Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use some object not entirely, but in separate pieces. The beginning of the study of this topic - share. Shares are equal parts into which an object is divided. After all, it is not always possible to express, for example, the length or price of a product as an integer; one should take into account parts or shares of any measure. Formed from the verb "to crush" - to divide into parts, and having Arabic roots, in the VIII century the word "fraction" itself appeared in Russian.
Fractional expressions have long been considered the most difficult section of mathematics. In the 17th century, when first textbooks in mathematics appeared, they were called "broken numbers", which was very difficult to display in people's understanding.
The modern form of simple fractional residues, parts of which are separated precisely by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His writings are dated 1202. But the purpose of this article is to simply and clearly explain to the reader how the multiplication of mixed fractions with different denominators occurs.
Multiplying fractions with different denominators
Initially, it is necessary to determine varieties of fractions:
- correct;
- wrong;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is easy to formulate independently: the result of multiplying simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the existing ones initially.
When multiplying simple fractions with different denominators for two or more factors, the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional bar will be the product of different numbers and, of course, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use ways to reduce fractional expressions. You can reduce only the numbers of the numerator with the numbers of the denominator; adjacent factors above or below the fractional bar cannot be reduced.
Along with simple fractional numbers, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses the multiplication of a number by ordinary fractional part, you can write down the rule for this action by the formula:
a * b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another option for solving the multiplication of a number by a fractional remainder. Simply divide the denominator by this number:
d* e/f = e/f: d.
It is useful to use this technique when the denominator is divided by a natural number without a remainder or, as they say, completely.
Convert mixed numbers to improper fractions and get the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way to represent a mixed fraction as an improper fraction, it can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and adding it to the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse. To select the integer part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator with a “corner”.
Multiplication of improper fractions produced in the usual way. When the entry goes under a single fractional line, as necessary, you need to reduce the fractions in order to reduce the numbers using this method and it is easier to calculate the result.
There are many assistants on the Internet to solve even complex mathematical problems in various program variations. A sufficient number of such services offer their help in calculating the multiplication of fractions with different numbers in the denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It is not difficult to work with it, the corresponding fields are filled in on the site page, the sign of the mathematical action is selected and “calculate” is pressed. The program counts automatically.
The topic of arithmetic operations with fractional numbers is relevant throughout the education of middle and senior schoolchildren. In high school, they are no longer considering the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations, obtained earlier, is applied in its original form. Well-learned basic knowledge gives full confidence in the successful solution of the most complex tasks.
In conclusion, it makes sense to cite the words of Leo Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his merits, but everyone can decrease his denominator - his opinion of himself, and by this decrease come closer to his perfection.
Note! Before writing a final answer, see if you can reduce the fraction you received.
Subtraction of fractions with the same denominators examples:
,
,
Subtracting a proper fraction from one.
If it is necessary to subtract from the unit a fraction that is correct, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.
An example of subtracting a proper fraction from one:
The denominator of the fraction to be subtracted = 7 , i.e., we represent the unit as an improper fraction 7/7 and subtract according to the rule for subtracting fractions with the same denominators.
Subtracting a proper fraction from a whole number.
Rules for subtracting fractions - correct from integer (natural number):
- We translate the given fractions, which contain an integer part, into improper ones. We get normal terms (it does not matter if they have different denominators), which we consider according to the rules given above;
- Next, we calculate the difference of the fractions that we received. As a result, we will almost find the answer;
- We perform the inverse transformation, that is, we get rid of the improper fraction - we select the integer part in the fraction.
Subtract a proper fraction from a whole number: we represent a natural number as a mixed number. Those. we take a unit in a natural number and translate it into the form of an improper fraction, the denominator is the same as that of the subtracted fraction.
Fraction subtraction example:
In the example, we replaced the unit with an improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.
Subtraction of fractions with different denominators.
Or, to put it another way, subtraction of different fractions.
Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to bring these fractions to the lowest common denominator (LCD), and only after that to subtract as with fractions with the same denominators.
The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of the given fractions.
Attention! If the numerator and denominator have common factors in the final fraction, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the result of the subtraction without reducing the fraction where possible is an unfinished solution to the example!
Procedure for subtracting fractions with different denominators.
- find the LCM for all denominators;
- put additional multipliers for all fractions;
- multiply all numerators by an additional factor;
- we write the resulting products in the numerator, signing a common denominator under all fractions;
- subtract the numerators of fractions, signing the common denominator under the difference.
In the same way, addition and subtraction of fractions is carried out in the presence of letters in the numerator.
Subtraction of fractions, examples:
Subtraction of mixed fractions.
At subtraction of mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.
The first option is to subtract mixed fractions.
If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract from it) ≥ the numerator of the fractional part of the subtrahend (we subtract it).
For example:
The second option is to subtract mixed fractions.
When the fractional parts different denominators. To begin with, we reduce the fractional parts to a common denominator, and then we subtract the integer part from the integer, and the fractional from the fractional.
For example:
The third option is to subtract mixed fractions.
The fractional part of the minuend is less than the fractional part of the subtrahend.
Example:
Because fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.
The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. So, we take a unit from the integer part and bring this unit to the form of an improper fraction with the same denominator and numerator = 18.
In the numerator from the right side we write the sum of the numerators, then we open the brackets in the numerator from the right side, that is, we multiply everything and give similar ones. We do not open brackets in the denominator. It is customary to leave the product in the denominators. We get:
In this lesson, we will consider the addition and subtraction of algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will be found in many topics of the algebra course, which you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, as well as analyze a number of typical examples.
Consider the simplest example for ordinary fractions.
Example 1 Add fractions: .
Solution:
Remember the rule for adding fractions. To begin with, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.
Definition
The smallest natural number that is divisible by both numbers and .
To find the LCM, it is necessary to decompose the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two 2s and two 3s: .
After finding the common denominator, it is necessary for each of the fractions to find an additional factor (in fact, divide the common denominator by the denominator of the corresponding fraction).
Then each fraction is multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.
We get: 
.
Answer:.
Consider now the addition of algebraic fractions with different denominators. First consider fractions whose denominators are numbers.
Example 2 Add fractions: .
Solution:
The solution algorithm is absolutely similar to the previous example. It is easy to find a common denominator for these fractions: and additional factors for each of them.
.
Answer:.
So let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:
1. Find the smallest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of this fraction).
3. Multiply the numerators by the appropriate additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with the same denominators.
Consider now an example with fractions, in the denominator of which there are literal expressions.
Example 3 Add fractions: .
Solution:
Since the literal expressions in both denominators are the same, you should find a common denominator for numbers. The final common denominator will look like: . So the solution to this example is:
Answer:.
Example 4 Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use the abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as a common denominator.
Answer:.
In general, when solving such examples, the most difficult task is to find a common denominator.
Let's look at a more complex example.
Example 5 Simplify: .
Solution:
When finding a common denominator, you must first try to factorize the denominators of the original fractions (to simplify the common denominator).
In this particular case:
Then it is easy to determine the common denominator: 
.
We determine additional factors and solve this example:
Answer:.
Now we will fix the rules for adding and subtracting fractions with different denominators.
Example 6 Simplify: .
Solution:
Answer:.
Example 7 Simplify: .
Solution:
.
Answer:.
Consider now an example in which not two, but three fractions are added (after all, the rules for addition and subtraction for more fractions remain the same).
Example 8 Simplify: .
§ 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), which contains all units and fractions of units of terms.
We will consider three cases in turn:
1. Addition of fractions with the same denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.
1. Addition of fractions with the same denominators.
Consider an example: 1 / 5 + 2 / 5 .
Take the segment AB (Fig. 17), take it as a unit and divide it into 5 equal parts, then the part AC of this segment will be equal to 1/5 of the segment AB, and the part of the same segment CD will be equal to 2/5 AB.
It can be seen from the drawing that if we take the segment AD, then it will be equal to 3/5 AB; but segment AD is precisely the sum of segments AC and CD. So, we can write:
1 / 5 + 2 / 5 = 3 / 5
Considering these terms and the resulting amount, 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 must add their numerators and leave the same denominator.
Consider an example:
2. Addition of fractions with different denominators.
Let's add 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 have been written; we have written it here for greater clarity.
Thus, to add fractions with different denominators, you must first bring them to the lowest common denominator, add their numerators and sign the common denominator.
Consider an example (we will write additional factors over the corresponding fractions):
3. Addition of mixed numbers.
Let's add the numbers: 2 3 / 8 + 3 5 / 6.
Let us first bring the fractional parts of our numbers to a common denominator and rewrite them again:
Now add the integer and fractional parts in sequence:
§ 88. Subtraction of fractions.
Subtraction of fractions is defined in the same way as subtraction of whole numbers. This is an action by which, given the sum of two terms and one of them, another term is found. Let's consider three cases in turn:
1. Subtraction of fractions with the same denominators.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.
1. Subtraction of fractions with the same denominators.
Consider 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 the AC part of this segment will be 1/15 of AB, and the AD part of the same segment will correspond to 13/15 AB. Let's set aside another segment ED, equal to 4/15 AB.
We need to subtract 4/15 from 13/15. In the drawing, this means that the segment ED must be subtracted from the 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, and the denominator remained the same.
Therefore, in order to subtract fractions with the same denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.
2. Subtraction of fractions with different denominators.
Example. 3/4 - 5/8
First, let's reduce these fractions to the smallest common denominator:
The intermediate link 6 / 8 - 5 / 8 is written here for clarity, but it can be skipped in the future.
Thus, in order to subtract a fraction from a fraction, you must first bring them to the smallest common denominator, then subtract the numerator of the subtrahend from the numerator of the minuend and sign the common denominator under their difference.
Consider an example:
3. Subtraction of mixed numbers.
Example. 10 3 / 4 - 7 2 / 3 .
Let's bring the fractional parts of the minuend and the 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 integer part of the reduced, split it into those parts in which the fractional part is expressed, and add to the fractional part of the reduced. And then the subtraction will be performed in the same way as in the previous example:
§ 89. Multiplication of fractions.
When studying the multiplication of fractions, we will consider the following questions:
1. Multiplying a fraction by an integer.
2. Finding a fraction of a given number.
3. Multiplication of 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 percentages of a given number. Let's consider them sequentially.
1. Multiplying a fraction by an integer.
Multiplying a fraction by an integer has the same meaning as multiplying an integer by an integer. Multiplying a fraction (multiplier) by an integer (multiplier) means composing the sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
So, if you need to multiply 1/9 by 7, then this can be done like this:
We easily got 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 an integer is equivalent to increasing this fraction as many times as there are units in the integer. And since the increase in the fraction is achieved either by increasing its numerator
or by decreasing its denominator 
, then we can either multiply the numerator by the integer, or divide the denominator by it, if such a division is possible.
From here we get the rule:
To multiply a fraction by an integer, you need to multiply the numerator by this integer and leave the denominator the same, or, if possible, divide the denominator by this number, leaving the numerator unchanged.
When multiplying, abbreviations are possible, for example:
2. Finding a fraction of a given number. There are many problems in which you have to find, or calculate, a part of a given number. The difference between these tasks 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 the method of solving them.
Task 1. I had 60 rubles; 1 / 3 of this money I spent on the purchase of books. How much did the books cost?
Task 2. The train must cover the distance between cities A and B, equal to 300 km. He has already covered 2/3 of that 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?
Here are some of the many problems that we have to deal with to find a fraction of a given number. They are usually called problems for finding a fraction of a given number.
Solution of problem 1. From 60 rubles. I spent 1 / 3 on books; So, to find the cost of books, you need to divide the number 60 by 3:
Problem 2 solution. The meaning of the problem is that you need to find 2 / 3 of 300 km. Calculate first 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, that is, multiply by 2:
100 x 2 = 200 (that's 2/3 of 300).
Solution of problem 3. Here you need to determine the number of brick houses, which are 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, that is, multiplied by 3:
100 x 3 = 300 (that's 3/4 of 400).
Based on the solution of these problems, we can derive the following rule:
To find the value of a fraction of a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.
3. Multiplication of 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 \u003d 5 + 5 + 5 + 5 \u003d 20). In this paragraph (paragraph 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, the multiplication consisted in finding the sum of identical terms.
Now we move on to multiplying a whole number by a fraction. Here we will meet with such, for example, multiplication: 9 2 / 3. It is quite obvious 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, to answer the question of what should be understood by multiplication by a fraction, how this action should be understood.
The meaning of multiplying an integer by a fraction is clear from the following definition: to multiply an integer (multiplier) by a fraction (multiplier) means to find this fraction of the multiplier.
Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it's easy to figure out that we end up with 6.
But now an interesting and important question arises: why such seemingly different actions as finding the sum of equal numbers and finding the fraction of a number are called the same word “multiplication” in arithmetic?
This happens because the previous action (repeating the number with terms several times) and the new action (finding the fraction of a number) give an answer to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by one and 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 fractional number: “1 m of cloth costs 50 rubles. How much will 3 / 4 m of such a 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 also change the numbers in it several 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 is a whole number multiplied by a fraction?
Let's take the numbers encountered in the last problem:
According to the definition, we must find 3 / 4 of 50. First we find 1 / 4 of 50, and then 3 / 4.
1/4 of 50 is 50/4;
3/4 of 50 is .
Hence.
Consider another example: 12 5 / 8 = ?
1/8 of 12 is 12/8,
5/8 of the number 12 is .
Hence,
From here we get the rule:
To multiply an integer by a fraction, you need to multiply the integer by the numerator of the fraction and make this product the numerator, and sign the denominator of the given fraction as the denominator.
We write this rule using letters:
To make this rule perfectly 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 must be remembered that before performing multiplication, you should do (if possible) cuts, For example:
4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying an integer by a fraction, that is, when multiplying a fraction by a fraction, you need to find the fraction in the multiplier from the first fraction (multiplier).
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 times 5/7. This means that you need to find 5 / 7 from 3 / 4 . Find first 1/7 of 3/4 and then 5/7
1/7 of 3/4 would be expressed like this:
5 / 7 numbers 3 / 4 will be expressed as follows:
Thus,
Another example: 5/8 times 4/9.
1/9 of 5/8 is ,
4/9 numbers 5/8 are .
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 rule can be written in general as follows:
When multiplying, it is necessary to make (if possible) reductions. Consider 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 those cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, then they are replaced by improper fractions. Multiply, for example, mixed numbers: 2 1/2 and 3 1/5. We turn each of them into an improper fraction and then we will multiply the resulting fractions according to the rule of multiplying a fraction by a fraction:
Rule. To multiply mixed numbers, you must first convert them to improper fractions and then multiply according to the rule of multiplying a fraction by a fraction.
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 when performing various practical calculations, we use all kinds of fractions. But one must keep in mind that many quantities admit not any, but natural subdivisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a penny, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of the ruble, it will be "10 kopecks, or a dime. You can take a quarter of the ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take, for example , 2/7 rubles because the ruble is not divided into sevenths.
The unit of weight measurement, i.e., the kilogram, primarily allows decimal subdivisions, for example, 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are uncommon.
In general our (metric) measures are decimal and allow decimal subdivisions.
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 "hundredths" division. Let's consider a few examples related 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 is 10 rubles. She went down by 1 ruble. 20 kop.
2. Savings banks pay out during the year to depositors 2/100 of the amount that is put into savings.
Example. 500 rubles are put into the cash desk, the income from this amount for the year is 10 rubles.
3. The number of graduates of one school was 5/100 of the total number of students.
EXAMPLE Only 1,200 students studied at the school, 60 of them graduated from school.
The hundredth of a number is called a percentage..
The word "percent" is borrowed from the Latin language and its root "cent" means one hundred. Together with the preposition (pro centum), this word means "for a hundred." The meaning of this 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 (they say centimeter).
For example, instead of saying that the plant produced 1/100 of all the products produced by it during the past month, we will say this: the plant produced one percent of the rejects during the past month. 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 of the amount put into savings.
3. The number of graduates of one school was 5 percent of the number of all students in the school.
To shorten the letter, it is customary to write the% sign instead of the word "percentage".
However, it must be remembered that the % sign is usually not written in calculations, 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 an integer with this icon.
You need to be able to replace an integer with the specified icon with a fraction with a denominator of 100:
Conversely, you need to get used to writing an integer with the indicated icon instead of a fraction with a denominator of 100:
7. Finding percentages of a given number.
Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch wood was there?
The meaning of this problem is that birch firewood was only a part of the firewood that was delivered to the school, and this part is expressed as a fraction of 30 / 100. So, we are faced with the task of finding a fraction of a number. To solve it, we must multiply 200 by 30 / 100 (tasks for finding the fraction of a number are solved by multiplying a number by a fraction.).
So 30% of 200 equals 60.
The fraction 30 / 100 encountered in this problem can be reduced by 10. It would be possible to perform this reduction from the very beginning; the solution to the problem would not change.
Task 2. There were 300 children of various ages in the camp. Children aged 11 were 21%, children aged 12 were 61% and finally 13 year olds were 18%. How many children of each age were in the camp?
In this problem, you need to perform three calculations, that is, successively find the number of children 11 years old, then 12 years old, and finally 13 years old.
So, here it will be necessary to find a fraction of a number three times. Let's do it:
1) How many children were 11 years old?
2) How many children were 12 years old?
3) How many children were 13 years old?
After solving the problem, it is useful to add the numbers found; their sum should be 300:
63 + 183 + 54 = 300
You should also pay attention to the fact that the sum of the percentages given in the condition of the problem is 100:
21% + 61% + 18% = 100%
This suggests that the total number of children in the camp was taken as 100%.
3 a da cha 3. The worker received 1,200 rubles per month. Of these, he spent 65% on food, 6% on an apartment and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% he saved. How much money was spent on the needs indicated in the task?
To solve this problem, you need to find a fraction of the number 1,200 5 times. Let's do it.
1) How much money is spent on food? The task says that this expense is 65% of all earnings, i.e. 65 / 100 of the number 1,200. Let's do the calculation:
2) How much money was paid for an apartment with heating? Arguing like 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 is spent on cultural needs?
5) How much money did the worker save?
For verification, it is useful to add 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 percentages given in the condition of the problem.
We have solved three problems. Despite the fact that these tasks were about different things (delivery of firewood for the school, the number of children of different ages, the expenses of the worker), they were solved in the same way. This happened because in all tasks it was necessary to find a few percent of the given numbers.
§ 90. Division of fractions.
When studying the division of fractions, we will consider the following questions:
1. Divide an integer by an integer.
2. Division of a fraction by an integer
3. Division of an integer by a fraction.
4. Division of a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number given its fraction.
7. Finding a number by its percentage.
Let's consider them sequentially.
1. Divide an integer by an integer.
As it was indicated in the section of integers, division is the action consisting in the fact that, given the product of two factors (the dividend) and one of these factors (the divisor), another factor is found.
The division of an integer by an integer we considered in the department of integers. We met there two cases of division: division without a remainder, or "entirely" (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 in the remainder). We can therefore say that in the realm of integers, exact division is not always possible, because the dividend is not always the product of the divisor and the integer. After the introduction of multiplication by a fraction, we can consider any case of division of integers as possible (only division by zero is excluded).
For example, dividing 7 by 12 means finding a number whose product times 12 would be 7. This 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 an integer by an integer, you need to make a fraction, the numerator of which is equal to the dividend, and the denominator is the divisor.
2. Division of a fraction by an integer.
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 such 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 the reduction of 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, we can state the rule: To divide a fraction by an integer, you need to divide the numerator of the fraction by that integer(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.
3. Division of an integer by a fraction.
Let it be required 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 by a proper fraction, the product must be less than the multiplicand. To make it clearer, let's write our actions as follows: 5: 1 / 2 = X , so x 1 / 2 \u003d 5.
We must find such a number X , which, when 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 5, and the whole number X twice as much, i.e. 5 2 \u003d 10.
So 5: 1 / 2 = 5 2 = 10
Let's check: 
Let's consider one more example. Let it be required to divide 6 by 2 / 3 . Let's first try to find the desired result using the drawing (Fig. 19).
Fig.19
Draw a segment AB, equal to 6 of some units, and divide each unit into 3 equal parts. In each unit, three-thirds (3 / 3) in the entire segment AB is 6 times larger, i.e. e. 18/3. We connect with the help of small brackets 18 obtained segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in b units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 integer units. Hence,
How to get this result without a drawing using only calculations? We will argue as follows: it is required to divide 6 by 2 / 3, i.e., it is required to answer the question, how many times 2 / 3 is contained in 6. Let's find out first: how many times is 1 / 3 contained in 6? In a whole unit - 3 thirds, and in 6 units - 6 times more, i.e. 18 thirds; to find this number, we must multiply 6 by 3. Hence, 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 did the following:
From here we get the rule for dividing an integer by a fraction. To divide an integer by a fraction, you need to multiply this integer by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.
We write the rule using letters:
To make this rule perfectly 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. Note that the same formula was obtained there.
When dividing, abbreviations are possible, for example:
4. Division of a fraction by a fraction.
Let it be required to divide 3/4 by 3/8. What will denote the number that will be obtained as a result of division? 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).
Take the segment AB, take it as a unit, 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 initial 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. We 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 the segment equal to 3/8 is contained in the segment equal to 3/4 exactly 2 times; So the result of the division can be written like this:
3 / 4: 3 / 8 = 2
Let's consider one more example. Let it be required to divide 15/16 by 3/32:
We can reason like this: we need to find a number that, after being multiplied 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 make up 15 / 16
1/32 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 into improper fractions, and then the resulting fractions should be divided according to the rules for dividing fractional numbers. Consider an example:
Convert mixed numbers to improper fractions:
Now let's split:
Thus, to divide mixed numbers, you need to convert them to improper fractions and then divide according to the rule for dividing fractions.
6. Finding a number given its fraction.
Among the various tasks on fractions, there are sometimes those in which the value of some fraction of an unknown number is given and it is required to find this number. This type of problem will be inverse to the problem of finding a 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 is given and it is required to find this number itself. This idea will become even clearer if we turn to the solution of 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 in this house?
Solution. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means that there are 3 times more windows in total, i.e.
The house had 150 windows.
Task 2. The shop sold 1,500 kg of flour, which is 3/8 of the total stock of flour in the shop. What was the store's initial supply of flour?
Solution. It can be seen from the condition of the problem that the sold 1,500 kg of flour make up 3/8 of the total stock; this means that 1/8 of this stock 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 stock).
Obviously, the entire stock will be 8 times larger. Hence,
500 8 \u003d 4,000 (kg).
The initial supply of flour in the store was 4,000 kg.
From the consideration of this problem, the following rule can be deduced.
To find a number by 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 well 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 studied the division of fractions, the above problems can be solved in 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 the problem of finding a number by its fraction in one action - division.
7. Finding a number by its percentage.
In these tasks, you will need to find a number, knowing a few percent of this 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 did I put in the savings bank? (Cash offices give depositors 2% of income per year.)
The meaning of the problem is that a certain amount of money was put by me in a savings bank and lay there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I put in. How much money did I deposit?
Therefore, knowing the part of this money, expressed in two ways (in rubles and in fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following tasks are solved by division:
So, 3,000 rubles were put into the savings bank.
Task 2. In two weeks, fishermen fulfilled the monthly plan by 64%, having prepared 512 tons of fish. What was their plan?
From the condition 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. How many tons of fish need to be harvested according to the plan, we do not know. The solution of the problem will consist in finding this number.
Such tasks are solved by dividing:
So, according to the plan, you need to prepare 800 tons of fish.
Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked the passing conductor how much of the journey they had already traveled. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?
It can be seen from the condition of the problem that 30% of the journey 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.
Take the fraction 2/3 and rearrange the numerator to the place of the denominator, we get 3/2. We got a fraction, the reciprocal of this one.
In order to get a fraction reciprocal of a given one, you need to put its numerator in the place of the denominator, and the denominator in the place of the numerator. In this way, we can get a fraction that is 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. Looking for the reciprocal of this, 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, inverse 3; 1 / 5, reverse 5
Since when finding reciprocals we also met with integers, in the future we will not talk about reciprocals, but about reciprocals.
Let's figure out how to write the reciprocal of a whole number. For fractions, this is solved simply: you need to put the denominator in the place of the numerator. In the same way, you can get the reciprocal of an integer, since any integer can have a denominator of 1. Therefore, the reciprocal of 7 will be 1 / 7, because 7 \u003d 7 / 1; for the number 10 the reverse is 1 / 10 since 10 = 10 / 1
This idea can be expressed in another way: the reciprocal of a given number is obtained by dividing one by the given number. This statement is true not only for integers, but also for fractions. Indeed, if you want to write a number that is the reciprocal of 5/9, then we can take 1 and divide it by 5/9, i.e.
Now let's point out one property mutually reciprocal numbers, which will be useful to us: the product of mutually reciprocal numbers is equal to one. Indeed:
Using this property, we can find reciprocals in the following way. Let's find the reciprocal of 8.
Let's denote it with the letter X , then 8 X = 1, hence X = 1 / 8 . Let's find another number, the inverse of 7/12, denote it by a 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 information about the division of fractions.
When we divide the number 6 by 3 / 5, then we do the following:
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 result is the same. So we can say that dividing one number by another can be replaced by multiplying the dividend by the reciprocal of the divisor.
The examples that we give below fully confirm this conclusion.
Consider the fraction $\frac63$. Its value is 2, because $\frac63 =6:3 = 2$. What happens if the numerator and denominator are multiplied by 2? $\frac63 \times 2=\frac(12)(6)$. Obviously, the value of the fraction has not changed, so $\frac(12)(6)$ is also equal to 2 as y. multiply the numerator and denominator by 3 and get $\frac(18)(9)$, or by 27 and get $\frac(162)(81)$ or by 101 and get $\frac(606)(303)$. In each of these cases, the value of the fraction that we get by dividing the numerator by the denominator is 2. This means that it has not changed.
The same pattern is observed in the case of other fractions. If the numerator and denominator of the fraction $\frac(120)(60)$ (equal to 2) is divided by 2 (result of $\frac(60)(30)$), or by 3 (result of $\frac(40)(20) $), or by 4 (the result of $\frac(30)(15)$) and so on, then in each case the value of the fraction remains unchanged and equal to 2.
This rule also applies to fractions that are not equal. whole number.
If the numerator and denominator of the fraction $\frac(1)(3)$ are multiplied by 2, we get $\frac(2)(6)$, that is, the value of the fraction has not changed. And in fact, if you divide the cake into 3 parts and take one of them, or divide it into 6 parts and take 2 parts, you will get the same amount of pie in both cases. Therefore, the numbers $\frac(1)(3)$ and $\frac(2)(6)$ are identical. Let's formulate a general rule.
The numerator and denominator of any fraction can be multiplied or divided by the same number, and the value of the fraction does not change.
This rule is very useful. For example, it allows in some cases, but not always, to avoid operations with large numbers.
For example, we can divide the numerator and denominator of the fraction $\frac(126)(189)$ by 63 and get the fraction $\frac(2)(3)$ which is much easier to calculate. One more example. The numerator and denominator of the fraction $\frac(155)(31)$ can be divided by 31 to get the fraction $\frac(5)(1)$ or 5, since 5:1=5.
In this example, we first encountered a fraction whose denominator is 1. Such fractions play an important role in calculations. It should be remembered that any number can be divided by 1 and its value will not change. That is, $\frac(273)(1)$ is equal to 273; $\frac(509993)(1)$ equals 509993 and so on. Therefore, we do not have to divide numbers by , since every integer can be represented as a fraction with a denominator of 1.
With such fractions, the denominator of which is equal to 1, you can perform the same arithmetic operations as with all other fractions: $\frac(15)(1)+\frac(15)(1)=\frac(30)(1) $, $\frac(4)(1) \times \frac(3)(1)=\frac(12)(1)$.
You may ask what is the use of representing an integer as a fraction, which will have a unit under the line, because it is more convenient to work with an integer. But the fact is that the representation of an integer as a fraction gives us the opportunity to perform various actions more efficiently when we are dealing with both integers and fractional numbers at the same time. For example, to learn add fractions with different denominators. Suppose we need to add $\frac(1)(3)$ and $\frac(1)(5)$.
We know that you can only add fractions whose denominators are equal. So, we need to learn how to bring fractions to such a form when their denominators are equal. In this case, we again need the fact that you can multiply the numerator and denominator of a fraction by the same number without changing its value.
First, we multiply the numerator and denominator of the fraction $\frac(1)(3)$ by 5. We get $\frac(5)(15)$, the value of the fraction has not changed. Then we multiply the numerator and denominator of the fraction $\frac(1)(5)$ by 3. We get $\frac(3)(15)$, again the value of the fraction has not changed. Therefore, $\frac(1)(3)+\frac(1)(5)=\frac(5)(15)+\frac(3)(15)=\frac(8)(15)$.
Now let's try to apply this system to the addition of numbers containing both integer and fractional parts.
We need to add $3 + \frac(1)(3)+1\frac(1)(4)$. First, we convert all the terms into fractions and get: $\frac31 + \frac(1)(3)+\frac(5)(4)$. Now we need to bring all the fractions to a common denominator, for this we multiply the numerator and denominator of the first fraction by 12, the second by 4, and the third by 3. As a result, we get $\frac(36)(12) + \frac(4 )(12)+\frac(15)(12)$, which is equal to $\frac(55)(12)$. If you want to get rid of improper fraction, it can be turned into a number consisting of an integer and a fractional part: $\frac(55)(12) = \frac(48)(12)+\frac(7)(12)$ or $4\frac(7)( 12)$.
All the rules that allow operations with fractions, which we have just studied, are also valid in the case of negative numbers. So, -1: 3 can be written as $\frac(-1)(3)$, and 1: (-3) as $\frac(1)(-3)$.
Since both dividing a negative number by a positive number and dividing a positive number by a negative result in negative numbers, in both cases we will get the answer in the form of a negative number. That is
$(-1) : 3 = \frac(1)(3)$ or $1 : (-3) = \frac(1)(-3)$. The minus sign when written this way refers to the entire fraction as a whole, and not separately to the numerator or denominator.
On the other hand, (-1) : (-3) can be written as $\frac(-1)(-3)$, and since dividing a negative number by a negative number gives a positive number, then $\frac(-1 )(-3)$ can be written as $+\frac(1)(3)$.
Addition and subtraction of negative fractions is carried out in the same way as the addition and subtraction of positive fractions. For example, what is $1- 1\frac13$? Let's represent both numbers as fractions and get $\frac(1)(1)-\frac(4)(3)$. Let's reduce the fractions to a common denominator and get $\frac(1 \times 3)(1 \times 3)-\frac(4)(3)$, i.e. $\frac(3)(3)-\frac(4) (3)$, or $-\frac(1)(3)$.
