How to multiply a fraction by a whole. Adding fractions with different denominators. Dividing a fraction by a number

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

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

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

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

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

Task. Find the meaning of the expression:

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

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

Also 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 because when adding, the numerator of a fraction produces a sum, not a product of numbers. Therefore, it is impossible to apply the basic 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 correct solution to the previous problem looks like this:

Correct solution:

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

In the middle and high school courses, students covered 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 a coordinate ray. If a unit segment is divided into 4 equal parts, label each part Latin letter, then the result can be 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), mixed number is converted to an improper fraction.

A correct fractional expression is always less than one, and an incorrect one is always 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 integer part in 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;
V specific example incomplete quotient - 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 the product fractional numbers with the same denominators.

It happens that after finding the result you need to reduce the fraction. IN mandatory you need 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 decimal 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 results in 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 irregular fraction 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 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.

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

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

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

Adding fractions with different denominators

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

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

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

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

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

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

Example 1. Let's add the fractions and

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

LCM (2 and 3) = 6

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

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

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

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

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

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

This completes the example. It turns out to add .

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

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

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

Please note that we have described this example too detailed. 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. While at school, we would have to write this example as follows:

But there is also reverse 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 highlight 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. The 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 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

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! Don't need 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 another 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! This is not common 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 a 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. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow 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. Right 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. And 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 of 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 of 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 an improper 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 multiplication common fraction per 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 whose numerator is one and the denominator is 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. 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.

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.

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.

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

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 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 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.

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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Welcome to OnlineMSchool.
My name is Dovzhik Mikhail Viktorovich. I am the owner and author of this site, I wrote all the theoretical material, and also developed online exercises and calculators that you can use to study mathematics.

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.

Pay attention! 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.

Pay attention! 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.

Pay attention! 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.

Pay attention! 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:

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

Please 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 upper 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. Dividing both numbers by a common divisor is called reducing the 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.