividing, abbreviations are possible, for example:

4. Division of a fraction by a fraction.

Let it be required to divide 3/4 by 3/8. What will denote the number that will be obtained as a result of division? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To understand this issue, let's make a drawing (Fig. 20).

Take the segment AB, take it as a unit, divide it into 4 equal parts and mark 3 such parts. Segment AC will be equal to 3/4 of segment AB. Let us now divide each of the four initial segments in half, then the segment AB will be divided into 8 equal parts and each such part will be equal to 1/8 of the segment AB. We connect 3 such segments with arcs, then each of the segments AD and DC will be equal to 3/8 of the segment AB. The drawing shows that the segment equal to 3/8 is contained in the segment equal to 3/4 exactly 2 times; So the result of the division can be written like this:

3 / 4: 3 / 8 = 2

Let's consider one more example. Let it be required to divide 15/16 by 3/32:

We can reason like this: we need to find a number that, after being multiplied by 3 / 32, will give a product equal to 15 / 16. Let's write the calculations like this:

15 / 16: 3 / 32 = X

3 / 32 X = 15 / 16

3/32 unknown number X make up 15 / 16

1/32 unknown number X is ,

32 / 32 numbers X make up .

Hence,

Thus, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second and make the first product the numerator and the second the denominator.

Let's write the rule using letters:

When dividing, abbreviations are possible, for example:

5. Division of mixed numbers.

When dividing mixed numbers, they must first be converted into improper fractions, and then the resulting fractions should be divided according to the rules for dividing fractional numbers. Consider an example:

Convert mixed numbers to improper fractions:

Now let's split:

Thus, to divide mixed numbers, you need to convert them to improper fractions and then divide according to the rule for dividing fractions.

6. Finding a number given its fraction.

Among the various tasks on fractions, there are sometimes those in which the value of some fraction of an unknown number is given and it is required to find this number. This type of problem will be inverse to the problem of finding a fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number is given and it is required to find this number itself. This idea will become even clearer if we turn to the solution of this type of problem.

Task 1. On the first day, glaziers glazed 50 windows, which is 1 / 3 of all windows of the built house. How many windows are in this house?

Decision. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means that there are 3 times more windows in total, i.e.

The house had 150 windows.

Task 2. The shop sold 1,500 kg of flour, which is 3/8 of the total stock of flour in the shop. What was the store's initial supply of flour?

Decision. It can be seen from the condition of the problem that the sold 1,500 kg of flour make up 3/8 of the total stock; this means that 1/8 of this stock will be 3 times less, i.e., to calculate it, you need to reduce 1500 by 3 times:

1,500: 3 = 500 (that's 1/8 of the stock).

Obviously, the entire stock will be 8 times larger. Hence,

500 8 \u003d 4,000 (kg).

The initial supply of flour in the store was 4,000 kg.

From the consideration of this problem, the following rule can be deduced.

To find a number by a given value of its fraction, it is enough to divide this value by the numerator of the fraction and multiply the result by the denominator of the fraction.

We solved two problems on finding a number given its fraction. Such problems, as is especially well seen from the last one, are solved by two actions: division (when one part is found) and multiplication (when the whole number is found).

However, after we have studied the division of fractions, the above problems can be solved in one action, namely: division by a fraction.

For example, the last task can be solved in one action like this:

In the future, we will solve the problem of finding a number by its fraction in one action - division.

7. Finding a number by its percentage.

In these tasks, you will need to find a number, knowing a few percent of this number.

Task 1. At the beginning of this year, I received 60 rubles from the savings bank. income from the amount I put into savings a year ago. How much money did I put in the savings bank? (Cash offices give depositors 2% of income per year.)

The meaning of the problem is that a certain amount of money was put by me in a savings bank and lay there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I put in. How much money did I deposit?

Therefore, knowing the part of this money, expressed in two ways (in rubles and in fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following tasks are solved by division:

So, 3,000 rubles were put into the savings bank.

Task 2. In two weeks, fishermen fulfilled the monthly plan by 64%, having prepared 512 tons of fish. What was their plan?

From the condition of the problem, it is known that the fishermen completed part of the plan. This part is equal to 512 tons, which is 64% of the plan. How many tons of fish need to be harvested according to the plan, we do not know. The solution of the problem will consist in finding this number.

Such tasks are solved by dividing:

So, according to the plan, you need to prepare 800 tons of fish.

Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked the passing conductor how much of the journey they had already traveled. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?

It can be seen from the condition of the problem that 30% of the journey from Riga to Moscow is 276 km. We need to find the entire distance between these cities, i.e., for this part, find the whole:

§ 91. Reciprocal numbers. Replacing division with multiplication.

Take the fraction 2/3 and rearrange the numerator to the place of the denominator, we get 3/2. We got a fraction, the reciprocal of this one.

In order to get a fraction reciprocal of a given one, you need to put its numerator in the place of the denominator, and the denominator in the place of the numerator. In this way, we can get a fraction that is the reciprocal of any fraction. For example:

3 / 4 , reverse 4 / 3 ; 5 / 6 , reverse 6 / 5

Two fractions that have the property that the numerator of the first is the denominator of the second and the denominator of the first is the numerator of the second are called mutually inverse.

Now let's think about what fraction will be the reciprocal of 1/2. Obviously, it will be 2 / 1, or just 2. Looking for the reciprocal of this, we got an integer. And this case is not isolated; on the contrary, for all fractions with a numerator of 1 (one), the reciprocals will be integers, for example:

1 / 3, inverse 3; 1 / 5, reverse 5

Since when finding reciprocals we also met with integers, in the future we will not talk about reciprocals, but about reciprocals.

Let's figure out how to write the reciprocal of a whole number. For fractions, this is solved simply: you need to put the denominator in the place of the numerator. In the same way, you can get the reciprocal of an integer, since any integer can have a denominator of 1. So the reciprocal of 7 will be 1 / 7, because 7 \u003d 7 / 1; for the number 10 the reverse is 1 / 10 since 10 = 10 / 1

This idea can be expressed in another way: the reciprocal of a given number is obtained by dividing one by the given number. This statement is true not only for integers, but also for fractions. Indeed, if you want to write a number that is the reciprocal of the fraction 5 / 9, then we can take 1 and divide it by 5 / 9, i.e.

Now let's point out one property mutually reciprocal numbers, which will be useful to us: the product of mutually reciprocal numbers is equal to one. Indeed:

Using this property, we can find reciprocals in the following way. Let's find the reciprocal of 8.

Let's denote it with the letter X , then 8 X = 1, hence X = 1 / 8 . Let's find another number, the inverse of 7/12, denote it by a letter X , then 7 / 12 X = 1, hence X = 1:7 / 12 or X = 12 / 7 .

We introduced here the concept of reciprocal numbers in order to slightly supplement information about the division of fractions.

When we divide the number 6 by 3 / 5, then we do the following:

Pay special attention to the expression and compare it with the given one: .

If we take the expression separately, without connection with the previous one, then it is impossible to solve the question of where it came from: from dividing 6 by 3/5 or from multiplying 6 by 5/3. In both cases the result is the same. So we can say that dividing one number by another can be replaced by multiplying the dividend by the reciprocal of the divisor.

The examples that we give below fully confirm this conclusion.

Decimal multiplication takes place in three stages.

Decimals are written in a column and multiplied as ordinary numbers.

We count the number of decimal places for the first decimal and the second. We add their number.

In the result obtained, we count from right to left as many digits as they turned out in the paragraph above and put a comma.

How to multiply decimals

We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. That is, we consider 3.11 as 311, and 0.01 as 1.

Received 311 . Now we count the number of signs (digits) after the decimal point for both fractions. The first decimal has two digits and the second has two. Total number of digits after commas:

We count from right to left 4 characters (numbers) of the resulting number. There are fewer digits in the result than you need to separate with a comma. In that case, you need left assign the missing number of zeros.

We are missing one digit, so we attribute one zero to the left.

When multiplying any decimal fraction on 10; 100; 1000 etc. the decimal point moves to the right as many digits as there are zeros after the one.

70.1 10 = 701
0.023 100 = 2.3
5.6 1000 = 5600

To multiply a decimal by 0.1; 0.01; 0.001, etc., it is necessary to move the comma to the left in this fraction by as many digits as there are zeros in front of the unit.

We count zero integers!

12 0.1 = 1.2
0.05 0.1 = 0.005
1.256 0.01 = 0.012 56

Multiplication of fractions

We will consider the multiplication of ordinary fractions in several possible ways.

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 a 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.

How to multiply a fraction by an integer rule

I. To multiply a decimal fraction by a natural number, you need to multiply it by this number, ignoring the comma, and in the resulting product, separate as many digits on the right as there were after the decimal point in the given fraction.

Examples. Perform multiplication: 1) 1.25 7; 2) 0.345 8; 3) 2.391 14.

Decision.

II. To multiply one decimal fraction by another, you need to perform the multiplication, ignoring the commas, and in the resulting result, separate as many digits with a comma on the right as there were after the commas in both factors together.

Examples. Perform multiplication: 1) 18.2 0.09; 2) 3.2 0.065; 3) 0.54 12.3.

Decision.

III. To multiply a decimal by 10, 100, 1000, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits.

Examples. Perform multiplication: 1) 3.25 10; 2) 0.637 100; 3) 4.307 1000; 4) 2.04 1000; 5) 0.00031 10000.

Decision.

IV. To multiply a decimal by 0.1; 0.01; 0.001, etc., you need to move the comma to the left by 1, 2, 3, etc. digits.

Examples. Perform multiplication: 1) 28.3 0.1; 2) 324.7 0.01; 3) 6.85 0.01; 4) 6179.5 0.001; 5) 92.1 0.0001.

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Multiplication of decimal fractions, rules, examples, solutions.

We turn to the study of the next action with decimal fractions, now we will comprehensively consider multiplying decimals. First, let's discuss the general principles of multiplying decimal fractions. After that, let's move on to multiplying a decimal fraction by a decimal fraction, show how the multiplication of decimal fractions by a column is performed, consider the solutions of examples. Next, we will analyze the multiplication of decimal fractions by natural numbers, in particular by 10, 100, etc. In conclusion, let's talk about multiplying decimal fractions by ordinary fractions and mixed numbers.

Let's say right away that in this article we will only talk about multiplying positive decimal fractions (see positive and negative numbers). The remaining cases are analyzed in the articles multiplication of rational numbers and multiplication of real numbers.

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General principles for multiplying decimals

Let's discuss the general principles that should be followed when performing multiplication with decimal fractions.

Since trailing decimals and infinite periodic fractions are the decimal form of common fractions, multiplying such decimals is essentially multiplying common fractions. In other words, multiplication of final decimals, multiplication of final and periodic decimal fractions, as well as multiplying periodic decimals comes down to multiplying ordinary fractions after converting decimal fractions to ordinary ones.

Consider examples of the application of the voiced principle of multiplying decimal fractions.

Perform the multiplication of decimals 1.5 and 0.75.

Let us replace the multiplied decimal fractions with the corresponding ordinary fractions. Since 1.5=15/10 and 0.75=75/100, then. You can reduce the fraction, and then select the whole part from the improper fraction, and it is more convenient to write the resulting ordinary fraction 1 125/1 000 as a decimal fraction 1.125.

It should be noted that it is convenient to multiply the final decimal fractions in a column, we will talk about this method of multiplying decimal fractions in the next paragraph.

Consider an example of multiplying periodic decimal fractions.

Compute the product of the periodic decimals 0,(3) and 2,(36) .

Let's convert periodic decimal fractions to ordinary fractions:

Then. You can convert the resulting ordinary fraction to a decimal fraction:

If there are infinite non-periodic fractions among the multiplied decimal fractions, then all multiplied fractions, including finite and periodic ones, should be rounded up to a certain digit (see rounding numbers), and then perform the multiplication of the final decimal fractions obtained after rounding.

Multiply the decimals 5.382… and 0.2.

First, we round off an infinite non-periodic decimal fraction, rounding can be done to hundredths, we have 5.382 ... ≈5.38. The final decimal fraction 0.2 does not need to be rounded to hundredths. Thus, 5.382… 0.2≈5.38 0.2. It remains to calculate the product of final decimal fractions: 5.38 0.2 \u003d 538 / 100 2 / 10 \u003d 1,076/1,000 \u003d 1.076.

Multiplication of decimal fractions by a column

Multiplication of finite decimal fractions can be performed by a column, similar to multiplication by a column of natural numbers.

Let's formulate multiplication rule for decimal fractions. To multiply decimal fractions by a column, you need:

ignoring commas, perform multiplication according to all the rules of multiplication by a column of natural numbers;
in the resulting number, separate as many digits on the right with a decimal point as there are decimal places in both factors together, and if there are not enough digits in the product, then the required number of zeros must be added on the left.

Consider examples of multiplying decimal fractions by a column.

Multiply the decimals 63.37 and 0.12.

Let's carry out the multiplication of decimal fractions by a column. First, we multiply the numbers, ignoring the commas:

It remains to put a comma in the resulting product. She needs to separate 4 digits on the right, since there are four decimal places in the factors (two in the fraction 3.37 and two in the fraction 0.12). There are enough numbers there, so you don’t have to add zeros on the left. Let's finish the record:

As a result, we have 3.37 0.12 = 7.6044.

Calculate the product of decimals 3.2601 and 0.0254 .

Having performed multiplication by a column without taking into account commas, we get the following picture:

Now in the product you need to separate 8 digits on the right with a comma, since the total number of decimal places of the multiplied fractions is eight. But there are only 7 digits in the product, therefore, you need to assign as many zeros on the left so that 8 digits can be separated by a comma. In our case, we need to assign two zeros:

This completes the multiplication of decimal fractions by a column.

Multiplying decimals by 0.1, 0.01, etc.

Quite often you have to multiply decimals by 0.1, 0.01, and so on. Therefore, it is advisable to formulate a rule for multiplying a decimal fraction by these numbers, which follows from the principles of multiplying decimal fractions discussed above.

So, multiplying a given decimal by 0.1, 0.01, 0.001, and so on gives a fraction that is obtained from the original one, if in its entry the comma is moved to the left by 1, 2, 3 and so on digits, respectively, and if there are not enough digits to move the comma, then you need to add the required number of zeros on the left.

For example, to multiply the decimal fraction 54.34 by 0.1, you need to move the decimal point to the left by 1 digit in the fraction 54.34, and you get the fraction 5.434, that is, 54.34 0.1 \u003d 5.434. Let's take another example. Multiply the decimal fraction 9.3 by 0.0001. To do this, we need to move the comma 4 digits to the left in the multiplied decimal fraction 9.3, but the record of the fraction 9.3 does not contain such a number of characters. Therefore, we need to add as many zeros in the record of the fraction 9.3 on the left so that we can easily transfer the comma to 4 digits, we have 9.3 0.0001 \u003d 0.00093.

Note that the announced rule for multiplying a decimal fraction by 0.1, 0.01, ... is also valid for infinite decimal fractions. For example, 0,(18) 0.01=0.00(18) or 93.938… 0.1=9.3938… .

Multiplying a decimal by a natural number

At its core multiplying decimals by natural numbers is no different from multiplying a decimal by a decimal.

It is most convenient to multiply a finite decimal fraction by a natural number by a column, while you should follow the rules for multiplying by a column of decimal fractions discussed in one of the previous paragraphs.

Calculate the product 15 2.27 .

Let's carry out the multiplication of a natural number by a decimal fraction in a column:

When multiplying a periodic decimal fraction by a natural number, the periodic fraction should be replaced with an ordinary fraction.

Multiply the decimal fraction 0,(42) by the natural number 22.

First, let's convert the periodic decimal to a common fraction:

Now let's do the multiplication: . This decimal result is 9,(3) .

And when multiplying an infinite non-periodic decimal fraction by a natural number, you must first round off.

Do the multiplication 4 2.145….

Rounding up to hundredths the original infinite decimal fraction, we will come to the multiplication of a natural number and a final decimal fraction. We have 4 2.145…≈4 2.15=8.60.

Multiplying a decimal by 10, 100, ...

Quite often you have to multiply decimal fractions by 10, 100, ... Therefore, it is advisable to dwell on these cases in detail.

Let's voice rule for multiplying a decimal by 10, 100, 1,000, etc. When multiplying a decimal fraction by 10, 100, ... in its entry, you need to move the comma to the right by 1, 2, 3, ... digits, respectively, and discard the extra zeros on the left; if there are not enough digits in the record of the multiplied fraction to transfer the comma, then you need to add the required number of zeros to the right.

Multiply the decimal 0.0783 by 100.

Let's transfer the fraction 0.0783 two digits to the right into the record, and we get 007.83. Dropping two zeros on the left, we get the decimal fraction 7.38. Thus, 0.0783 100=7.83.

Multiply the decimal fraction 0.02 by 10,000.

To multiply 0.02 by 10,000 we need to move the comma 4 digits to the right. Obviously, in the record of the fraction 0.02 there are not enough digits to transfer the comma to 4 digits, so we will add a few zeros to the right so that the comma can be transferred. In our example, it is enough to add three zeros, we have 0.02000. After moving the comma, we get the entry 00200.0 . Dropping the zeros on the left, we have the number 200.0, which is equal to the natural number 200, which is the result of multiplying the decimal fraction 0.02 by 10,000.

The stated rule is also valid for multiplying infinite decimal fractions by 10, 100, ... When multiplying periodic decimal fractions, you need to be careful with the period of the fraction that is the result of multiplication.

Multiply the periodic decimal 5.32(672) by 1000 .

Before multiplication, we write the periodic decimal fraction as 5.32672672672 ..., this will allow us to avoid mistakes. Now let's move the comma to the right by 3 digits, we have 5 326.726726 ... . Thus, after multiplication, a periodic decimal fraction is obtained 5 326, (726) .

5.32(672) 1000=5326,(726) .

When multiplying infinite non-periodic fractions by 10, 100, ..., you must first round the infinite fraction to a certain digit, and then carry out the multiplication.

Multiplying a Decimal by a Common Fraction or a Mixed Number

To multiply a finite decimal fraction or an infinite periodic decimal fraction by an ordinary fraction or a mixed number, you need to represent the decimal fraction as an ordinary fraction, and then carry out the multiplication.

Multiply the decimal fraction 0.4 by the mixed number.

Since 0.4=4/10=2/5 and then. The resulting number can be written as a periodic decimal fraction 1.5(3) .

When multiplying an infinite non-periodic decimal fraction by a common fraction or a mixed number, the common fraction or mixed number should be replaced by a decimal fraction, then round the multiplied fractions and finish the calculation.

Since 2/3 \u003d 0.6666 ..., then. After rounding the multiplied fractions to thousandths, we come to the product of two final decimal fractions 3.568 and 0.667. Let's do the multiplication in a column:

The result obtained should be rounded to thousandths, since the multiplied fractions were taken with an accuracy of thousandths, we have 2.379856≈2.380.

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Multiplication of ordinary fractions: rules, examples, solutions.

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Multiplying a common fraction by a common fraction

That is, the formula corresponds to the multiplication of ordinary fractions a / b and c / d.

Note that with the help of the voiced multiplication rule, you can multiply both regular and improper fractions, and fractions with the same denominators, and fractions with different denominators.

Consider examples of multiplying common fractions.

Multiply the common fraction 7/11 by the common fraction 9/8.

Here is a summary of the solution: .

Multiply fractions 4/15 and 55/6.

Let's apply the rule of multiplication of ordinary fractions: .

To clarify, let's go back to the previous example.

Calculate the product of fractions of the form .

By the formula for multiplying ordinary fractions, we have .

Since 4=2 2 , 55=5 11 , 15=3 5 and 6=2 3 , then . Now we cancel the common prime factors: .

It remains only to calculate the products in the numerator and denominator, and then select the integer part from the improper fraction: .

It should be noted that the multiplication of fractions is characterized by a commutative property, that is, the multiplied fractions can be interchanged: .

Multiplying a fraction by a natural number

With the help of letters, the rule for multiplying a fraction a / b by a natural number n has the form .

The formula follows from the formula for multiplying two ordinary fractions of the form . Indeed, representing a natural number as a fraction with a denominator of 1, we obtain .

Consider examples of multiplying a fraction by a natural number.

Multiply the fraction 2/27 by 5.

Multiplying the numerator 2 by the number 5 gives 10, therefore, by virtue of the rule of multiplying a fraction by a natural number, the product of 2/27 by 5 is equal to the fraction 10/27.

The whole solution can be conveniently written as follows: .

When multiplying a fraction by a natural number, the resulting fraction often has to be reduced, and if it is also incorrect, then represent it as a mixed number.

Multiply the fraction 5/12 by the number 8.

Here is the whole solution: .

Note that the reduction could be done by replacing the numbers in the numerator and denominator by their expansions into prime factors. In this case, the solution would look like this:

Multiply three or more fractions

Consider examples of multiplication of several fractions and natural numbers.

Multiply three common fractions 1/20, 12/5, 3/7 and 5/8.

Multiply five numbers .

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Multiplication and division of fractions.

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Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

Multiplication of mixed fractions.

The second way to multiply a fraction by a natural number.

Multilevel fractions.

Multiplying a common fraction by a common fraction

Multiplying a fraction by a natural number

Multiply three or more fractions

Multiplication of fractions

Multiplying a fraction by a fraction

Multiplying a fraction by a natural number

Multiplication of mixed numbers

Another way to multiply a fraction by a natural number

Multiplication of mixed numbers: rules, examples, solutions.

Multiplication of mixed numbers.

Multiplication of a mixed number and a natural number

Multiplication of fractional numbers

Division of a fraction by a natural number

How to multiply decimals

Multiplication of fractions

Multiplying a fraction by a fraction

Multiplying a fraction by a natural number

Multiplication of mixed numbers

Another way to multiply a fraction by a natural number

How to multiply a fraction by an integer rule

Multiplication of decimal fractions, rules, examples, solutions.

General principles for multiplying decimals

Multiplication of decimal fractions by a column

Multiplying decimals by 0.1, 0.01, etc.

Multiplying a decimal by a natural number

Multiplying a decimal by 10, 100, ...

Multiplying a Decimal by a Common Fraction or a Mixed Number

Multiplication of ordinary fractions: rules, examples, solutions.

Multiplying a common fraction by a common fraction

Multiplying a fraction by a natural number

Multiply three or more fractions

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