How to multiply numbers with different denominators. Rules for multiplying and dividing fractions by whole numbers

Multiplying a whole number by a fraction is not a difficult task. But there are subtleties that you probably understood at school, but have since forgotten.

How to multiply a whole number by a fraction - a few terms

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

The numerator is the top part of the fraction - what we are dividing. The denominator is lower. This is what we divide by.
A proper fraction is one whose numerator is less than its denominator. An improper fraction is one whose numerator is greater than or equal to its denominator.

How to multiply a whole number by a fraction

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

Reduction

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

Improper fractions

But suppose we multiply four by two fifths. It turned out to be eight-fifths. This is an improper fraction.
It definitely needs to be brought to the correct form. To do this, you need to select an entire part from it.
Here you need to use division with a remainder. We get one and three as a remainder.
One whole and three fifths is our proper fraction.

Bringing thirty-five eighths to the correct form is a little more difficult. The closest number to thirty-seven that is divisible by eight is thirty-two. When divided we get four. Subtract thirty-two from thirty-five and we get three. Result: four whole and three eighths.

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

Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. Good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.

To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.

Designation:

From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.

As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.

By definition we have:

Multiplying fractions with whole parts and negative fractions

If present in fractions whole part, they must be converted into incorrect ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:

Plus by minus gives minus;
Two negatives make an affirmative.

Until now, these rules have only been encountered in addition and subtraction. negative fractions when it was necessary to get rid of an entire part. For a work, they can be generalized in order to “burn” several disadvantages at once:

We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, because there was no pair for it, we take it out of the limits of multiplication. The result is a negative fraction.

Task. Find the meaning of the expression:

We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:

Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).

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

Reducing fractions on the fly

Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:

Task. Find the meaning of the expression:

By definition we have:

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

Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs due to the fact that when adding the numerator of a fraction, the sum appears, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since in this property we're talking about specifically about multiplying numbers.

There are simply no other reasons for reducing fractions, so the right decision the previous task looks like this:

Correct solution:

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

In this article we will look at multiplying mixed numbers. First, we will outline the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next we'll talk about multiplying a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and a common fraction.

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Multiplying mixed numbers.

Multiplying 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 it down mixed number multiplication rule:

First, the mixed numbers being multiplied must be replaced by improper fractions;
Secondly, you need to use the rule for multiplying fractions by fractions.

Let's look at examples of applying this rule when multiplying a mixed number by a mixed number.

Perform multiplication of mixed numbers and .

First, let's represent the mixed numbers being multiplied as improper fractions: And . Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: . Applying the rule for multiplying fractions, we get . The resulting fraction is irreducible (see reducible and irreducible fractions), but it is improper (see proper and improper fractions), therefore, to obtain the final answer, it remains to isolate the whole part from the improper fraction: .

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

To strengthen the skills of multiplying mixed numbers, consider solving 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 reducing a fraction: replace all the numbers in the fraction with their decompositions into prime factors, and perform a reduction of identical factors.

Multiplying a mixed number and a natural number

After replacing a mixed number with an improper fraction, multiplying a mixed number and a natural number leads to the multiplication of an ordinary fraction and a natural number.

Multiply a mixed number and the natural number 45.

A mixed number is equal to a fraction, then . Let's replace the numbers in the resulting fraction with their decompositions into prime factors, perform a reduction, and then select the whole part: .

Multiplication of a mixed number and a natural number is sometimes conveniently carried out 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, .

Calculate the product.

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

Multiplying mixed numbers and fractions It is most convenient to reduce it to the multiplication of ordinary fractions by representing the mixed number being multiplied as an improper fraction.

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

Replacing the mixed number with a fraction, we get .

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Multiplying fractions

§ 140. Definitions. 1) Multiplying a fraction by an integer is defined in the same way as multiplying integers, namely: to multiply a number (multiplicand) by an integer (factor) means to compose 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) Multiplying a number (multiplicand) by a fraction (factor) means finding this fraction of the multiplicand.

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

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

For example:

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

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

§ 141. The expediency of these definitions. To understand the advisability of introducing the last two definitions of multiplication into arithmetic, let’s take the following problem:

Task. A train, moving uniformly, covers 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 one definition of multiplication that is indicated in integer arithmetic (the 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 you need to multiply 40 km by this number of hours.

If a given number of hours is expressed as a fraction (for example, an hour), 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 40 km will need to be multiplied by the integer contained in the mixed number, and to the result add another fraction of 40 km, which is in the mixed number.

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

you need to multiply 40 km by a given number of hours, whatever it may be.

Thus, if the problem is represented in general view So:

A train, moving uniformly, covers v km in an hour. How many kilometers will the train travel in t hours?

then, whatever the numbers v and t are, we can give 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, finding 5% (i.e. five hundredths) of a given number means the same thing as multiplying a given number by or by ; finding 125% of a given number means the same as multiplying this number by or by, etc.

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

Multiplication by a proper fraction decreases the number, and multiplication by an improper fraction increases the number 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 a whole number. Let a fraction be multiplied by 5. This means increased by 5 times. To increase a fraction by 5 times, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).

That's why:
Rule 1. To multiply a fraction by a whole number, you need to multiply the numerator by this whole number, but 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 a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this 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 multiplying 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 outlined are contained in one, which in general can be expressed as follows:
4) Multiplication of mixed numbers.

Rule 4th. 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 during multiplication. When multiplying fractions, if possible, it is necessary to make a preliminary reduction, as can be seen from the following examples:

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

§ 145. Changing a product with changing 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:
to multiply several fractions, you need to multiply their numerators with each other and the denominators with each other 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 of the factors of the number are integers or mixed, if only we consider the integer as a fraction with a denominator of one, and we turn the mixed numbers into improper fractions. For example:
§ 147. Basic properties of multiplication. Those properties of multiplication that we indicated for integers (§ 56, 57, 59) also apply to the multiplication of fractional numbers. Let us indicate these properties.

1) The product does not change when the factors are changed.

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 terms differ only in the order of the integer factors, and the product of integers does not change when the places of the factors are changed.

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 the following conclusion can be drawn:

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

For example:
3) Distributive law of multiplication (relative to addition). To multiply a sum by a number, you can multiply each term separately by that number and add the results.

This law was explained by us (§ 59) as applied to integers. 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 relative to addition) remains true even when the letters represent fractional numbers. Let's consider three cases.

1) Let us first assume that the factor m is an integer, for example m = 3 (a, b, c – any numbers). According to the definition of multiplication by an integer, we can write (limiting ourselves to three terms for simplicity):

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

Based on the associative law of addition, we can omit all parentheses on the right side; By applying the commutative law of addition, and then again the associative 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.

This means that the distributive law is confirmed in this case.

Multiplying and dividing fractions

Last time we learned how to add and subtract fractions (see lesson “Adding and Subtracting Fractions”). The most difficult part of those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.

To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.

By definition we have:

Multiplying fractions with whole parts and negative fractions

If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:

Plus by minus gives minus;
Two negatives make an affirmative.

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

We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, because there was no pair for it, we take it out of the limits of multiplication. The result is a negative fraction.

Task. Find the meaning of the expression:

We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:

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

Reducing fractions on the fly

Task. Find the meaning of the expression:

By definition we have:

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

However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs due to the fact that when adding the numerator of a fraction, the sum appears, and not the product of numbers. Therefore, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.

There are simply no other reasons for reducing fractions, so the correct solution to the previous problem looks like this:

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

Multiplying fractions.

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

Multiplying a common fraction by a fraction.

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

Let's look at an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.

Multiplying a fraction by a number.

First, let's remember the rule, any number can be represented as a fraction \(\bf n = \frac \) .

Let's use this rule when multiplying.

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

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

Multiplying mixed fractions.

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

Multiplication of reciprocal fractions and numbers.

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

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

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

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

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

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

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

Example #4:
Calculate the product of two mutually inverse fractions: a) \(\frac \times \frac \)

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

Solution:
a) to answer the first question, let's give an example. The fraction \(\frac \) is proper, its inverse fraction will be equal to \(\frac \) - an improper fraction. Answer: no.

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

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

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

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

Example #7:
Can two mutually reciprocal numbers be at the same time mixed numbers?

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

Multiplying a decimal by a natural number

Presentation for the lesson

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

Introduce the multiplication rule to students in a fun way decimal per natural number, per digit unit and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge when solving examples and problems.
Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their own work and the work of each other.
Cultivate interest in mathematics, activity, mobility, and communication skills.

Equipment: interactive whiteboard, a poster with a cyphergram, posters with statements by mathematicians.

Organizational moment.
Oral arithmetic – generalization of previously studied material, preparation for studying new material.
Explanation of new material.
Homework assignment.
Mathematical physical education.
Generalization and systematization of acquired knowledge in game form using a computer.
Grading.

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

Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster with verbal counting on adding and subtracting decimal fractions, as a result of which the children receive the following code 523914687. )

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

Guys, we know how to multiply natural numbers. Today we will look at multiplication decimal numbers to a natural number. Multiplying a decimal fraction by a natural number can be considered as a sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 ·3 = 5.21 + 5.21 + 5.21 = 15.63 So, 5.21 ·3 = 15.63. Presenting 5.21 as a common fraction to a natural number, we get

And in this case we got the same result: 15.63. Now, ignoring the comma, instead of the number 5.21, take the number 521 and multiply it by this natural number. Here we must remember that in one of the factors the comma has been moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now in this example we move the comma to the left two places. Thus, by how many times one of the factors was increased, by how many times the product was decreased. Based on the similarities of these methods, we will draw a conclusion.

To multiply a decimal fraction by a natural number, you need to:
1) without paying attention to the comma, multiply natural numbers;
2) in the resulting product, separate as many digits from the right with a comma as there are in the decimal fraction.

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

To multiply a decimal fraction by digit units 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many places as there are zeros in the digit unit.

I finish my explanation by expressing the decimal fraction as a percentage. I introduce the rule:

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

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

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

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

6. And now you have rested a little, you can solve the tasks. Open your textbook to page 205, № 1029. In this task you need to calculate the value of the expressions:

The tasks appear on the computer. As they are solved, a picture appears with the image of a boat that floats away when fully assembled.

By solving this task on a computer, the rocket gradually folds up; after solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the Kazakh soil from the Baikonur Cosmodrome they take off to the stars spaceships. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.

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

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In the course of secondary and high school Students studied the topic “Fractions”. However, this concept is much broader than what is given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.

What is a fraction?

Historically, fractional numbers arose out of the need to measure. As practice shows, there are often examples of determining the length of a segment and the volume of a rectangular rectangle.

Initially, students are introduced to the concept of a share. For example, if you divide a watermelon into 8 parts, then each person will get one-eighth of the watermelon. This one part of eight is called a share.

A share equal to ½ of any value is called half; ⅓ - third; ¼ - a quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. A common fraction is divided into a numerator and a denominator. Between them is the fraction bar, or fraction bar. The fractional line can be drawn as either a horizontal or an oblique line. In this case, it denotes the division sign.

The denominator represents how many equal parts the quantity or object is divided into; and the numerator is how many identical shares are taken. The numerator is written above the fraction line, the denominator is written below it.

It is most convenient to show ordinary fractions on coordinate ray. If a unit segment is divided into 4 equal parts, label each part Latin letter, then the result can be an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of a given segment.

Types of fractions

Fractions can be ordinary, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.

A proper fraction is a number whose numerator is less than its denominator. Accordingly, an improper fraction is a number whose numerator is greater than its denominator. The second type is usually written as a mixed number. This expression consists of an integer and a fractional part. For example, 1½. 1 is an integer part, ½ is a fractional part. However, if you need to carry out some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.

Correct fractional expression always less than one, and incorrect - greater than or equal to 1.

As for this expression, we mean a record in which any number is represented, the denominator of the fractional expression of which can be expressed in terms of one with several zeros. If the fraction is proper, then the whole part is decimal notation will be equal to zero.

To write a decimal fraction, you must first write the whole part, separate it from the fraction using a comma, and then write the fraction expression. It must be remembered that after the decimal point the numerator must contain the same number of digital characters as there are zeros in the denominator.

Example. Express the fraction 7 21 / 1000 in decimal notation.

Algorithm for converting an improper fraction to a mixed number and vice versa

It is incorrect to write an improper fraction in the answer to a problem, so it needs to be converted to a mixed number:

divide the numerator by the existing denominator;
in a specific example, an incomplete quotient is a whole;
and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.

Example. Convert improper fraction to mixed number: 47 / 5.

Solution. 47: 5. The partial quotient is 9, the remainder = 2. So, 47 / 5 = 9 2 / 5.

Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:

the integer part is multiplied by the denominator of the fractional expression;
the resulting product is added to the numerator;
the result is written in the numerator, the denominator remains unchanged.

Example. Present the number in mixed form as an improper fraction: 9 8 / 10.

Solution. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.

Answer: 98 / 10.

Multiplying fractions

Various algebraic operations can be performed on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, multiplying fractions with different denominators is no different from multiplying fractions with the same denominators.

It happens that after finding the result you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, one cannot say that an improper fraction in an answer is an error, but it is also difficult to call it a correct answer.

Example. Find the product of two ordinary fractions: ½ and 20/18.

As can be seen from the example, after finding the product, a reducible fractional notation is obtained. Both the numerator and the denominator in this case are divided by 4, and the result is the answer 5 / 9.

Multiplying decimal fractions

The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:

two decimal fractions must be written one under the other so that the rightmost digits are one under the other;
you need to multiply the written numbers, despite the commas, that is, as natural numbers;
count the number of digits after the decimal point in each number;
in the result obtained after multiplication, you need to count from the right as many digital symbols as are contained in the sum in both factors after the decimal point, and put a separating sign;
if there are fewer numbers in the product, then you need to write as many zeros in front of them to cover this number, put a comma and add the whole part equal to zero.

Example. Calculate the product of two decimal fractions: 2.25 and 3.6.

Solution.

Multiplying mixed fractions

To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:

convert mixed numbers into improper fractions;
find the product of the numerators;
find the product of denominators;
write down the result;
simplify the expression as much as possible.

Example. Find the product of 4½ and 6 2/5.

Multiplying a number by a fraction (fractions by a number)

In addition to finding the product of two fractions and mixed numbers, there are tasks where you need to multiply by a fraction.

So, to find the product of a decimal fraction and a natural number, you need:

write the number under the fraction so that the rightmost digits are one above the other;
find the product despite the comma;
in the resulting result, separate the integer part from the fractional part using a comma, counting from the right the number of digits that are located after the decimal point in the fraction.

To multiply a common fraction by a number, you need to find the product of the numerator and the natural factor. If the answer produces a fraction that can be reduced, it should be converted.

Example. Calculate the product of 5 / 8 and 12.

Solution. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.

Answer: 7 1 / 2.

As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.

Multiplication of fractions also concerns finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the whole part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the resulting result as much as possible.

Example. Find the product of 9 5 / 6 and 9.

Solution. 9 5 / 6 x 9 = 9 x 9 + (5 x 9) / 6 = 81 + 45 / 6 = 81 + 7 3 / 6 = 88 1 / 2.

Answer: 88 1 / 2.

Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0.001

The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the decimal point to the right by as many digits as there are zeros after the one in the factor.

Example 1. Find the product of 0.065 and 1000.

Solution. 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2. Find the product of 3.9 and 1000.

Solution. 3.9 x 1000 = 3.900 x 1000 = 3900.

Answer: 3900.

If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma in the resulting product to the left by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written before the natural number.

Example 1. Find the product of 56 and 0.01.

Solution. 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2. Find the product of 4 and 0.001.

Solution. 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

So, finding the product of different fractions should not cause difficulties, except perhaps calculating the result; in this case, you simply cannot do without a calculator.

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

Multiplying a common fraction by a fraction.

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

\(\bf \frac(a)(b) \times \frac(c)(d) = \frac(a \times c)(b \times d)\\\)

\(\frac(6)(7) \times \frac(2)(3) = \frac(6 \times 2)(7 \times 3) = \frac(12)(21) = \frac(4 \ times 3)(7 \times 3) = \frac(4)(7)\\\)

The fraction \(\frac(12)(21) = \frac(4 \times 3)(7 \times 3) = \frac(4)(7)\\\) was reduced by 3.

Multiplying a fraction by a number.

First, let's remember the rule, any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .

Let's use this rule when multiplying.

\(5 \times \frac(4)(7) = \frac(5)(1) \times \frac(4)(7) = \frac(5 \times 4)(1 \times 7) = \frac (20)(7) = 2\frac(6)(7)\\\)

Improper fraction \(\frac(20)(7) = \frac(14 + 6)(7) = \frac(14)(7) + \frac(6)(7) = 2 + \frac(6)( 7)= 2\frac(6)(7)\\\) converted to a mixed fraction.

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

\(\frac(2)(5) \times 3 = \frac(2 \times 3)(5) = \frac(6)(5) = 1\frac(1)(5)\\\\\) \(\bf \frac(a)(b) \times c = \frac(a \times c)(b)\\\)

Multiplying mixed fractions.

Example:
\(2\frac(1)(4) \times 3\frac(5)(6) = \frac(9)(4) \times \frac(23)(6) = \frac(9 \times 23) (4 \times 6) = \frac(3 \times \color(red) (3) \times 23)(4 \times 2 \times \color(red) (3)) = \frac(69)(8) = 8\frac(5)(8)\\\)

Multiplication of reciprocal fractions and numbers.

The fraction \(\bf \frac(a)(b)\) is the inverse of the fraction \(\bf \frac(b)(a)\), provided a≠0,b≠0.
The fractions \(\bf \frac(a)(b)\) and \(\bf \frac(b)(a)\) are called reciprocal fractions. The product of reciprocal fractions is equal to 1.
\(\bf \frac(a)(b) \times \frac(b)(a) = 1 \\\)

Example:
\(\frac(5)(9) \times \frac(9)(5) = \frac(45)(45) = 1\\\)

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

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

Example #1:
Calculate the product: a) \(\frac(8)(9) \times \frac(7)(11)\) b) \(\frac(2)(15) \times \frac(10)(13)\ )

Solution:
a) \(\frac(8)(9) \times \frac(7)(11) = \frac(8 \times 7)(9 \times 11) = \frac(56)(99)\\\\ \)
b) \(\frac(2)(15) \times \frac(10)(13) = \frac(2 \times 10)(15 \times 13) = \frac(2 \times 2 \times \color( red) (5))(3 \times \color(red) (5) \times 13) = \frac(4)(39)\)

Example #2:
Calculate the products of a number and a fraction: a) \(3 \times \frac(17)(23)\) b) \(\frac(2)(3) \times 11\)

Solution:
a) \(3 \times \frac(17)(23) = \frac(3)(1) \times \frac(17)(23) = \frac(3 \times 17)(1 \times 23) = \frac(51)(23) = 2\frac(5)(23)\\\\\)
b) \(\frac(2)(3) \times 11 = \frac(2)(3) \times \frac(11)(1) = \frac(2 \times 11)(3 \times 1) = \frac(22)(3) = 7\frac(1)(3)\)

Example #3:
Write the reciprocal of the fraction \(\frac(1)(3)\)?
Answer: \(\frac(3)(1) = 3\)

Example #4:
Calculate the product of two reciprocal fractions: a) \(\frac(104)(215) \times \frac(215)(104)\)

Solution:
a) \(\frac(104)(215) \times \frac(215)(104) = 1\)

Example #5:
Can reciprocal fractions be:
a) simultaneously with proper fractions;
b) simultaneously improper fractions;
c) simultaneously natural numbers?

Solution:
a) to answer the first question, let's give an example. The fraction \(\frac(2)(3)\) is proper, its inverse fraction will be equal to \(\frac(3)(2)\) - an improper fraction. Answer: no.

b) in almost all enumerations of fractions this condition is not met, but there are some numbers that fulfill the condition of being simultaneously an improper fraction. For example, the improper fraction is \(\frac(3)(3)\), its inverse fraction is equal to \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions when the numerator and denominator are equal.

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

Example #6:
Do the product of mixed fractions: a) \(4 \times 2\frac(4)(5)\) b) \(1\frac(1)(4) \times 3\frac(2)(7)\)

Solution:
a) \(4 \times 2\frac(4)(5) = \frac(4)(1) \times \frac(14)(5) = \frac(56)(5) = 11\frac(1 )(5)\\\\ \)
b) \(1\frac(1)(4) \times 3\frac(2)(7) = \frac(5)(4) \times \frac(23)(7) = \frac(115)( 28) = 4\frac(3)(7)\)

Example #7:
Can two reciprocals be mixed numbers at the same time?

Let's look at an example. Let's take a mixed fraction \(1\frac(1)(2)\), find its inverse fraction, to do this we convert it into an improper fraction \(1\frac(1)(2) = \frac(3)(2) \) . Its inverse fraction will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two fractions that are mutually inverse cannot be mixed numbers at the same time.