The rule for adding fractions with a common denominator. Adding and subtracting fractions
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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 an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or shares of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
Modern look simple fractional remainders, the parts of which are separated by a horizontal line, were first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how multiplication occurs mixed fractions With different denominators.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next you need to remember how multiplication occurs fractional numbers with the same denominators. The very rule of this process is easy to formulate independently: the result of multiplication 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 essence, new denominator there is a square of one of the originally existing ones.
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 resulting number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.
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 methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, 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 multiplication of a number by ordinary fractional part, the rule for this action can be written as:
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 solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Translate mixed numbers into improper fractions and obtain 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 of representing a mixed fraction as an improper fraction, it can also be represented as general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse side. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations programs. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - 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’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider 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 give complete confidence in successful decision most complex tasks.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
§ 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, 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 consider 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 link 6 / 8 - 5 / 8 is written here for clarity, but can be skipped from now on.
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 what is being subtracted is greater than the fractional part of what is being reduced. 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 there are units 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 much in total brick houses?
Here are some of those numerous tasks to find parts of a given number that we encounter. 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 the previous paragraph, such problems were solved; so it’s easy to figure out that we’ll end up with 6.
But now there is an interesting and important question: Why are such seemingly different operations, such as finding the sum of equal numbers and finding the fraction of a number, called in arithmetic by the same word “multiplication”?
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 factor, 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 to 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 "percentage" is borrowed from Latin language 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 its products as defective, we will say this: over the past month the plant produced one percent defective. 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 in the camp different ages. 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 the total number of children in the camp was 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 task?
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 tasks dealt with various 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 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 section on integers, division is an action consisting 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 in this case The numerator of 6 is divisible by 3, so the numerator should be halved.
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 we need 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, 2 each; 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 did the following:
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 various tasks on fractions, 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 meaning 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 talk not about reciprocal fractions, but about reciprocal numbers.
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 number for an integer, since any integer can have a denominator of 1. This means that the inverse number 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 note 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.
The study of subtracting fractions with different denominators is found in school subject Algebra in the eighth grade and it sometimes causes difficulties for children to understand. To subtract fractions with different denominators, use the following formula:
The procedure for subtracting fractions is similar to addition, since it completely copies the principle of operation.
First, we calculate the most small number, which is a multiple of both one and the other denominator.
Secondly, we multiply the numerator and denominator of each fraction by a certain number that will allow us to bring the denominator to a given minimum common denominator.
Thirdly, the subtraction procedure itself occurs, when in the end the denominator is duplicated, and the numerator of the second fraction is subtracted from the first.
Example: 8/3 2/4 = 8/3 1/2 = 16/6 3/6 = 13/6 = 2 whole 1/6
First you need to bring them to the same denominator, and then subtract. For example, 1/2 - 1/4 = 2/4 - 1/4 = 1/4. Or, more difficult, 1/3 - 1/5 = 5/15 - 3/15 = 2/15. Do you need to explain how fractions are reduced to a common denominator?
For operations such as addition or subtraction ordinary fractions with different denominators, a simple rule applies - the denominators of these fractions are reduced to one number, and the action itself is performed with the numbers in the numerator. That is, the fractions receive a common denominator and seem to be combined into one. Finding a common denominator for arbitrary fractions usually comes down to simply multiplying each fraction by the denominator of the other fraction. But in simpler cases, you can immediately find factors that will bring the denominators of the fractions to the same number.
Example of subtracting fractions: 2/3 - 1/7 = 2*7/3*7 - 1*3/7*3 = 14/21 - 3/21 = (14-3)/21 = 11/21
Many adults have already forgotten how to subtract fractions with different denominators, but this action relates to elementary mathematics.
To subtract fractions with different denominators, you need to bring them to a common denominator, that is, find the least common multiple of the denominators, then multiply the numerators by additional factors, equal to the ratio least common multiple and denominator.
Fraction signs are preserved. Once the fractions have the same denominators, you can subtract, and then, if possible, reduce the fraction.
Elena, you decided to repeat school course mathematics?)))
To subtract fractions with different denominators, they must first be reduced to the same denominator and then subtracted. The simplest option: Multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction. We get two fractions with the same denominators. Now we subtract the numerator of the second fraction from the numerator of the first fraction, and they have the same denominator.
For example, three-fifths subtracting two sevenths is equal to twenty-one thirty-fifths subtracting ten thirty-fifths and this is equal to eleven thirty-fifths.
If the denominators are large numbers, then you can find their least common multiple, i.e. a number that will be divisible by one and the other denominator. And bring both fractions to a common denominator (least common multiple)
How to subtract fractions with different denominators is a very simple task - we bring the fractions to a common denominator and then do the subtraction in the numerator.
Many people encounter difficulties when there are integers next to these fractions, so I wanted to show how to do this with the following example:
subtracting fractions from whole part and with different denominators
first we subtract the whole parts 8-5 = 3 (the three remains near the first fraction);
we bring the fractions to a common denominator 6 (if the numerator of the first fraction is greater than the second, we do the subtraction and write it next to the whole part, in our case we move on);
we decompose the whole part 3 into 2 and 1;
We write 1 as a fraction 6/6;
6/6+3/6-4/6 write under common denominator 6 and do the actions in the numerator;
We write down the result found 2 5/6.
It is important to remember that fractions are subtracted if they have the same denominator. Therefore, when we have fractions with different denominators in difference, they simply need to be brought to a common denominator, which is not difficult to do. We simply have to factor the numerator of each fraction and calculate the least common multiple, which must not equal zero. Don’t forget to also multiply the numerators by the resulting additional factors, but here is an example for convenience:
If you want to subtract fractions with unlike denominators, you will first have to find the common denominator for the two fractions. And then subtract the second from the numerator of the first fraction. A new fraction is obtained, with a new meaning.
As far as I remember from the 3rd grade mathematics course, to subtract fractions with different denominators, you first need to calculate the common denominator and reduce it to it, and then simply subtract the numerators from each other and the denominator remains the same.
To subtract fractions with unlike denominators, we first have to find the lowest common denominator of those fractions.
Let's look at an example:
Divide the larger number 25 by the smaller 20. It is not divisible. This means we multiply the denominator 25 by such a number, the resulting sum can be divided by 20. This number will be 4. 25x4=100. 100:20=5. Thus we found the lowest common denominator - 100.
Now we need to find the additional factor for each fraction. To do this, divide the new denominator by the old one.
Multiply 9 by 4 = 36. Multiply 7 by 5 = 35.
Having a common denominator, we carry out the subtraction as shown in the example and get the result.
This lesson will cover addition and subtraction. 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 appear in many topics in the algebra course that 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, and also analyze whole line typical examples.
Let's consider simplest example for ordinary fractions.
Example 1. Add fractions: .
Solution:
Let's remember the rule for adding fractions. To begin, 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 prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two twos and two threes: .
After finding the common denominator, you need to find an additional factor for each fraction (in fact, divide the common denominator by the denominator of the corresponding fraction).
Each fraction is then 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:.
Let us now consider the addition of algebraic fractions with different denominators. First, let's look at 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 the common denominator of 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 lowest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of the given fraction).
3. Multiply the numerators by the corresponding additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with like denominators.
Let us now consider an example with fractions whose denominator contains letter expressions.
Example 3. Add fractions: .
Solution:
Since the letter expressions in both denominators are the same, you should find a common denominator for the numbers. The final common denominator will look like: . Thus, the solution to this example looks like:.
Answer:.
Example 4. Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as the 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 factor 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 let's establish the rules for adding and subtracting fractions with different denominators.
Example 6. Simplify: .
Solution:
Answer:.
Example 7. Simplify: .
Solution:
.
Answer:.
Let us now consider an example in which not two, but three fractions are added (after all, the rules of addition and subtraction for more fractions remain the same).
Example 8. Simplify: .