Multiplying fractions with the same denominator. Drawing up a system of equations

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

Multiplying a common fraction by a fraction.

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

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

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.

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

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

Multiplying a fraction by a number.

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

Let's use this rule when multiplying.

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

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

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

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

Multiplying mixed fractions.

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.

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

Multiplication of reciprocal fractions and numbers.

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

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

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 receive the work mixed fractions you need to convert them into improper fractions 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) $\frac(8)(9) \times \frac(7)(11)$ b) \(\frac(2)(15) \times \frac(10)(13)\ )

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

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

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

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

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

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

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

Solution:
a) to answer the first question, let's give an example. The fraction $\frac(2)(3)$ is proper, its inverse fraction will be equal to $\frac(3)(2)$ - not proper fraction. Answer: no.

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

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

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

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

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

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

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 single segment is divided into 4 equal parts, each part is designated by a Latin letter, then the result can be excellent visual material. So, point A shows a share equal to 1/4 of the total unit segment, and point B marks 2/8 of this segment.

Types of fractions

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

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

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

As for this expression, we mean a record in which any number is represented, the denominator of the fractional expression of which can be expressed in terms of one with several zeros. If the fraction is proper, then the 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;
in a specific example, an incomplete quotient is a whole;
and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.

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

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

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

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

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

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

Answer: 98 / 10.

Multiplying fractions

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

It happens that after finding the result you need to reduce the fraction. 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 produces a fraction that can be reduced, it should be converted.

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

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

Answer: 7 1 / 2.

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

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

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

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

Answer: 88 1 / 2.

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

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

Example 1. Find the product of 0.065 and 1000.

Solution. 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2. Find the product of 3.9 and 1000.

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

Answer: 3900.

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

Example 1. Find the product of 56 and 0.01.

Solution. 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2. Find the product of 4 and 0.001.

Solution. 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

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

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 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.

Operations with fractions

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 .

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

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 .

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 the same;
If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

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 we look for the least common multiple (LCM) of the denominators of both fractions. 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

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

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

LCM (2 and 3) = 6

Now 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 diagram we provided above.

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

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

Step 2. Divide the LCM by the denominator of each fraction and get an additional 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 highlight its whole part

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 the same. 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 the same:

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:

The answer was an improper fraction. If the example is completed, then it is customary to get rid of the improper fraction. Let's get rid of the improper fraction in the answer. To do this, let’s select its entire part:

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 the same;

If the answer turns out to be an improper fraction, then you need to highlight its whole part.

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:

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. It would be necessary to make it simpler and more aesthetically pleasing. What can be done? You can shorten this fraction. Recall that reducing a fraction is dividing the numerator and denominator by the largest common divisor numerator and denominator.

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

GCD should not be confused with NOC. The most common mistake of many beginners. GCD is the greatest common divisor. We find it to reduce a fraction.

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

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

So, we find GCD for numbers 20 and 30:

GCD (20 and 30) = 10

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

We received a beautiful 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 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

The answer turned out to be a regular fraction, but it would be good if it was shortened. To reduce this fraction, it must be divided by the gcd of the numerator and denominator. So, let’s find the gcd of numbers 105 and 450:

GCD for (105 and 150) is 15

Now we divide the numerator and denominator of our answer by gcd:

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 number a is a number that, when multiplied by a 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, multiply a 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.

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

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

Multiplying common fractions

Let's look at an example.

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

Multiplying two ordinary fractions

Rule for multiplying ordinary fractions:

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

Example 1

Perform multiplication of common fractions $\frac(3)(7)$ and $\frac(5)(11)$.

Solution.

Let's use the rule for multiplying ordinary fractions:

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

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

If multiplying fractions results in a reducible or improper fraction, you need to simplify it.

Example 2

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

Solution.

We use the rule for multiplying ordinary fractions:

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

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

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

Short solution:

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

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

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

Example 3

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

Solution.

Let's use the formula for multiplying ordinary fractions:

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

Obviously, the numerator and denominator contain numbers that can be reduced in pairs to the numbers $2$, $3$ and $5$. Let's factor the numerator and denominator into simple factors and make a reduction:

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

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

When multiplying fractions, you can apply the commutative law:

Multiplying a common fraction by a natural number

The rule for multiplying a common fraction by a natural number:

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

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

Example 4

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

Solution.

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

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

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

Do not forget to check the result of multiplication by the reducibility of the fraction or by an improper fraction.

Example 5

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

Solution.

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

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

By dividing by the number $3$) we can determine that the resulting fraction can be reduced:

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

The result was an incorrect fraction. Let's select the whole part:

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

Short solution:

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

Fractions could also be reduced by replacing the numbers in the numerator and denominator with their factorizations into prime factors. In this case, the solution could be written as follows:

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

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

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

Dividing fractions

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

Dividing two ordinary fractions

Rule for dividing ordinary fractions: Obviously, the numerator and denominator of the resulting fraction can be factorized and reduced:

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

As a result, we get an improper fraction, from which we select the whole part:

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

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

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

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 .

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 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.

Example 1. Let's add the fractions and

LCM (2 and 3) = 6

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

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:

Please note that we have described this example in too much detail. In educational institutions it is 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 another side to the coin. 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

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:

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

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

Example 1. Find the meaning of the expression:

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

LCM (3 and 4) = 12

Now let's return to fractions and

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

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:

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.

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

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

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

Reciprocal numbers

Now we will get acquainted with a 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