In the course of secondary and high school Students studied the topic “Fractions”. However, this concept is much broader than what is given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.
What is a fraction?
It so happened historically that 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 an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of a given segment.
Types of fractions
Fractions can be ordinary, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.
A proper fraction is a number whose numerator is less than its denominator. Accordingly, an improper fraction is a number whose numerator is greater than its denominator. The second type is usually written as a mixed number. This expression consists of an integer and a fractional part. For example, 1½. 1 is an integer part, ½ is a fractional part. However, if you need to carry out some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.
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 record decimal, you must first write the whole part, separate it from the fractional part using a comma, and then write down the fractional 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 multiplying fractions with 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 of a decimal fraction and a natural number, you need:
- write the number under the fraction so that the rightmost digits are one above the other;
- find the product despite the comma;
- in the resulting result, separate the integer part from the fractional part using a comma, counting from the right the number of digits that are located after the decimal point in the fraction.
To multiply a common fraction by a number, you need to find the product of the numerator and the natural factor. If the answer produces a fraction that can be reduced, it should be converted.
Example. Calculate the product of 5 / 8 and 12.
Solution. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.
Answer: 7 1 / 2.
As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.
Multiplication of fractions also concerns finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the whole part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the resulting result as much as possible.
Example. Find the product of 9 5 / 6 and 9.
Solution. 9 5 / 6 x 9 = 9 x 9 + (5 x 9) / 6 = 81 + 45 / 6 = 81 + 7 3 / 6 = 88 1 / 2.
Answer: 88 1 / 2.
Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0.001
It follows from the previous paragraph next rule. 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.
§ 87. Addition of fractions.
Adding fractions has many similarities to adding whole numbers. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), containing all the units and fractions of the units of the terms.
We will consider three cases sequentially:
1. Addition of fractions with like denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.
1. Addition of fractions with like denominators.
Consider an example: 1/5 + 2/5.
Let's take segment AB (Fig. 17), take it as one and divide it into 5 equal parts, then part AC of this segment will be equal to 1/5 of segment AB, and part of the same segment CD will be equal to 2/5 AB.
From the drawing it is clear that if we take the segment AD, it will be equal to 3/5 AB; but the segment AD is precisely the sum of the segments AC and CD. So we can write:
1 / 5 + 2 / 5 = 3 / 5
Considering these terms and the resulting sum, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.
From this we get the following rule: To add fractions with the same denominators, you need to add their numerators and leave the same denominator.
Let's look at an example:
2. Addition of fractions with different denominators.
Let's add the fractions: 3 / 4 + 3 / 8 First they need to be reduced to the lowest common denominator:
The intermediate link 6/8 + 3/8 could not be written; we've written it here for clarity.
Thus, in order to add fractions with different denominators, you must first reduce them to the lowest common denominator, add their numerators and label the common denominator.
Let's look at an example (we will write additional factors above the corresponding fractions):
3. Addition of mixed numbers.
Let's add the numbers: 2 3/8 + 3 5/6.
Let’s first bring the fractional parts of our numbers to a common denominator and rewrite them again:
Now we add the integer and fractional parts sequentially:
§ 88. Subtraction of fractions.
Subtracting fractions is defined in the same way as subtracting whole numbers. This is an action with the help of which, given the sum of two terms and one of them, another term is found. Let us consider three cases in succession:
1. Subtracting fractions with like denominators.
2. Subtracting fractions with different denominators.
3. Subtraction of mixed numbers.
1. Subtracting fractions with like denominators.
Let's look at an example:
13 / 15 - 4 / 15
Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then part AC of this segment will represent 1/15 of AB, and part AD of the same segment will correspond to 13/15 AB. Let us set aside another segment ED equal to 4/15 AB.
We need to subtract the fraction 4/15 from 13/15. In the drawing, this means that segment ED must be subtracted from segment AD. As a result, segment AE will remain, which is 9/15 of segment AB. So we can write:
The example we made shows that the numerator of the difference was obtained by subtracting the numerators, but the denominator remained the same.
Therefore, to subtract fractions with like denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.
2. Subtracting fractions with different denominators.
Example. 3/4 - 5/8
First, let's reduce these fractions to the lowest common denominator:
The intermediate 6 / 8 - 5 / 8 is written here for clarity, but can be skipped later.
Thus, in order to subtract a fraction from a fraction, you must first reduce them to the lowest common denominator, then subtract the numerator of the minuend from the numerator of the minuend and sign the common denominator under their difference.
Let's look at an example:
3. Subtraction of mixed numbers.
Example. 10 3/4 - 7 2/3.
Let us reduce the fractional parts of the minuend and subtrahend to the lowest common denominator:
We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the whole part of the minuend, split it into those parts in which the fractional part is expressed, and add it to the fractional part of the minuend. And then the subtraction will be performed in the same way as in the previous example:
§ 89. Multiplication of fractions.
When studying fraction multiplication, we will consider the following questions:
1. Multiplying a fraction by a whole number.
2. Finding the fraction of a given number.
3. Multiplying a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding the percentage of a given number. Let's consider them sequentially.
1. Multiplying a fraction by a whole number.
Multiplying a fraction by a whole number has the same meaning as multiplying a whole number by an integer. To multiply a fraction (multiplicand) by an integer (factor) means to create a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
This means that if you need to multiply 1/9 by 7, then it can be done like this:
We easily obtained the result, since the action was reduced to adding fractions with the same denominators. Hence,
Consideration of this action shows that multiplying a fraction by a whole number is equivalent to increasing this fraction by as many times as the number of units contained in the whole number. And since increasing a fraction is achieved either by increasing its numerator
or by reducing its denominator 
, then we can either multiply the numerator by an integer or divide the denominator by it, if such division is possible.
From here we get the rule:
To multiply a fraction by a whole number, you multiply the numerator by that whole number and leave the denominator the same, or, if possible, divide the denominator by that number, leaving the numerator unchanged.
When multiplying, abbreviations are possible, for example:
2. Finding the fraction of a given number. There are many problems in which you have to find, or calculate, part of a given number. The difference between these problems and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce a method for solving them.
Task 1. I had 60 rubles; I spent 1/3 of this money on buying books. How much did the books cost?
Task 2. The train must travel a distance between cities A and B equal to 300 km. He has already covered 2/3 of this distance. How many kilometers is this?
Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there in total?
These are some of the many problems we encounter to find a part of a given number. They are usually called problems to find the fraction of a given number.
Solution to problem 1. From 60 rub. I spent 1/3 on books; This means that to find the cost of books you need to divide the number 60 by 3:
Solving problem 2. The point of the problem is that you need to find 2/3 of 300 km. Let's first calculate 1/3 of 300; this is achieved by dividing 300 km by 3:
300: 3 = 100 (that's 1/3 of 300).
To find two-thirds of 300, you need to double the resulting quotient, i.e., multiply by 2:
100 x 2 = 200 (that's 2/3 of 300).
Solving problem 3. Here you need to determine the number of brick houses that make up 3/4 of 400. Let’s first find 1/4 of 400,
400: 4 = 100 (that's 1/4 of 400).
To calculate three quarters of 400, the resulting quotient must be tripled, i.e. multiplied by 3:
100 x 3 = 300 (that's 3/4 of 400).
Based on the solution to these problems, we can derive the following rule:
To find the value of a fraction from a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.
3. Multiplying a whole number by a fraction.
Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 = 5+5 +5+5 = 20). In this paragraph (point 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.
In both cases, multiplication consisted of finding the sum of identical terms.
Now we move on to multiplying a whole number by a fraction. Here we will encounter, for example, multiplication: 9 2 / 3. It is clear that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.
Because of this, we will have to give a new definition of multiplication, i.e., in other words, answer the question of what should be understood by multiplication by a fraction, how this action should be understood.
The meaning of multiplying a whole number by a fraction is clear from the following definition: multiplying an integer (multiplicand) by a fraction (multiplicand) means finding this fraction of the multiplicand.
Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. IN previous paragraph such problems were solved; so it’s easy to figure out that we’ll end up with 6.
But now an interesting and important question arises: why are such various actions How is finding the sum of equal numbers and finding the fraction of a number called by the same word “multiplication” in arithmetic?
This happens because the previous action (repeating a number with terms several times) and the new action (finding the fraction of a number) give answers to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by the same action.
To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?
This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).
Let’s take the same problem, but in it the amount of cloth will be expressed as a fraction: “1 m of cloth costs 50 rubles. How much will 3/4 m of such cloth cost?”
This problem also needs to be solved by multiplying the number of rubles (50) by the number of meters (3/4).
You can change the numbers in it several more times, without changing the meaning of the problem, for example, take 9/10 m or 2 3/10 m, etc.
Since these problems have the same content and differ only in numbers, we call the actions used in solving them the same word - multiplication.
How do you multiply a whole number by a fraction?
Let's take the numbers encountered in the last problem:
According to the definition, we must find 3/4 of 50. Let's first find 1/4 of 50, and then 3/4.
1/4 of 50 is 50/4;
3/4 of the number 50 is .
Hence.
Let's consider another example: 12 5 / 8 =?
1/8 of the number 12 is 12/8,
5/8 of the number 12 is .
Hence,
From here we get the rule:
To multiply a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.
Let's write this rule using letters:
To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for multiplying a number by a quotient, which was set out in § 38
It is important to remember that before performing multiplication, you should do (if possible) reductions, For example:
4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying a whole number by a fraction, i.e., when multiplying a fraction by a fraction, you need to find the fraction that is in the factor from the first fraction (the multiplicand).
Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.
How do you multiply a fraction by a fraction?
Let's take an example: 3/4 multiplied by 5/7. This means you need to find 5/7 of 3/4. Let's first find 1/7 of 3/4, and then 5/7
1/7 of the number 3/4 will be expressed as follows:
5/7 numbers 3/4 will be expressed as follows:
Thus,
Another example: 5/8 multiplied by 4/9.
1/9 of 5/8 is ,
4/9 of the number 5/8 is .
Thus, 
From these examples the following rule can be deduced:
To multiply a fraction by a fraction, you need to multiply the numerator by the numerator, and the denominator by the denominator, and make the first product the numerator, and the second product the denominator of the product.
This is the rule in general view can be written like this:
When multiplying, it is necessary to make (if possible) reductions. Let's look at examples:
5. Multiplication of mixed numbers. Since mixed numbers can easily be replaced by improper fractions, this circumstance is usually used when multiplying mixed numbers. This means that in cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, they are replaced by improper fractions. Let's multiply, for example, mixed numbers: 2 1/2 and 3 1/5. Let's turn each of them into an improper fraction and then multiply the resulting fractions according to the rule for multiplying a fraction by a fraction:
Rule. To multiply mixed numbers, you must first convert them into improper fractions and then multiply them according to the rule for multiplying fractions by fractions.
Note. If one of the factors is an integer, then the multiplication can be performed based on the distribution law as follows:
6. The concept of interest. When solving problems and performing various practical calculations, we use all kinds of fractions. But it must be borne in mind that many quantities allow not just any, but natural divisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a kopeck, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of a ruble, it will be "10 kopecks, or a ten-kopeck piece. You can take a quarter of a ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take it, for example , 2/7 of a ruble because the ruble is not divided into sevenths.
The unit of weight, i.e. the kilogram, primarily allows for decimal divisions, for example 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are not common.
In general, our (metric) measures are decimal and allow decimal divisions.
However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (uniform) method of subdividing quantities. Many years of experience have shown that such a well-justified division is the “hundredth” division. Let us consider several examples relating to the most diverse areas of human practice.
1. The price of books has decreased by 12/100 of the previous price.
Example. The previous price of the book was 10 rubles. It decreased by 1 ruble. 20 kopecks
2. Savings banks pay depositors 2/100 of the amount deposited for savings during the year.
Example. 500 rubles are deposited in the cash register, the income from this amount for the year is 10 rubles.
3. The number of graduates from one school was 5/100 of the total number of students.
EXAMPLE There were only 1,200 students at the school, of which 60 graduated.
The hundredth part of a number is called a percentage.
The word "percent" is borrowed from Latin and its root "cent" means one hundred. Together with the preposition (pro centum), this word means “for a hundred.” The meaning of such an expression follows from the fact that initially in ancient Rome interest was the money that the debtor paid to the lender “for every hundred.” The word “cent” is heard in such familiar words: centner (one hundred kilograms), centimeter (say centimeter).
For example, instead of saying that over the past month the plant produced 1/100 of all products produced by it was defective, we will say this: over the past month the plant produced one percent of defects. Instead of saying: the plant produced 4/100 more products than the established plan, we will say: the plant exceeded the plan by 4 percent.
The above examples can be expressed differently:
1. The price of books has decreased by 12 percent of the previous price.
2. Savings banks pay depositors 2 percent per year on the amount deposited in savings.
3. The number of graduates from one school was 5 percent of all school students.
To shorten the letter, it is customary to write the % symbol instead of the word “percentage”.
However, you need to remember that in calculations the % sign is usually not written; it can be written in the problem statement and in the final result. When performing calculations, you need to write a fraction with a denominator of 100 instead of a whole number with this symbol.
You need to be able to replace an integer with the indicated icon with a fraction with a denominator of 100:
Conversely, you need to get used to writing an integer with the indicated symbol instead of a fraction with a denominator of 100:
7. Finding the percentage of a given number.
Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch firewood was there?
The meaning of this problem is that birch firewood made up only part of the firewood that was delivered to the school, and this part is expressed in the fraction 30/100. This means that we have a task to find a fraction of a number. To solve it, we must multiply 200 by 30/100 (problems of finding the fraction of a number are solved by multiplying the number by the fraction.).
This means that 30% of 200 equals 60.
The fraction 30/100 encountered in this problem can be reduced by 10. It would be possible to do this reduction from the very beginning; the solution to the problem would not have changed.
Task 2. There were 300 children of various ages in the camp. Children 11 years old made up 21%, children 12 years old made up 61% and finally 13 year old children made up 18%. How many children of each age were there in the camp?
In this problem you need to perform three calculations, i.e. sequentially find the number of children 11 years old, then 12 years old and finally 13 years old.
This means that here you will need to find the fraction of the number three times. Let's do it:
1) How many 11-year-old children were there?
2) How many 12-year-old children were there?
3) How many 13-year-old children were there?
After solving the problem, it is useful to add the numbers found; their sum should be 300:
63 + 183 + 54 = 300
It should also be noted that the sum of the percentages given in the problem statement is 100:
21% + 61% + 18% = 100%
This suggests that total number children in the camp were taken as 100%.
3 a d a h a 3. The worker received 1,200 rubles per month. Of this, he spent 65% on food, 6% on apartments and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% saved. How much money was spent on the needs indicated in the problem?
To solve this problem you need to find the fraction of 1,200 5 times. Let's do this.
1) How much money was spent on food? The problem says that this expense is 65% of total earnings, i.e. 65/100 of the number 1,200. Let’s do the calculation:
2) How much money did you pay for an apartment with heating? Reasoning similarly to the previous one, we arrive at the following calculation:
3) How much money did you pay for gas, electricity and radio?
4) How much money was spent on cultural needs?
5) How much money did the worker save?
To check, it is useful to add up the numbers found in these 5 questions. The amount should be 1,200 rubles. All earnings are taken as 100%, which is easy to check by adding up the percentage numbers given in the problem statement.
We solved three problems. Despite the fact that these problems dealt with different things (delivery of firewood for the school, the number of children of different ages, the worker's expenses), they were solved in the same way. This happened because in all problems it was necessary to find several percent of given numbers.
§ 90. Division of fractions.
As we study division of fractions, we will consider the following questions:
1. Divide an integer by an integer.
2. Dividing a fraction by a whole number
3. Dividing a whole number by a fraction.
4. Dividing a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number from its given fraction.
7. Finding a number by its percentage.
Let's consider them sequentially.
1. Divide an integer by an integer.
As was indicated in the department of integers, division is the action that consists in the fact that, given the product of two factors (dividend) and one of these factors (divisor), another factor is found.
We looked at dividing an integer by an integer in the section on integers. We encountered two cases of division there: division without a remainder, or “entirely” (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 remainder). We can therefore say that in the field of integers, exact division is not always possible, because the dividend is not always the product of the divisor by the integer. After introducing multiplication by a fraction, we can consider any case of dividing integers possible (only division by zero is excluded).
For example, dividing 7 by 12 means finding a number whose product by 12 would be equal to 7. Such a number is the fraction 7 / 12 because 7 / 12 12 = 7. Another example: 14: 25 = 14 / 25, because 14 / 25 25 = 14.
Thus, to divide a whole number by a whole number, you need to create a fraction whose numerator is equal to the dividend and the denominator is equal to the divisor.
2. Dividing a fraction by a whole number.
Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find a second factor that, when multiplied by 3, would give the given product 6/7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6/7 by 3 times.
We already know that reducing a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore you can write:
In this case, the numerator 6 is divisible by 3, so the numerator should be reduced by 3 times.
Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:
Based on this, a rule can be made: To divide a fraction by a whole number, you need to divide the numerator of the fraction by that whole number.(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.
3. Dividing a whole number by a fraction.
Let it be necessary to divide 5 by 1/2, i.e., find a number that, after multiplying by 1/2, will give the product 5. Obviously, this number must be greater than 5, since 1/2 is a proper fraction, and when multiplying a number the product of a proper fraction must be less than the product being multiplied. To make this clearer, let's write our actions as follows: 5: 1 / 2 = X , which means x 1 / 2 = 5.
We must find such a number X , which, if multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is equal to 5, and the whole number X twice as much, i.e. 5 2 = 10.
So 5: 1 / 2 = 5 2 = 10
Let's check: 
Let's look at another example. Let's say you want to divide 6 by 2/3. Let's first try to find the desired result using the drawing (Fig. 19).
Fig.19
Let us draw a segment AB equal to 6 units, and divide each unit into 3 equal parts. In each unit, three thirds (3/3) of the entire segment AB is 6 times larger, i.e. e. 18/3. Using small brackets, we connect the 18 resulting segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in 6 units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 whole units. Hence,
How to get this result without a drawing using calculations alone? Let's reason like this: we need to divide 6 by 2/3, i.e. we need to answer the question how many times 2/3 is contained in 6. Let's find out first: how many times 1/3 is contained in 6? In a whole unit there are 3 thirds, and in 6 units there are 6 times more, i.e. 18 thirds; to find this number we must multiply 6 by 3. This means that 1/3 is contained in b units 18 times, and 2/3 is contained in b units not 18 times, but half as many times, i.e. 18: 2 = 9. Therefore , when dividing 6 by 2/3 we have completed the following actions:
From here we get the rule for dividing a whole number by a fraction. To divide a whole number by a fraction, you need to multiply this whole number by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.
Let's write the rule using letters:
To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for dividing a number by a quotient, which was set out in § 38. Please note that the same formula was obtained there.
When dividing, abbreviations are possible, for example:
4. Dividing a fraction by a fraction.
Let's say we need to divide 3/4 by 3/8. What will the number that results from division mean? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To understand this issue, let's make a drawing (Fig. 20).
Let's take a segment AB, take it as one, divide it into 4 equal parts and mark 3 such parts. Segment AC will be equal to 3/4 of segment AB. Let us now divide each of the four original segments in half, then the segment AB will be divided into 8 equal parts and each such part will be equal to 1/8 of the segment AB. Let us connect 3 such segments with arcs, then each of the segments AD and DC will be equal to 3/8 of the segment AB. The drawing shows that a segment equal to 3/8 is contained in a segment equal to 3/4 exactly 2 times; This means that the result of division can be written as follows:
3 / 4: 3 / 8 = 2
Let's look at another example. Let's say we need to divide 15/16 by 3/32:
We can reason like this: we need to find a number that, after multiplying by 3/32, will give a product equal to 15/16. Let's write the calculations like this:
15 / 16: 3 / 32 = X
3 / 32 X = 15 / 16
3/32 unknown number X are 15/16
1/32 of an unknown number X is ,
32 / 32 numbers X make up .
Hence,
Thus, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second, and make the first product the numerator, and the second the denominator.
Let's write the rule using letters:
When dividing, abbreviations are possible, for example:
5. Division of mixed numbers.
When dividing mixed numbers, they must first be converted to improper fractions and then divide the resulting fractions according to the rules for dividing fractional numbers. Let's look at an example:
Let's convert mixed numbers to improper fractions:
Now let's divide:
Thus, to divide mixed numbers, you need to convert them into improper fractions and then divide using the rule for dividing fractions.
6. Finding a number from its given fraction.
Among the various fraction problems, sometimes there are those in which the value of some fraction of an unknown number is given and you need to find this number. This type of problem will be the inverse of the problem of finding the fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number was given and it was required to find this number itself. This idea will become even clearer if we turn to solving this type of problem.
Task 1. On the first day, the glaziers glazed 50 windows, which is 1/3 of all the windows of the built house. How many windows are there in this house?
Solution. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means there are 3 times more windows in total, i.e.
The house had 150 windows.
Task 2. The store sold 1,500 kg of flour, which is 3/8 of the total flour stock the store had. What was the store's initial supply of flour?
Solution. From the conditions of the problem it is clear that 1,500 kg of flour sold constitute 3/8 of the total stock; This means that 1/8 of this reserve will be 3 times less, i.e. to calculate it you need to reduce 1500 by 3 times:
1,500: 3 = 500 (this is 1/8 of the reserve).
Obviously, the entire supply will be 8 times larger. Hence,
500 8 = 4,000 (kg).
The initial stock of flour in the store was 4,000 kg.
From consideration of this problem, the following rule can be derived.
To find a number from a given value of its fraction, it is enough to divide this value by the numerator of the fraction and multiply the result by the denominator of the fraction.
We solved two problems on finding a number given its fraction. Such problems, as is especially clearly seen from the last one, are solved by two actions: division (when one part is found) and multiplication (when the whole number is found).
However, after we have learned the division of fractions, the above problems can be solved with one action, namely: division by a fraction.
For example, the last task can be solved in one action like this:
In the future, we will solve problems of finding a number from its fraction with one action - division.
7. Finding a number by its percentage.
In these problems you will need to find a number knowing a few percent of that number.
Task 1. At the beginning of this year I received 60 rubles from the savings bank. income from the amount I put into savings a year ago. How much money have I put in the savings bank? (The cash desks give depositors a 2% return per year.)
The point of the problem is that I put a certain amount of money in a savings bank and stayed there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I deposited. How much money did I put in?
Consequently, knowing part of this money, expressed in two ways (in rubles and fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following problems are solved by division:
This means that 3,000 rubles were deposited in the savings bank.
Task 2. Fishermen fulfilled the monthly plan by 64% in two weeks, harvesting 512 tons of fish. What was their plan?
From the conditions of the problem it is known that the fishermen completed part of the plan. This part is equal to 512 tons, which is 64% of the plan. We don’t know how many tons of fish need to be prepared according to the plan. Finding this number will be the solution to the problem.
Such problems are solved by division:
This means that according to the plan, 800 tons of fish need to be prepared.
Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked a passing conductor how much of the journey they had already covered. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?
From the problem conditions it is clear that 30% of the route from Riga to Moscow is 276 km. We need to find the entire distance between these cities, i.e., for this part, find the whole:
§ 91. Reciprocal numbers. Replacing division with multiplication.
Let's take the fraction 2/3 and replace the numerator in place of the denominator, we get 3/2. We got the inverse of this fraction.
In order to obtain a fraction that is the inverse of a given fraction, you need to put its numerator in place of the denominator, and the denominator in place of the numerator. In this way we can get the reciprocal of any fraction. For example:
3/4, reverse 4/3; 5/6, reverse 6/5
Two fractions that have the property that the numerator of the first is the denominator of the second, and the denominator of the first is the numerator of the second, are called mutually inverse.
Now let's think about what fraction will be the reciprocal of 1/2. Obviously, it will be 2 / 1, or just 2. By looking for the inverse fraction of the given one, we got an integer. And this case is not isolated; on the contrary, for all fractions with a numerator of 1 (one), the reciprocals will be integers, for example:
1/3, reverse 3; 1/5, reverse 5
Since in finding reciprocal fractions we also encountered integers, in what follows we will not talk about reciprocal fractions, but about reciprocal numbers X.
Let's figure out how to write the inverse of an integer. For fractions, this can be solved simply: you need to put the denominator in place of the numerator. In the same way, you can get the inverse of an integer, since any integer can have a denominator of 1. This means that the inverse of 7 will be 1/7, because 7 = 7/1; for the number 10 the inverse will be 1/10, since 10 = 10/1
This idea can be expressed differently: the reciprocal of a given number is obtained by dividing one by a given number. This statement is true not only for whole numbers, but also for fractions. In fact, if we need to write the inverse of the fraction 5/9, then we can take 1 and divide it by 5/9, i.e.
Now let's point out one thing property reciprocal numbers, which will be useful to us: the product of reciprocal numbers is equal to one. Indeed:
Using this property, we can find reciprocal numbers in the following way. Let's say we need to find the inverse of 8.
Let's denote it by the letter X , then 8 X = 1, hence X = 1/8. Let's find another number that is the inverse of 7/12 and denote it by the letter X , then 7/12 X = 1, hence X = 1: 7 / 12 or X = 12 / 7 .
We introduced here the concept of reciprocal numbers in order to slightly supplement the information about dividing fractions.
When we divide the number 6 by 3/5, we do the following:
Please pay Special attention to the expression and compare it with the given one: .
If we take the expression separately, without connection with the previous one, then it is impossible to solve the question of where it came from: from dividing 6 by 3/5 or from multiplying 6 by 5/3. In both cases the same thing happens. Therefore we can say that dividing one number by another can be replaced by multiplying the dividend by the inverse of the divisor.
The examples we give below fully confirm this conclusion.
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 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 get the product of mixed fractions, you need to convert them into an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter whether they are the same or different denominators For fractions, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.
How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction into an improper fraction and then find the product using the rules of multiplication.
How to multiply a number by a fraction?
Answer: we multiply the number with the numerator, but leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac(8)(9) \times \frac(7)(11)\) 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 \(\frac(3)(2)\) - an improper fraction. Answer: no.
b) in almost all enumerations of fractions this condition is not met, but there are some numbers that fulfill the condition of being simultaneously an improper fraction. For example, the improper fraction is \(\frac(3)(3)\), its inverse fraction is equal to \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions when the numerator and denominator are equal.
c) natural numbers are numbers that we use when counting, for example, 1, 2, 3, …. If we take the number \(3 = \frac(3)(1)\), then its inverse fraction will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not a natural number. If we go through all the numbers, the reciprocal of the number is always a fraction, except for 1. If we take the number 1, then its reciprocal fraction will be \(\frac(1)(1) = \frac(1)(1) = 1\). Number 1 is a natural number. Answer: they can simultaneously be natural numbers only in one case, if this is the number 1.
Example #6:
Do the product of mixed fractions: a) \(4 \times 2\frac(4)(5)\) b) \(1\frac(1)(4) \times 3\frac(2)(7)\)
Solution:
a) \(4 \times 2\frac(4)(5) = \frac(4)(1) \times \frac(14)(5) = \frac(56)(5) = 11\frac(1 )(5)\\\\ \)
b) \(1\frac(1)(4) \times 3\frac(2)(7) = \frac(5)(4) \times \frac(23)(7) = \frac(115)( 28) = 4\frac(3)(7)\)
Example #7:
Can two reciprocals be mixed numbers at the same time?
Let's look at an example. Let's take a mixed fraction \(1\frac(1)(2)\), find its inverse fraction, to do this we convert it into an improper fraction \(1\frac(1)(2) = \frac(3)(2) \) . Its inverse fraction will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two fractions that are mutually inverse cannot be mixed numbers at the same time.
In this article we will look at multiplying mixed numbers. First, we will outline the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next we'll talk about multiplying a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and common fraction.
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Multiplying mixed numbers.
Multiplying mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.
Let's write it down mixed number multiplication rule:
- First, the mixed numbers being multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule for multiplying fractions by fractions.
Let's look at examples of applying this rule when multiplying a mixed number by a mixed number.
Perform multiplication of mixed numbers and .
First, let's represent the mixed numbers being multiplied as improper fractions: 
And 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule for multiplying fractions, we get 
. The resulting fraction is irreducible (see reducible and irreducible fractions), but it is improper (see proper and improper fractions), therefore, to obtain the final answer, it remains to isolate the whole part from the improper fraction: .
Let's write the entire solution in one line: .
.
To strengthen the skills of multiplying mixed numbers, consider solving another example.
Do the multiplication.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it’s time to remember about reducing a fraction: replace all the numbers in the fraction with their decompositions into prime factors, and perform a reduction of identical factors.
Multiplying a mixed number and a natural number
After replacing a mixed number with an improper fraction, multiplying a mixed number and a natural number leads to the multiplication of an ordinary fraction and a natural number.
Multiply a mixed number and the natural number 45.
A mixed number is equal to a fraction, then 
. Let's replace the numbers in the resulting fraction with their decompositions into prime factors, perform a reduction, and then select the whole part: .
.
Multiplication of a mixed number and a natural number is sometimes conveniently carried out using the distributive property of multiplication relative to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, 
.
Calculate the product.
Let's replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .
Multiplying mixed numbers and fractions It is most convenient to reduce it to the multiplication of ordinary fractions by representing the mixed number being multiplied as an improper fraction.
Multiply the mixed number by the common fraction 4/15.
Replacing the mixed number with a fraction, we get 
.
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Multiplying fractions
§ 140. Definitions. 1) Multiplying a fraction by an integer is defined in the same way as multiplying integers, namely: to multiply a number (multiplicand) by an integer (factor) means to compose a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
So multiplying by 5 means finding the sum:
2) Multiplying a number (multiplicand) by a fraction (factor) means finding this fraction of the multiplicand.
Thus, we will now call finding a fraction of a given number, which we considered before, multiplication by a fraction.
3) To multiply a number (multiplicand) by a mixed number (factor) means to multiply the multiplicand first by the integer number of the multiplier, then by the fraction of the multiplier, and add the results of these two multiplications together.
For example:
The number obtained after multiplication in all these cases is called work, i.e. the same as when multiplying integers.
From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.
§ 141. The expediency of these definitions. To understand the advisability of introducing the last two definitions of multiplication into arithmetic, let’s take the following problem:
Task. A train, moving uniformly, covers 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?
If we remained with that one definition of multiplication, which is indicated in integer arithmetic (the addition of equal terms), then our problem would have three various solutions, namely:
If the given number of hours is an integer (for example, 5 hours), then to solve the problem you need to multiply 40 km by this number of hours.
If a given number of hours is expressed as a fraction (for example, an hour), then you will have to find the value of this fraction from 40 km.
Finally, if the given number of hours is mixed (for example, hours), then 40 km will need to be multiplied by the integer contained in the mixed number, and to the result add another fraction of 40 km, which is in the mixed number.
The definitions we have given allow us to give one general answer to all these possible cases:
you need to multiply 40 km by a given number of hours, whatever it may be.
Thus, if the problem is presented in general form as follows:
A train, moving uniformly, covers v km in an hour. How many kilometers will the train travel in t hours?
then, no matter what the numbers v and t are, we can give one answer: the desired number is expressed by the formula v · t.
Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, finding 5% (i.e. five hundredths) of a given number means the same thing as multiplying a given number by or by ; finding 125% of a given number means the same as multiplying this number by or by, etc.
§ 142. A note about when a number increases and when it decreases from multiplication.
Multiplication by a proper fraction decreases the number, and multiplication by an improper fraction increases the number if this improper fraction is greater than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so .
§ 143. Derivation of multiplication rules.
1) Multiplying a fraction by a whole number. Let a fraction be multiplied by 5. This means increased by 5 times. To increase a fraction by 5 times, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).
That's why:
Rule 1. To multiply a fraction by a whole number, you need to multiply the numerator by this whole number, but leave the denominator the same; instead, you can also divide the denominator of the fraction by the given whole number (if possible), and leave the numerator the same.
Comment. The product of a fraction and its denominator is equal to its numerator.
So:
Rule 2. To multiply a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator, and make the first product the numerator, and the second the denominator of the product.
Comment. This rule can also be applied to multiplying a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:
Thus, the three rules now outlined are contained in one, which in general can be expressed as follows:
4) Multiplication of mixed numbers.
Rule 4th. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For example:
§ 144. Reduction during multiplication. When multiplying fractions, if possible, it is necessary to make a preliminary reduction, as can be seen from the following examples:
Such a reduction can be done because the value of a fraction will not change if its numerator and denominator are reduced by the same number of times.
§ 145. Changing a product with changing factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .
So, if in the example:
to multiply several fractions, you need to multiply their numerators with each other and the denominators with each other and make the first product the numerator, and the second the denominator of the product.
Comment. This rule can also be applied to such products in which some of the factors of the number are integers or mixed, if only we consider the integer as a fraction with a denominator of one, and we turn mixed numbers into improper fractions. For example:
§ 147. Basic properties of multiplication. Those properties of multiplication that we indicated for integers (§ 56, 57, 59) also apply to the multiplication of fractional numbers. Let us indicate these properties.
1) The product does not change when the factors are changed.
For example:
Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their terms differ only in the order of the integer factors, and the product of integers does not change when the places of the factors are changed.
2) The product will not change if any group of factors is replaced by their product.
For example:
The results are the same.
From this property of multiplication the following conclusion can be drawn:
to multiply a number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, etc.
For example:
3) Distributive law of multiplication (relative to addition). To multiply a sum by a number, you can multiply each term separately by that number and add the results.
This law was explained by us (§ 59) as applied to integers. It remains true without any changes for fractional numbers.
Let us show, in fact, that the equality
(a + b + c + .)m = am + bm + cm + .
(the distributive law of multiplication relative to addition) remains true even when the letters represent fractional numbers. Let's consider three cases.
1) Let us first assume that the factor m is an integer, for example m = 3 (a, b, c – any numbers). According to the definition of multiplication by an integer, we can write (limiting ourselves to three terms for simplicity):
(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).
Based on the associative law of addition, we can omit all the parentheses on the right side; By applying the commutative law of addition, and then again the associative law, we can obviously rewrite the right-hand side as follows:
(a + a + a) + (b + b + b) + (c + c + c).
(a + b + c) * 3 = a * 3 + b * 3 + c * 3.
This means that the distributive law is confirmed in this case.
Multiplying and dividing fractions
Last time we learned how to add and subtract fractions (see lesson “Adding and Subtracting Fractions”). The most difficult part of those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. Good news is that these operations are even simpler than addition and subtraction. First, let's look at simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out because there was no pair for it, we take it outside the limits of multiplication. The result is a negative fraction.
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 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
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 main property of a fraction, since in this property we're talking about specifically about multiplying numbers.
There are simply no other reasons for reducing fractions, so correct solution the previous task looks like this:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
Multiplying fractions.
To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.
Multiplying a common fraction by a fraction.
To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.
Let's look at an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.
Multiplying a fraction by a number.
First, let's remember the rule, any number can be represented as a fraction \(\bf n = \frac \) .
Let's use this rule when multiplying.
The improper fraction \(\frac = \frac = \frac + \frac = 2 + \frac = 2\frac \\\) was converted to a mixed fraction.
In other words, When multiplying a number by a fraction, we multiply the number by the numerator and leave the denominator unchanged. Example:
Multiplying mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. We multiply the numerator with the numerator, and multiply the denominator with the denominator.
Multiplication of reciprocal fractions and numbers.
Related questions:
How to multiply a fraction by a fraction?
Answer: The product of ordinary fractions is the multiplication of a numerator with a numerator, a denominator with a denominator. To get the product of mixed fractions, you need to convert them into an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter whether fractions have the same or different denominators, multiplication occurs according to the rule of finding the product of a numerator with a numerator, a denominator with a denominator.
How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction into an improper fraction and then find the product using the rules of multiplication.
How to multiply a number by a fraction?
Answer: we multiply the number with the numerator, but leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac \times \frac \) b) \(\frac \times \frac \)
Example #2:
Calculate the products of a number and a fraction: a) \(3 \times \frac \) b) \(\frac \times 11\)
Example #3:
Write the reciprocal of the fraction \(\frac \)?
Answer: \(\frac = 3\)
Example #4:
Calculate the product of two mutually inverse fractions: a) \(\frac \times \frac \)
Example #5:
Can reciprocal fractions be:
a) simultaneously with proper fractions;
b) simultaneously improper fractions;
c) simultaneously natural numbers?
Solution:
a) to answer the first question, let's give an example. The fraction \(\frac \) is proper, its inverse fraction will be equal to \(\frac \) - an improper fraction. Answer: no.
b) in almost all enumerations of fractions this condition is not met, but there are some numbers that fulfill the condition of being simultaneously an improper fraction. For example, an improper fraction is \(\frac \) , its inverse fraction is equal to \(\frac \). We get two improper fractions. Answer: not always under certain conditions when the numerator and denominator are equal.
c) natural numbers are numbers that we use when counting, for example, 1, 2, 3, …. If we take the number \(3 = \frac \), then its inverse fraction will be \(\frac \). The fraction \(\frac \) is not a natural number. If we go through all the numbers, the reciprocal of the number is always a fraction, except for 1. If we take the number 1, then its reciprocal fraction will be \(\frac = \frac = 1\). Number 1 is a natural number. Answer: they can simultaneously be natural numbers only in one case, if this is the number 1.
Example #6:
Do the product of mixed fractions: a) \(4 \times 2\frac \) b) \(1\frac \times 3\frac \)
Solution:
a) \(4 \times 2\frac = \frac \times \frac = \frac = 11\frac \\\\ \)
b) \(1\frac \times 3\frac = \frac \times \frac = \frac = 4\frac \)
Example #7:
Can two reciprocals be mixed numbers at the same time?
Let's look at an example. Let's take a mixed fraction \(1\frac \), find its inverse fraction, to do this we convert it into an improper fraction \(1\frac = \frac \) . Its inverse fraction will be equal to \(\frac \) . The fraction \(\frac\) is a proper fraction. Answer: Two fractions that are mutually inverse cannot be mixed numbers at the same time.
Multiplying a decimal by a natural number
Presentation for the lesson
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.
- In a fun way, introduce to students the rule for multiplying a decimal fraction by a natural number, by a place value unit, and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge when solving examples and problems.
- Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their own work and the work of each other.
- Cultivate interest in mathematics, activity, mobility, and communication skills.
Equipment: interactive board, a poster with a cyphergram, posters with statements by mathematicians.
- Organizing time.
- Oral arithmetic – generalization of previously studied material, preparation for studying new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of acquired knowledge in a playful way using a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not be teaching it alone, but with my friend. And my friend is also unusual, you will see him now. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, buddy? Komposha replies: “My name is Komposha.” Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is hung on the board with an oral calculation for adding and subtracting decimal fractions, as a result of which the children receive the following code 523914687. )
Komposha helps decipher the received code. The result of decoding is the word MULTIPLICATION. Multiplication is keyword topics of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how to multiply natural numbers. Today we will look at multiplication decimal numbers to a natural number. Multiplying a decimal fraction by a natural number can be considered as a sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 ·3 = 5.21 + 5.21 + 5.21 = 15.63 So, 5.21 ·3 = 15.63. Presenting 5.21 as a common fraction to a natural number, we get
And in this case we got the same result: 15.63. Now, ignoring the comma, instead of the number 5.21, take the number 521 and multiply it by this natural number. Here we must remember that in one of the factors the comma has been moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now in this example we move the comma to the left two places. Thus, by how many times one of the factors was increased, by how many times the product was decreased. Based on the similarities of these methods, we will draw a conclusion.
To multiply a decimal fraction by a natural number, you need to:
1) without paying attention to the comma, multiply natural numbers;
2) in the resulting product, separate as many digits from the right with a comma as there are in the decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 ·3 = 15.63 and 7.624 ·15 = 114.34. Then I show multiplication by a round number 12.6 · 50 = 630. Next, I move on to multiplying a decimal fraction by a place value unit. I show the following examples: 7.423 · 100 = 742.3 and 5.2 · 1000 = 5200. So, I introduce the rule for multiplying a decimal fraction by a digit unit:
To multiply a decimal fraction by digit units 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many places as there are zeros in the digit unit.
I finish my explanation by expressing the decimal fraction as a percentage. I introduce the rule:
To express a decimal fraction as a percentage, you must multiply it by 100 and add the % sign.
I’ll give an example on a computer: 0.5 100 = 50 or 0.5 = 50%.
4. At the end of the explanation I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, we are doing a mathematical physical education session together with Komposha to consolidate the topic. Everyone stands up, shows the solved examples to the class, and they must answer whether the example was solved correctly or incorrectly. If the example is solved correctly, then they raise their arms above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and stretch their fingers.
6. And now you have rested a little, you can solve the tasks. Open your textbook to page 205, № 1029. In this task you need to calculate the value of the expressions:
The tasks appear on the computer. As they are solved, a picture appears with the image of a boat that floats away when fully assembled.
By solving this task on a computer, the rocket gradually folds up; after solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the soil of Kazakhstan, from the Baikonur Cosmodrome, they take off to the stars spaceships. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.
How far will a passenger car travel in 4 hours if the speed of the passenger car is 74.8 km/h.
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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…”)
This operation is much nicer than addition-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:
For example:
Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...
To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:
For example:
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. Note practical advice, and there will be fewer of them (errors)!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.
2. In examples with different types fractions - go 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!).
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.
So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.
Calculate:
Have you decided?
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, written with 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.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.