§ 87. Addition of fractions.
Adding fractions has many similarities to adding integers. 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. Adding 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 the fractions: 3/4 + 3/8 First they need to be reduced to the smallest 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 (multiplicand) 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 same denominator 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 there is an interesting and important question: why such at first glance various activities, 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 is the rule in general view can be written like this:
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 measurement for weight, i.e., the kilogram, allows, first of all, 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 "percentage" is borrowed from Latin and its root "cent" means one hundred. Together with the preposition (pro centum), this word means "for a hundred." The meaning of 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 was indicated in the section on 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 unknown date 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 to improper fractions, then divide the resulting fractions 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 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 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, glaziers glazed 50 windows, which is 1 / 3 of all windows of the built house. How many windows are in this house?
Decision. 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 store sold 1,500 kg of flour, which is 3/8 of the store's total stock of flour. What was the store's initial supply of flour?
Decision. 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 (that's 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 it 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 current 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. So 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 given number . This statement is true not only for integers, but also for fractions. Indeed, if you want to write a number, inverse fraction 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.
The study of the issue of subtracting fractions with different denominators is found in the school subject Algebra in the eighth grade and it sometimes makes children difficult 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 action.
First, we calculate the smallest number that is a multiple of both one and the other denominator.
Secondly, we multiply the numerator and denominator of each fraction by a certain number, which will allow us to bring the denominator to the given minimum common denominator.
Thirdly, the subtraction procedure itself takes place, when, as a result, 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 integer 1/6
First you need to bring them to the same denominator, and then subtract them. For example, 1/2 - 1/4 = 2/4 - 1/4 = 1/4. Or, harder, 1/3 - 1/5 = 5/15 - 3/15 = 2/15. Do you need to explain how fractions are reduced to a common denominator?
In operations such as adding or subtracting ordinary fractions with different denominators, a simple rule applies - the denominators of these fractions are reduced to one number, and the operation itself is performed with the numbers in the numerator. That is, fractions get a common denominator and seem to be combined into one. Finding a common denominator for arbitrary fractions usually comes down to simply multiplying each of the fractions by the denominator of the other fraction. But in simpler cases, you can immediately find factors that will bring the denominators of fractions to the same number.
Fraction subtraction example: 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 belongs 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 of the least common multiple and the denominator.
The signs of the fractions are preserved. After the fractions have the same denominators, you can subtract, and then, if possible, reduce the fraction.
Elena, did you decide to repeat the school mathematics course?)))
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. 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 subtract two sevenths is equal to twenty-one thirty-fifths subtracted 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 both one and the other denominator. And bring both fractions to a common denominator (least common multiple)
How to subtract fractions with different denominators the task is very simple - we bring the fractions to a common denominator and then do the subtraction in the numerator.
A lot of people face difficulties when there are integers next to these fractions, so I wanted to show how to do this with the following example:
subtraction of fractions with an integer part and with different denominators
first we subtract the whole parts 8-5 = 3 (the triple 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 subtract and write near the integer part, in our case we move on);
we decompose the integer part 3 into 2 and 1;
1 is written as a fraction 6/6;
6/6+3/6-4/6 we write under the common denominator 6 and do the actions in the numerator;
write down the found result 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 the difference, they need to be brought simply to a common denominator, which is not difficult to do. We just have to factor each fraction's numerator and calculate the least common multiple, which must not be zero. Do not forget to also multiply the numerators by the additional factors obtained, but here is an example for convenience:
If you want to subtract fractions with different denominators, then first you have to find a common denominator for these two fractions. And then subtract the second from the numerator of the first fraction. It turns out a new fraction, with a new value.
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 bring it to it, and then the numerators are simply subtracted from each other and the denominator remains that common.
To subtract fractions with different denominators, we first have to find the smallest common denominator of these fractions.
Let's look at an example:
Divide more 25 to less than 20. Not divisible. So we multiply the denominator 25 by such a number that the resulting sum can be divided by 20. This number will be 4. 25x4 \u003d 100. 100:20=5. Thus, we found the lowest common denominator - 100.
Now we need to find an additional factor for each fraction. To do this, we divide the new denominator by the old one.
Multiply 9 by 4 = 36. Multiply 7 by 5 = 35.
Having a common denominator, we subtract, as shown in the example, and get the result.
In the article, we will show how to solve fractions with simple clear examples. Let's understand what a fraction is and consider solving fractions!
concept fractions is introduced into the course of mathematics starting from the 6th grade of secondary school.
Fractions look like: ±X / Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:
In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, from which 4 parts were taken, i.e. 4/7.
If the part of dividing one number by another is not a whole number, it is written as a fraction.
For example, the expression 4:2 \u003d 2 gives an integer, but 4:7 is not completely divisible, so this expression is written as a fraction 4/7.
In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written with a slash.
If the numerator is less than the denominator, the fraction is correct, if vice versa, it is incorrect. A fraction can contain an integer.
For example, 5 whole 3/4.
This entry means that in order to get the whole 6, one part of four is not enough.
If you want to remember how to solve fractions for 6th grade you need to understand that solving fractions basically comes down to understanding a few simple things.
- A fraction is essentially an expression for a fraction. I.e numeric expression how much of a given value is from one whole. For example, the fraction 3/5 expresses that if we divide something whole into 5 parts and the number of parts or parts of this whole is three.
- A fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2= 1 whole 1/2.
- Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole, but fractional. With them, you can perform all the same operations as with numbers. Counting fractions is not more difficult, and further we will show this with specific examples.
How to solve fractions. Examples.
A variety of arithmetic operations are applicable to fractions.
Bringing a fraction to a common denominator
For example, you need to compare the fractions 3/4 and 4/5.
To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible without remainder by each of the denominators of the fractions
Least common denominator(4.5) = 20
Then the denominator of both fractions is reduced to the lowest common denominator
Answer: 15/20
Addition and subtraction of fractions
If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference of fractions is considered the same way, the only difference is that the numerators are subtracted.
For example, you need to find the sum of fractions 1/2 and 1/3
Now find the difference between the fractions 1/2 and 1/4
Multiplication and division of fractions
Here the solution of fractions is simple, everything is quite simple here:
- Multiplication - numerators and denominators of fractions are multiplied among themselves;
- Division - first we get a fraction, the reciprocal of the second fraction, i.e. swap its numerator and denominator, after which we multiply the resulting fractions.
For example:
On this about how to solve fractions, all. If you have any questions about solving fractions, something is not clear, then write in the comments and we will answer you.
If you are a teacher, it is possible to download the presentation for elementary school(http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) will come in handy.
The numerator, and that by which it is divided is the denominator.
To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.
To calculate the product of fractions, first multiply the numerator of one fractions to another numerator. Write the result to the numerator of the new fractions. Then multiply the denominators as well. Specify the final value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).
To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the steps, first “flip” the divisor, if it’s more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 × 5 = 5; 3 × 1 = 3).
Sources:
- Basic tasks for fractions
Fractional numbers allow you to express in different form the exact value of the quantity. You can do the same with fractions. mathematical operations, as with integers: subtraction, addition, multiplication and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution require reduction of the fractional part of the result.
You will need
- - calculator
Instruction
Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given values will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.
Rewrite them through the separator ":" and continue the usual division.
To get the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.
note
Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.
When writing fractional numbers, the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by a single whole number. AT this example dividing by 2 is possible. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.
As you know from mathematics, a fractional number consists of a numerator and a denominator. The numerator is at the top and the denominator at the bottom.
It is quite simple to perform mathematical operations on the addition or subtraction of fractional quantities with the same denominator. You just need to be able to add or subtract the numbers in the numerator (top), and the same bottom number remains unchanged.
For example, let's take the fractional number 7/9, here:
- the number "seven" on top is the numerator;
- the number "nine" below is the denominator.
Example 1. Addition:
5/49 + 4/49 = (5+4) / 49 =9/49.
Example 2. Subtraction:
6/35−3/35 = (6−3) / 35 = 3/35.
Subtraction of simple fractional values \u200b\u200bthat have a different denominator
To perform a mathematical operation to subtract values that have a different denominator, you must first bring them to a common denominator. When performing this task, it is necessary to adhere to the rule that this common denominator must be the smallest of all options.
Example 3
Given two simple quantities with different denominators (lower numbers): 7/8 and 2/9.
Subtract the second from the first value.
The solution consists of several steps:
1. Find the common lower number, i.e. that which is divisible both by the lower value of the first fraction and the second. This will be the number 72, since it is a multiple of the numbers "eight" and "nine".
2. The bottom digit of each fraction has increased:
- the number "eight" in the fraction 7/8 increased nine times - 8*9=72;
- the number "nine" in the fraction 2/9 has increased eight times - 9*8=72.
3. If the denominator (lower number) has changed, then the numerator (upper number) must also change. According to the existing mathematical rule, the upper figure must be increased by exactly the same amount as the lower one. I.e:
- the numerator "seven" in the first fraction (7/8) is multiplied by the number "nine" - 7*9=63;
- the numerator "two" in the second fraction (2/9) is multiplied by the number "eight" - 2*8=16.
4. As a result of the actions, we got two new values, which, however, are identical to the original ones.
- first: 7/8 = 7*9 / 8*9 = 63/72;
- second: 2/9 = 2*8 / 9*8 = 16/72.
5. Now it is allowed to subtract one fractional number from another:
7/8−2/9 = 63/72−16/72 =?
6. Performing this action, we return to the topic of subtracting fractions with the same lower numbers (denominators). And this means that the subtraction action will be carried out from above, in the numerator, and the lower figure is transferred without changes.
63/72−16/72 = (63−16) / 72 = 47/72.
7/8−2/9 = 47/72.
Example 4
Let's complicate the problem by taking several fractions for solving with different, but multiple digits at the bottom.
Values given: 5/6; 1/3; 1/12; 7/24.
They must be taken away from each other in this sequence.
1. We bring the fractions in the above way to a common denominator, which will be the number "24":
- 5/6 = 5*4 / 6*4 = 20/24;
- 1/3 = 1*8 / 3*8 = 8/24;
- 1/12 = 1*2 / 12*2 = 2/24.
7/24 - we leave this last value unchanged, since the denominator is the total number "24".
2. Subtract all values:
20/24−8/2−2/24−7/24 = (20−8−2−7)/24 = 3/24.
3. Since the numerator and denominator of the resulting fraction are divisible by one number, they can be reduced by dividing by the number "three":
3:3 / 24:3 = 1/8.
4. We write the answer like this:
5/6−1/3−1/12−7/24 = 1/8.
Example 5
Given three fractions with non-multiple denominators: 3/4; 2/7; 1/13.
You need to find the difference.
1. We bring the first two numbers to a common denominator, it will be the number "28":
- ¾ \u003d 3 * 7 / 4 * 7 \u003d 21/28;
- 2/7 = 2*4 / 7*4 = 8/28.
2. Subtract the first two fractions between each other:
¾−2/7 = 21/28−8/28 = (21−8) / 28 = 13/28.
3. Subtract the third given fraction from the resulting value:
4. We bring the numbers to a common denominator. If it is not possible to find the same denominator more than the easy way, then you just need to perform the actions by multiplying successively all the denominators by each other, not forgetting to increase the value of the numerator by the same figure. In this example, we do this:
- 13/28 \u003d 13 * 13 / 28 * 13 \u003d 169/364, where 13 is the lower digit from 5/13;
- 5/13 \u003d 5 * 28 / 13 * 28 \u003d 140/364, where 28 is the lower digit from 13/28.
5. Subtract the resulting fractions:
13/28−5/13 = 169/364−140/364 = (169−140) / 364 = 29/364.
Answer: ¾-2/7-5/13 = 29/364.
Mixed fractional numbers
In the examples discussed above, only proper fractions were used.
As an example:
- 8/9 is a proper fraction;
- 9/8 is wrong.
An improper fraction cannot be turned into a proper fraction, but it is possible to turn it into mixed. Why is the top number (numerator) divided by the bottom number (denominator) to get a number with a remainder. The whole number obtained by division is written down in this way, the remainder is written in the numerator at the top, and the denominator, which is at the bottom, remains the same. To make it clearer, consider specific example:
Example 6
We convert the improper fraction 9/8 into the proper one.
To do this, the number "nine" is divided by "eight", we get as a result mixed fraction with integer and remainder:
9: 8 = 1 and 1/8 (in another way it can be written as 1 + 1/8), where:
- the number 1 is the integer resulting from the division;
- another number 1 - the remainder;
- the number 8 is the denominator, which has remained unchanged.
An integer is also called a natural number.
The remainder and denominator are a new, but already correct fraction.
When writing the number 1, it is written before the correct fraction 1/8.
Subtracting mixed numbers with different denominators
From the above, we give the definition of a mixed fractional number: "Mixed number is a value that is equal to the sum of a whole number and the correct common fraction. In this case, the whole part is called natural number , and the number that is in the remainder is its fractional part».
Example 7
Given: two mixed fractional quantities, consisting of a whole number and a proper fraction:
- the first value is 9 and 4/7, that is, (9 + 4/7);
- the second value is 3 and 5/21, i.e. (3+5/21).
It is required to find the difference between these values.
1. To subtract 3+5/21 from 9+4/7, you must first subtract integer values from each other:
4/7−5/21 = 4*3 / 7*3−5/21 =12/21−5/21 = (12−5) / 21 = 7/21.
3. The result of the difference between two mixed numbers will consist of a natural (integer) number 6 and a proper fraction 7/21 = 1/3:
(9 + 4/7) - (3 + 5/21) = 6 + 1/3.
Mathematicians of all countries have agreed that the "+" sign when writing mixed quantities can be omitted and only the whole number in front of the fraction without any sign can be left.