Multiplying fractions numerator 12. Multiplying ordinary fractions: rules, examples, solutions

GET OVER THESE RAKES ALREADY! 🙂

Multiplying and dividing fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very “not very. »
And for those who “very much so. ")

This operation is much more pleasant than addition and subtraction! Because it's easier. As a reminder, 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). That is:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

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

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

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

How can I make this fraction look decent? Yes, very simple! Use two-point division:

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

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And also very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

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

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Take practical advice into account, and there will be fewer of them (mistakes)!

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types fractions - move on to ordinary fractions.

3. We reduce all fractions until they stop.

4. Multi-storey fractional expressions reduce to ordinary ones using division through two points (watch the order of division!).

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions.

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

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak. Here they are, the answers, separated by semicolons.

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

Now we draw conclusions. If everything worked out, I’m happy for you! Basic 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.

All these (and more!) examples are discussed in Special Section 555 “Fractions”. WITH detailed explanations what, why and how. This analysis helps a lot with a lack of knowledge and skills!

Yes, and on the second problem there is something there.) Quite practical advice, how to become more attentive. Yes Yes! Advice that can be applied every.

In addition to knowledge and attentiveness, success requires a certain automaticity. Where can I get it? I hear a heavy sigh... Yes, only in practice, nowhere else.

You can go to the website 321start.ru for training. There in the “Try” option there are 10 examples for everyone. With instant verification. For registered users - 34 examples from simple to severe. This is only in fractions.

If you like this site.

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

Here you can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

And here you can get acquainted with functions and derivatives.

Rule 1.

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

Rule 2.

To multiply a fraction by a fraction:

1. find the product of the numerators and the product of the denominators of these fractions

2. Write the first product as the numerator, and the second as the denominator.

Rule 3.

In order to multiply mixed numbers, you need to write them as improper fractions, and then use the rule for multiplying fractions.

Rule 4.

To divide one fraction by another, you must multiply the dividend by the reciprocal of the divisor.

Example 1.

Calculate

Example 2.

Calculate

Example 3.

Calculate

Example 4.

Calculate

Mathematics. Other materials

Raising a number to a rational power. (

Raising a number to a natural power. (

Generalized interval method for solving algebraic inequalities (Author A.V. Kolchanov)

Method for replacing factors when solving algebraic inequalities (Author Kolchanov A.V.)

Signs of divisibility (Lungu Alena)

Test yourself on the topic ‘Multiplication and Division’ ordinary fractions’

Multiplying fractions

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

Multiplying a common fraction by a fraction

This is the simplest case in which you need to use the following rules for multiplying fractions.

To multiply fraction by 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 to see if the fractions can be reduced. Reducing fractions in calculations will make your calculations much easier.

Multiplying a fraction by a natural number

To make a 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 not proper fraction, do not forget to turn it into a mixed number, that is, highlight the whole part.

Multiplying mixed numbers

To multiply mixed numbers, you must first turn 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 when making calculations it is more convenient to use another method of multiplying a common 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, this version of the rule is more convenient to use if the denominator of the fraction is divisible by a natural number without a remainder.

Dividing a fraction by a number

What is the fastest way to divide a fraction by a number? Let's analyze the theory, draw a conclusion, and use examples to see how dividing a fraction by a number can be done using a new short rule.

Typically, dividing a fraction by a number follows the rule for dividing fractions. We multiply the first number (fraction) by the inverse of the second. Since the second number is an integer, its inverse is a fraction, the numerator of which is equal to one, and the denominator is equal to the given number. Schematically, dividing a fraction by a natural number looks like this:

From this we conclude:

To divide a fraction by a number, you need to multiply the denominator by that number and leave the numerator the same. The rule can be formulated even more briefly:

When dividing a fraction by a number, the number goes into the denominator.

Divide a fraction by a number:

To divide a fraction by a number, we rewrite the numerator unchanged, and multiply the denominator by this number. We reduce 6 and 3 by 3.

When dividing a fraction by a number, we rewrite the numerator and multiply the denominator by that number. We reduce 16 and 24 by 8.

When dividing a fraction by a number, the number goes into the denominator, so we leave the numerator the same and multiply the denominator by the divisor. We reduce 21 and 35 by 7.

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. 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.

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.

Task. Find the meaning of the expression:

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 outside the limits of multiplication. The result is a negative fraction.

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:

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 complete reduction It was not possible to achieve this, but the total amount of calculations still decreased.

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

You can't do that!

The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.

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

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

Dividing fractions.

Dividing a fraction by a natural number.

Examples of dividing a fraction by a natural number

Dividing a natural number by a fraction.

Examples of dividing a natural number by a fraction

Division of ordinary fractions.

Examples of dividing ordinary fractions

Dividing mixed numbers.

convert mixed fractions to improper fractions;
multiply the first fraction by the reciprocal of the second;
reduce the resulting fraction;
If you get an improper fraction, convert the improper fraction into a mixed fraction.

Examples of dividing mixed numbers

1 1 2: 2 2 3 = 1 2 + 1 2: 2 3 + 2 3 = 3 2: 8 3 = 3 2 3 8 = 3 3 2 8 = 9 16

2 1 7: 3 5 = 2 7 + 1 7: 3 5 = 15 7: 3 5 = 15 7 5 3 = 15 5 7 3 = 5 5 7 = 25 7 = 7 3 + 4 7 = 3 4 7

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Fractions. Multiplying and dividing fractions.

Multiplying a common fraction by a fraction.

To multiply ordinary fractions, you need to multiply the numerator by the numerator (we get the numerator of the product) and the denominator by the denominator (we get the denominator of the product).

Formula for multiplying fractions:

Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.

Note! There is no need to look for a common denominator here!!

Dividing a common fraction by a fraction.

Dividing an ordinary fraction by a fraction occurs like this: you turn the second fraction over (i.e., change the numerator and denominator) and after that the fractions are multiplied.

Formula for dividing ordinary fractions:

Multiplying a fraction by a natural number.

Note! When multiplying a fraction by a natural number, the numerator of the fraction is multiplied by our natural number, and the denominator of the fraction is left the same. If the result of the product is an improper fraction, then be sure to highlight the whole part, turning the improper fraction into a mixed fraction.

Dividing fractions involving natural numbers.

It's not as scary as it seems. As with addition, we convert the whole number into a fraction with one in the denominator. For example:

Multiplying mixed fractions.

Rules for multiplying fractions (mixed):

convert mixed fractions to improper fractions;
multiplying the numerators and denominators of fractions;
reduce the fraction;
If you get an improper fraction, then we convert the improper fraction into a mixed fraction.

Note! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.

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

It may be more convenient to use the second method of multiplying a common fraction by a number.

Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multistory fractions.

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

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

Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.

2. In tasks with different types of fractions, go to the type of ordinary fractions.

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

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

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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 top part fractions are what we divide. 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.
She definitely needs to be brought to the right kind. 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.

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

3) To multiply a number (multiplicand) by a mixed number (factor) means to multiply the multiplicand first by the whole 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 the 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 that one definition of multiplication, which is indicated in integer arithmetic (the addition of equal terms), then our problem would have three various 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 give allow for all these possible cases give one general answer:

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, no matter what 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 whole number (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 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 mean 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 the parentheses on the right side; By applying the commutative law of addition, and then again the associative law, we can obviously rewrite right side So:

(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.

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

Task. Find the meaning of the expression:

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

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.

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!

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 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:

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 they are the same or different denominators For fractions, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.

How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction 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) simultaneously 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 reciprocal numbers exist 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 in this work, please download the full version.

In a fun way, introduce to students the rule for multiplying a decimal fraction by a natural number, by a place value unit, and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge when solving examples and problems.
To develop and activate students’ logical thinking, 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, poster with a cyphergram, posters with statements by mathematicians.

Organizing time.
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 a playful way 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 multiplying decimal numbers by 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 decimal for a natural number, you need:
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 soil of Kazakhstan, 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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Lesson content

Adding fractions with like denominators

There are two types of addition of fractions:

Adding fractions with like denominators
Adding fractions with different denominators

First, let's learn the addition of fractions with like denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged. For example, let's add the fractions and . Add the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2. Add fractions and .

The answer turned out to be an improper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:

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

Example 3. Add fractions and .

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

This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a drawing. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

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

To add fractions with the same denominator, you need to add their numerators and leave the denominator unchanged;

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the fractions must be the same. But they are not always the same.

For example, fractions can be added because they have same denominators.

But fractions cannot be added right away, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will look at only one of them, since the other methods may seem complicated for a beginner.

The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

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

Example 1. Let's add the fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now let's return to fractions and . First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:

Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:

Look carefully at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's take this example to the end:

This completes the example. It turns out to add .

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

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

The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).

Please note that we have described this example in too much detail. IN educational institutions It’s not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:

But there is also back side medals. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

Find the LCM of the denominators of fractions;
Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction;
Multiply the numerators and denominators of fractions by their additional factors;
Add fractions that have the same denominators;
If the answer turns out to be an improper fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the instructions given above.

Step 1. Find the LCM of the denominators of the fractions

Find the LCM of the denominators of both fractions. The denominators of fractions are the numbers 2, 3 and 4

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it above the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We get the second additional factor 4. We write it above the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We get the third additional factor 3. We write it above the third fraction:

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

We multiply the numerators and denominators by their additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it up:

The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is moved to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of the new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an improper fraction, then select the whole part of it

Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:

We received an answer

Subtracting fractions with like denominators

There are two types of subtraction of fractions:

Subtracting fractions with like denominators
Subtracting fractions with different denominators

First, let's learn how to subtract fractions with like denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, but leave the denominator the same.

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

This example can be easily understood if we remember the pizza, which is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction you need to subtract the numerators of the remaining fractions:

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

To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

Subtracting fractions with different denominators

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

The common denominator is found using the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written above the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written above the second fraction.

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

Example 1. Find the meaning of the expression:

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

First we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now let's return to fractions and

Let's find an additional factor for the first fraction. To do this, divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. Write a four above the first fraction:

We do the same with the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a three over the second fraction:

Now we are ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:

We received an answer

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

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

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

The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2. Find the value of an expression

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

Let's find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it above the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it above the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it above the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler. What can be done? You can shorten this fraction.

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

So, we find the gcd of numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found gcd, that is, by 10

We received an answer

Multiplying a fraction by a number

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

Example 1. Multiply a fraction by the number 1.

Multiply the numerator of the fraction by the number 1

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

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

This notation can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The answer was an improper fraction. Let's highlight the whole part of it:

The expression can be understood as taking two quarters 4 times. For example, if you take 4 pizzas, you will get two whole pizzas

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

Multiplying fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.

Example 1. Find the value of the expression.

We received an answer. It is advisable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what pizza looks like when divided into three parts:

One piece of this pizza and the two pieces we took will have the same dimensions:

In other words, we are talking about the same size pizza. Therefore the value of the expression is

Example 2. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer was an improper fraction. Let's highlight the whole part of it:

Example 3. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer turned out to be a regular fraction, but it would be good if it was shortened. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.

So, let’s find the gcd of numbers 105 and 450:

Now we divide the numerator and denominator of our answer by the gcd that we have now found, that is, by 15

Representing a whole number as a fraction

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

Reciprocal numbers

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

Definition. Reverse to numbera is a number that, when multiplied bya gives one.

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

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

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

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let’s multiply the fraction by itself, only upside down:

What will happen as a result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number , since when you multiply 5 by you get one.

The reciprocal of a number can also be found for any other integer.

You can also find the reciprocal of any other fraction. To do this, just turn it over.

Dividing a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How much pizza will each person get?

It can be seen that after dividing half the pizza, two equal pieces were obtained, each of which constitutes a pizza. So everyone gets a pizza.

Division of fractions is done using reciprocals. Reciprocal numbers allow you to replace division with multiplication.

To divide a fraction by a number, you need to multiply the fraction by the inverse of the divisor.

Using this rule, we will write down the division of our half of the pizza into two parts.

So, you need to divide the fraction by the number 2. Here the dividend is the fraction and the divisor is the number 2.

To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is the fraction. So you need to multiply by