) and the denominator by the denominator (we get the denominator of the product).
Fraction multiplication formula:
For instance:
Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.
Division of an ordinary fraction by a fraction.
Division of fractions involving a natural number.
It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For instance:
Multiplication of mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper;
- multiply the numerators and denominators of fractions;
- we reduce the fraction;
- if we get an improper fraction, then we convert the improper fraction to a mixed one.
Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the multiplication rule ordinary fractions.
The second way to multiply a fraction by a natural number.
It is more convenient to use the second method of multiplying an ordinary fraction by a number.
Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the above example, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multilevel fractions.
In high school, three-story (or more) fractions are often found. Example:
To bring such a fraction to its usual form, division through 2 points is used:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.
2. In tasks with different kinds fractions - go to the form of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.
§ 87. Addition of fractions.
Fraction addition has many similarities to whole number addition. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), which contains all the units and fractions of units of the terms.
We will consider three cases in sequence:
1. Adding fractions with same denominators.
2. Adding fractions with different denominators.
3. Addition of mixed numbers.
1. Adding 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 by 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.
The drawing shows that if you take the segment AD, then it will be equal to 3/5 AB; but the segment AD is just 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 from the addition of the numerators of the terms, and the denominator remained unchanged.
From here we get the following rule: to add fractions with the same denominator, add their numerators and leave the same denominator.
Let's consider an example:
2. Adding fractions with different denominators.
We 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 have been written; we wrote it here for clarity.
Thus, in order 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.
Add the numbers: 2 3/8 + 3 5/6.
First, we bring the fractional parts of our numbers to a common denominator and rewrite them again:
Now let's add the whole and fractional parts sequentially:
§ 88. Subtraction of fractions.
Subtracting fractions is defined in the same way as subtracting whole numbers. This is an action by which, for a given sum of two terms and one of them, another term is found. Let us consider three cases in sequence:
1. Subtraction of fractions with the same denominator.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.
1. Subtraction of fractions with the same denominator.
Let's consider an example:
13 / 15 - 4 / 15
Take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then a part of the AC of this segment will be 1/15 of AB, and a part of AD of the same segment will correspond to 13/15 AB. Let's put aside the segment ED, equal to 4/15 AB.
We need to subtract 4/15 from 13/15. In the drawing, this means that you need to subtract the segment ED from the segment AD. As a result, the segment AE will remain, which is 9/15 of the segment AB. So we can write:
Our example shows that the numerator of the difference is obtained by subtracting the numerators, but the denominator remains the same.
Therefore, to subtract fractions with the same denominator, you need to subtract the numerator of the subtracted from the numerator of the decremented and leave the same denominator.
2. Subtraction of fractions with different denominators.
Example. 3/4 - 5/8
First, we bring these fractions to the lowest common denominator:
Intermediate 6/8 - 5/8 is written here for clarity, but can be omitted hereafter.
Thus, in order to subtract a fraction from a fraction, you must first bring them to the lowest common denominator, then subtract the numerator of the subtracted from the numerator of the reduced one and sign the common denominator under their difference.
Let's consider an example:
3. Subtraction of mixed numbers.
Example. 10 3/4 - 7 2/3.
Let us bring the fractional parts of the reduced and subtracted to the lowest common denominator:
We subtract the whole from the whole and the fraction from the fraction. But there are times when the fractional part of the subtracted is greater than the fractional part of the reduced. In such cases, you need to take one unit from the whole part of the diminished one, split it into those parts in which the fractional part is expressed, and add it to the fractional part of the diminished one. And then the subtraction will be done 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. Multiplication of a fraction by an integer.
2. Finding the fraction of a given number.
3. Multiplication of an integer by a fraction.
4. Multiplication of 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. Multiplication of a fraction by an integer.
Multiplying a fraction by an integer has the same meaning as multiplying an integer by an integer. Multiplying a fraction (multiplier) by an integer (multiplier) means making up the sum of the same terms, in which each term is equal to the multiplier, 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 whole number. And since an increase in the fraction is achieved either by increasing its numerator
or by decreasing 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 an integer, multiply the numerator by that integer 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 the solution of which you have to find, or calculate, a part of a given number. The difference between these tasks from others is that they give the number of some objects or units of measurement and it is required to find a part of this number, which is also indicated here by a certain fraction. To make it easier to understand, we will first give examples of such problems, and then we will introduce you to the way to solve them.
Objective 1. I had 60 rubles; I spent 1/3 of this money on the purchase of books. How much did the books cost?
Objective 2. The train must travel the distance between cities A and B, equal to 300 km. He has already covered 2/3 of this distance. How many kilometers is it?
Task 3. There are 400 houses in the village, of which 3/4 are brick, the rest are wooden. How many brick houses are there?
Here are some of the many problems of finding a fraction of a given number that we have to face. They are usually called problems of finding the fraction of a given number.
Solution to Problem 1. From 60 rubles. I spent on books 1/3; So, to find the cost of books, you need to divide the number 60 by 3:
Solution to Problem 2. The meaning of the problem is that you need to find 2/3 of 300 km. Let's calculate first 1/3 of 300; this is achieved by dividing 300 km by 3:
300: 3 = 100 (this is 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 (this is 2/3 of 300).
Solution to problem 3. Here you need to determine the number of brick houses, which are 3/4 of 400. Let's find first 1/4 of 400,
400: 4 = 100 (this is 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 (this is 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 an integer by a fraction.
Earlier (§ 26) it was established that the multiplication of integers must be understood as the addition of the same terms (5 x 4 = 5 + 5 + 5 + 5 = 20). In this paragraph (item 1), it was established that multiplying a fraction by an integer means finding the sum of the same terms equal to this fraction.
In both cases, multiplication consisted of finding the sum of the same terms.
We now turn to integer multiplication by a fraction. Here we will meet such, for example, multiplication: 9 2/3. It is quite obvious that the previous definition of multiplication does not fit this case. This can be seen from the fact that we cannot replace such multiplication by adding numbers equal to each other.
Due to this, we will have to give a new definition of multiplication, that is, 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 an integer by a fraction is clarified from the following definition: multiplying an integer (multiplier) by a fraction (multiplier) means finding this fraction of the multiplier.
Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such tasks were solved; so it’s easy to figure out that we’ll end up with 6.
But now an interesting and important question arises: why such seemingly different actions, such as finding the sum of equal numbers and finding the fraction of a number, are called in arithmetic by the same word "multiplication"?
This happens because the previous action (repetition of the number by the summands 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 problems are solved by the same action.
To understand this, consider the following problem: “1 meter 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).
It is possible and several more times, without changing the meaning of the problem, to change the numbers in it, for example, take 9/10 m or 2 3/10 m, etc.
Since these tasks have the same content and differ only in numbers, we call the actions used to solve them by the same word - multiplication.
How is an integer multiplied by a fraction done?
Let's take the numbers encountered in the last problem:
According to the definition, we have to find 3/4 of 50. First we find 1/4 of 50, and then 3/4.
1/4 of the number 50 is 50/4;
3/4 of the number 50 is.
Hence.
Consider another example: 12 5/8 =?
1/8 of 12 is 12/8,
5/8 of the number 12 are.
Hence,
From here we get the rule:
To multiply an integer 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 viewed as a quotient. Therefore, it is useful to compare the rule found with the rule for multiplying a number by a quotient, which was presented in § 38
It must be remembered that before performing the multiplication, you should do (if possible) reductions, For example:
4. Multiplication of 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 factor from the first fraction (multiplication).
Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.
How is the multiplication of a fraction by a fraction done?
Let's take an example: 3/4 times 5/7. This means that you need to find 5/7 of 3/4. Find first 1/7 of 3/4, and then 5/7
1/7 of 3/4 will be expressed like this:
5/7 of 3/4 will be expressed like this:
In this way,
Another example: 5/8 times 4/9.
1/9 of 5/8 is,
4/9 of the number 5/8 is.
In this way, 
Considering these examples, the following rule can be inferred:
To multiply a fraction by a fraction, you need to multiply the numerator by the numerator, and the denominator by the denominator, and make the first product the numerator, and the second, the denominator of the product.
This is the rule in general view can be written like this:
When multiplying, it is necessary to do (if possible) reductions. Let's consider some 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 multiplier, or the factor, or both factors are expressed by mixed numbers, then they are replaced with incorrect fractions. Let's multiply, for example, the mixed numbers: 2 1/2 and 3 1/5. Let's convert each of them into an irregular 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 into improper fractions and then multiply them 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 performing various practical calculations, we use all kinds of fractions. But it must be borne in mind that many quantities allow not any, but natural subdivisions 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 - 3 kopecks. You can take 1/10 of a ruble, it will be "10 kopecks, or a dime. You can take a quarter of a ruble, that is, 25 kopecks, half a ruble, that is, 50 kopecks (fifty kopecks). But they practically do not take, for example , 2/7 rubles because the ruble is not divided into sevenths.
The unit of measurement of weight, that is, the kilogram, allows first of all decimal divisions, 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 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-proven division is the "hundredth" division. Consider a few examples from a wide variety of areas of human practice.
1. The price of books has dropped by 12/100 of the previous price.
Example. The previous price of the book is 10 rubles. It dropped by 1 ruble. 20 kopecks
2. Savings banks pay out to depositors 2/100 of the amount allocated for savings during the year.
Example. The cashier has 500 rubles, 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 the school.
One hundredth of a number is called a percentage..
The word "percent" is borrowed from the Latin language and its root "cent" means one hundred. Together with the preposition (pro centum), this word means "over a hundred." The meaning of this expression follows from the fact that originally in ancient Rome, interest was called money that was paid by the debtor to the lender "for every hundred." The word "cent" is heard in such familiar words: centner (one hundred kilograms), centimeter (said centimeter).
For example, instead of saying that the plant for the past month gave scrap 1/100 of all its products, we will say this: the plant for the past month gave one percent of scrap. Instead of saying: the plant produced 4/100 more than the established plan, we will say: the plant exceeded the plan by 4 percent.
The above examples can be stated differently:
1. The price of books has dropped 12 percent from the previous price.
2. Savings banks pay out to depositors 2 percent per year of the amount allocated for savings.
3. The number of graduates from one school was 5 percent of all students in the school.
To shorten the letter, it is customary to write the% symbol instead of the word "percentage".
However, it should be remembered 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 an integer with this sign.
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 sign instead of a fraction with a denominator of 100:
7. Finding the percentage of a given number.
Objective 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How many birch firewood was there?
The meaning of this task 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. This means that we are faced with the task of finding the fraction of a number. To solve it, we must multiply 200 by 30/100 (the problems of finding the fraction of a number are solved by multiplying the number by a fraction.).
This means that 30% of 200 equals 60.
The fraction 30/100, encountered in this problem, can be reduced by 10. One could have performed this reduction from the very beginning; the solution to the problem would not have changed.
Objective 2. There were 300 children of various ages in the camp. Children 11 years old accounted for 21%, children 12 years old accounted for 61% and finally 13 year old children accounted for 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 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 interest 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 case 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% - for 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 the number 1 200 5 times. Let's do it.
1) How much money was spent on food? The problem says that this expense is 65% of the total earnings, that is, 65/100 of the number 1200. Let's make the calculation:
2) How much money was paid for an apartment with heating? Reasoning 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 was spent on cultural needs?
5) How much money did the worker save?
It is helpful to add the numbers found in these 5 questions to test. 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 problem statement.
We have solved three problems. Despite the fact that these problems dealt with different things (delivery of firewood for the school, the number of children of different ages, the worker's expenses), they were solved in the same way. This happened because in all problems it was necessary to find a few percent of the given numbers.
§ 90. Division of fractions.
When studying the division of fractions, we will consider the following issues:
1. Division of an integer by an integer.
2. Division of a fraction by an integer
3. Division of an integer into a fraction.
4. Division of a fraction into a fraction.
5. Division of mixed numbers.
6. Finding a number for a given fraction.
7. Finding the number by its percentage.
Let's consider them sequentially.
1. Division of an integer by an integer.
As it was indicated in the section of integers, division is an action consisting in the fact that for a given product of two factors (divisible) and one of these factors (divisor), another factor is found.
We looked at the division of an integer by an integer in the department of integers. We encountered two cases of division there: division without remainder, or "entirely" (150: 10 = 15), and division with remainder (100: 9 = 11 and 1 in remainder). We can, therefore, say that in the field of whole numbers, exact division is not always possible, because the dividend is not always the product of the divisor by an integer. After the introduction of multiplication by a fraction, we can consider any case of division of integers possible (only division by zero is excluded).
For example, dividing 7 by 12 means finding a number whose product by 12 would be 7. That number is 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 compose a fraction, the numerator of which is 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, which from multiplication by 3 would give the given product 6/7. Obviously, it should be three times less than this piece. This means that the task set before us was to reduce the fraction 6/7 by 3 times.
We already know that decreasing a fraction can be performed either by decreasing its numerator, or by increasing its denominator. Therefore, one can write:
In this case, the numerator of 6 is divisible by 3, so the numerator should be reduced by 3 times.
Let's take another example: divide 5/8 by 2. Here the numerator of 5 is not evenly divisible by 2, so you have to multiply the denominator by this number:
Based on this, we can formulate a rule: to divide a fraction by an integer, you need to divide the numerator of the fraction by this 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 into a fraction.
Suppose it is required to divide 5 by 1/2, that is, 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 regular fraction, and when multiplying the number for a regular fraction, the product must be less than the multiplicable. To make it clearer, let's write our actions as follows: 5: 1/2 = X , so x 1/2 = 5.
We have to find such a number X , which, if multiplied by 1/2, would give 5. Since multiplying some number by 1/2 - this 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 take another example. Suppose you want to divide 6 by 2/3. Let's first try to find the desired result using the drawing (Fig. 19).
Fig. 19
Let's draw a segment AB, equal to some 6 units, and divide each unit into 3 equal parts. In each unit, three-thirds (3/3) in the entire segment AB is 6 times more, 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 6 units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 whole units. Hence,
How can you get this result without a blueprint using only calculations? We will argue as follows: it is required to divide 6 by 2/3, that is, it is required to answer the question how many times 2/3 are contained in 6. Let's find out first: how many times 1/3 is contained in 6? In a whole unit - 3 thirds, and in 6 units - 6 times more, that is, 18 thirds; to find this number, we must multiply 6 by 3. This means that 1/3 is contained in 6 units 18 times, and 2/3 are contained in 6 not 18 times, but half as many times, that is, 18: 2 = 9. Therefore , when dividing 6 by 2/3, we did the following:
From this we get the rule for dividing an integer by a fraction. To divide an integer into a fraction, you need to multiply this integer by the denominator of the given fraction and, having made 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 viewed as a quotient. Therefore, it is useful to compare the rule found with the rule for dividing a number by a quotient, which was presented in § 38. Note that the same formula was obtained there.
When dividing, abbreviations are possible, for example:
4. Division of a fraction into a fraction.
Suppose you want to divide 3/4 by 3/8. What will be the number that will be the result of division? It will answer the question of 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. The AC segment will be equal to 3/4 of the AB segment. Let us now divide each of the four initial segments in half, then the AB segment will be divided into 8 equal parts and each such part will be equal to 1/8 of the AB segment. 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 the segment equal to 3/8 is contained in the segment equal to 3/4 exactly 2 times; hence, the result of division can be written as follows:
3 / 4: 3 / 8 = 2
Let's take another example. Let's divide 15/16 by 3/32:
We can reason like this: you 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 into improper fractions, and then dividing the resulting fractions according to the rules for dividing fractional numbers. Let's consider an example:
Let's convert the 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 by the rule of division of fractions.
6. Finding a number for a given fraction.
Among the various problems on fractions, sometimes there are 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 with respect to the problem of finding the fraction of a given number; there a number was given and it was required to find a certain 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.
Objective 1. On the first day, the glaziers glazed 50 windows, which is 1/3 of all windows in 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 windows in the house, which means that there are 3 times more windows in total, i.e.
The house had 150 windows.
Objective 2. The store sold 1,500 kg of flour, which is 3/8 of the store's total flour supply. What was the store's original flour supply?
Solution. It can be seen from the problem statement 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, that is, 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 = 4000 (kg).
The original store 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 for 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 have solved two problems of finding a number from a given fraction. Such problems, as is especially clearly seen from the latter, 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 step like this:
In the future, we will solve the problem of finding a number by its fraction in one action - division.
7. Finding the number by its percentage.
In these tasks, you will need to find a number, knowing a few percent of this number.
Objective 1. At the beginning of this year, I received 60 rubles from a savings bank. income from the amount I put on savings a year ago. How much money did I put in a savings bank? (Cash desks give contributors 2% income per year.)
The meaning of the problem is that a certain amount of money was deposited by me in a savings bank and remained 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 put in?
Therefore, knowing a part of this money, expressed in two ways (in rubles and in fraction), we have to find the entire, so far unknown, amount. This is an ordinary task of finding a number given its fraction. The following tasks are solved by division:
This means that 3000 rubles were put into the savings bank.
Objective 2. The fishermen fulfilled the monthly plan by 64% in two weeks, having harvested 512 tons of fish. What was their plan?
It is known from the problem statement that the fishermen have fulfilled part of the plan. This part is equal to 512 tons, which is 64% of the plan. We do not 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 tasks are solved by dividing:
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 the passing conductor what part of the way they had already passed. To this the conductor replied: "We have already covered 30% of the entire route." What is the distance from Riga to Moscow?
It can be seen from the problem statement that 30% of the route from Riga to Moscow is 276 km. We need to find the entire distance between these cities, that is, for a given part, find the whole:
§ 91. Mutually reciprocal numbers. Replacing division by multiplication.
Take the fraction 2/3 and move the numerator to the denominator, so you get 3/2. We got the inverse of this fraction.
In order to get the inverse of the given fraction, 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 the reciprocal of any fraction. For instance:
3/4, reverse 4/3; 5/6, reverse 6/5
Two fractions with 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 which fraction will be the inverse of 1/2. Obviously, it will be 2/1, or just 2. Looking for the inverse of the given fraction, we got an integer. And this case is not an isolated one; on the contrary, for all fractions with numerator 1 (one), integers will be inverse, for example:
1/3, reverse 3; 1/5, reverse 5
Since when looking for reciprocal fractions, we also met with integers, in what follows we will talk not about reciprocal fractions, but about reciprocal numbers.
Let's figure out how to write the reciprocal of an integer. For fractions, this can be solved simply: you need to put the denominator in the place of the numerator. In the same way, you can get the reciprocal for an integer, since any integer can have a denominator 1. Hence, the reciprocal 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 thought can be expressed in another way: the inverse of a given number is obtained by dividing one by a given number. This statement is true not only for integers, but also for fractions. Indeed, if we want to write the reciprocal of the 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. Suppose you need to find the inverse of 8.
Let us denote it by 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 mutually reciprocal numbers in order to slightly supplement the information on the division of fractions.
When we divide the number 6 by 3/5, then we do the following:
Pay close attention to the expression and compare it to 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 we give below fully support this conclusion.
In the middle and high school course, students studied the topic "Fractions". However, this concept is much broader than given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.
What is a fraction?
It so happened historically that fractional numbers appeared because of the need to measure. As practice shows, there are often examples for determining the length of a segment, the volume of a rectangular rectangle.
Initially, students are introduced to such a concept as a share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of a watermelon. This one part of eight is called a share.
A share equal to ½ of any value is called a half; ⅓ - third; ¼ - a quarter. Entries like 5/8, 4/5, 2/4 are called common fractions. An ordinary fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A fractional bar can be drawn as either a horizontal or a slanted line. In this case, it stands for the division sign.
The denominator represents how many equal shares the value, object is divided into; and the numerator is how many equal shares are taken. The numerator is written above the fractional bar, the denominator below it.
It is most convenient to show ordinary fractions on a coordinate ray. If a single segment is divided into 4 equal parts, each part is designated by a Latin letter, then as a result you can get an excellent visual material. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.
Varieties of fractions
Fractions are common, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.
A proper fraction is a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer part and a fractional part. For example, 1½. 1 - integer part, ½ - fractional. However, if you need to perform some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.
A correct fractional expression is always less than one, and an incorrect one is always greater than or equal to 1.
As for this expression, they understand a record in which any number is represented, the denominator of the fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the integer part in the decimal notation will be zero.
To burn decimal, you must first write the integer part, separate it from the fractional part with a comma, and then write the fractional expression. It must be remembered that after the comma the numerator must contain as many numeric characters as there are zeros in the denominator.
Example. Represent the fraction 7 21 / 1000 in decimal notation.
Algorithm for converting an improper fraction to a mixed number and vice versa
It is incorrect to write down an improper fraction in the answer of the problem, so it must be converted to a mixed number:
- divide the numerator by the existing denominator;
- v specific example incomplete quotient - whole;
- and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.
Example. Convert improper fraction to mixed number: 47 / 5 .
Solution. 47: 5. The incomplete quotient is 9, the remainder = 2. Hence, 47 / 5 = 9 2 / 5.
Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:
- the integer part is multiplied by the denominator of the fractional expression;
- the resulting product is added to the numerator;
- the result is written in the numerator, the denominator remains unchanged.
Example. Express the number in mixed form as an improper fraction: 9 8 / 10 .
Solution. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.
Answer: 98 / 10.
Multiplication of ordinary fractions
You can perform various algebraic operations on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product of fractional numbers with the same denominators.
It happens that after finding the result, you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, it cannot be said that an improper fraction in the answer is a mistake, but it is also difficult to call it the correct answer.
Example. Find the product of two ordinary fractions: ½ and 20/18.
As can be seen from the example, after finding the product, a reducible fractional notation is obtained. Both the numerator and the denominator in this case are divisible by 4, and the result is the answer 5 / 9.
Multiplying decimal fractions
The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:
- two decimal fractions must be written under each other so that the rightmost digits are one under the other;
- you need to multiply the written numbers, despite the commas, that is, as natural numbers;
- count the number of digits after the comma in each of the numbers;
- in the result obtained after multiplication, you need to count as many digital characters on the right as are contained in the sum in both factors after the decimal point, and put a separating sign;
- if there are fewer digits in the product, then so many zeros must be written in front of them to cover this number, put a comma and assign an integer part equal to zero.
Example. Calculate the product of two decimals: 2.25 and 3.6.
Solution.
Multiplication of mixed fractions
To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:
- convert mixed numbers to improper fractions;
- find the product of numerators;
- find the product of the denominators;
- write down the result;
- simplify the expression as much as possible.
Example. Find the product of 4½ and 6 2 / 5.
Multiplying a number by a fraction (fractions by a number)
In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply by a fraction.
So, to find the product of a decimal fraction and a natural number, you need:
- write the number under the fraction so that the rightmost digits are one above the other;
- find the work, despite the comma;
- in the result obtained, separate the integer part from the fractional part using a comma, counting to the right the number of characters that is after the decimal point in the fraction.
To multiply an ordinary fraction by a number, you should find the product of the numerator and the natural factor. If the answer is a reducible fraction, it should be converted.
Example. Calculate the product of 5 / 8 and 12.
Solution. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.
Answer: 7 1 / 2.
As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.
Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the result as much as possible.
Example. Find the product of 9 5 / 6 and 9.
Solution. 9 5 / 6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45 / 6 \u003d 81 + 7 3 / 6 \u003d 88 1 / 2.
Answer: 88 1 / 2.
Multiplication by factors 10, 100, 1000 or 0.1; 0.01; 0.001
The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digit characters as there are zeros in the multiplier after one.
Example 1. Find the product of 0.065 and 1000.
Solution. 0.065 x 1000 = 0065 = 65.
Answer: 65.
Example 2. Find the product of 3.9 and 1000.
Solution. 3.9 x 1000 = 3.900 x 1000 = 3900.
Answer: 3900.
If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma to the left in the resulting product by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written in front of a natural number.
Example 1. Find the product of 56 and 0.01.
Solution. 56 x 0.01 = 0056 = 0.56.
Answer: 0,56.
Example 2. Find the product of 4 and 0.001.
Solution. 4 x 0.001 = 0004 = 0.004.
Answer: 0,004.
So, finding the product of various fractions should not cause difficulties, except perhaps the calculation of the result; In this case, you simply cannot do without a calculator.
In this article, we will analyze multiplication of mixed numbers. First, we will voice the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next, we will talk about the multiplication of a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and an ordinary fraction.
Page navigation.
Multiplication of mixed numbers.
Multiplication of mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.
Let's write down multiplication rule for mixed numbers:
- First, the mixed numbers to be multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule of multiplying a fraction by a fraction.
Consider examples of applying this rule when multiplying a mixed number by a mixed number.
Perform mixed number multiplication and .
First, we represent the multiplied mixed numbers as improper fractions: 
and 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule of multiplication of fractions, we get 
. The resulting fraction is irreducible (see reducible and irreducible fractions), but it is incorrect (see regular and improper fractions), therefore, to get the final answer, it remains to extract the integer part from the improper fraction: .
Let's write the whole solution in one line: .
.
To consolidate the skills of multiplying mixed numbers, consider the solution of another example.
Do the multiplication.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it's time to remember about fraction reduction: we will replace all the numbers in the fraction with their expansions into prime factors, and we will perform the reduction of the same factors.
Multiplication of a mixed number and a natural number
After replacing the mixed number with an improper fraction, multiplying a mixed number and a natural number is reduced to the multiplication of an ordinary fraction and a natural number.
Multiply the mixed number and the natural number 45 .
A mixed number is a fraction, then 
. Let's replace the numbers in the resulting fraction with their expansions into prime factors, make a reduction, after which we select the integer part: .
.
Multiplication of a mixed number and a natural number is sometimes conveniently done using the distributive property of multiplication with respect to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, 
.
Compute the product.
We replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .
Multiplying a mixed number and a common fraction it is most convenient to reduce to the multiplication of ordinary fractions, representing the multiplied mixed number as an improper fraction.
Multiply the mixed number by the common fraction 4/15.
Replacing the mixed number with a fraction, we get 
.
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Multiplication of fractional numbers
§ 140. Definitions. 1) The multiplication of a fractional number by an integer is defined in the same way as the multiplication of integers, namely: to multiply some number (multiplier) by an integer (factor) means to make a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.
So multiplying by 5 means finding the sum:
2) To multiply some number (multiplier) by a fraction (multiplier) means to find this fraction of the multiplicand.
Thus, finding a fraction of a given number, which we considered before, we will now call multiplication by a fraction.
3) To multiply some number (multiplier) by a mixed number (factor) means to multiply the multiplicand first by the integer of the factor, then by the fraction of the factor, and add the results of these two multiplications together.
For instance:
The number obtained after multiplication is in all these cases called product, i.e., in the same way as when multiplying integers.
From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.
§ 141. Expediency of these definitions. To understand the expediency of introducing the last two definitions of multiplication into arithmetic, let us take the following problem:
Task. The train, moving evenly, travels 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?
If we had remained with that one definition of multiplication, which is indicated in the arithmetic of integers (addition of equal terms), then our problem would have three various solutions, namely:
If the given number of hours is an integer (for example, 5 hours), then to solve the problem, 40 km must be multiplied by this number of hours.
If a given number of hours is expressed as a fraction (for example, hours), then you will have to find the value of this fraction from 40 km.
Finally, if the given number of hours is mixed (for example, hours), then it will be necessary to multiply 40 km by an integer contained in the mixed number, and add to the result such a fraction from 40 km as is in the mixed number.
The definitions we have given allow us to give one general answer to all these possible cases:
40 km must be multiplied by the given number of hours, whatever it may be.
Thus, if the problem is presented in general form as follows:
A train moving uniformly travels v km per hour. How many kilometers will the train cover in t hours?
then, whatever the numbers v and t, we can express one answer: the desired number is expressed by the formula v · t.
Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, to find 5% (i.e. five hundredths) of a given number means the same as multiplying the given number by or by; finding 125% of a given number is the same as multiplying that number by or by , etc.
§ 142. A note about when a number increases and when it decreases from multiplication.
From multiplication by a proper fraction, the number decreases, and from multiplication by an improper fraction, the number increases if this improper fraction is greater than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so,.
§ 143. Derivation of multiplication rules.
1) Multiplying a fraction by an integer. Let the fraction be multiplied by 5. This means to increase by 5 times. To increase a fraction by 5, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).
So:
Rule 1. To multiply a fraction by an integer, you must multiply the numerator by this integer, and leave the denominator the same; instead, you can also divide the denominator of the fraction by the given integer (if possible), and leave the numerator the same.
Comment. The product of a fraction and its denominator is equal to its numerator.
So:
Rule 2. To multiply an integer by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to the multiplication of a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:
Thus, the three rules now stated are contained in one, which can be expressed in general terms as follows:
4) Multiplication of mixed numbers.
Rule 4. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For instance:
§ 144. Reduction in multiplication. When multiplying fractions, if possible, a preliminary reduction should be done, as can be seen from the following examples:
Such a reduction can be done because the value of a fraction will not change if the numerator and denominator are reduced by the same number of times.
§ 145. Change of product with change of factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .
So, if in the example:
in order to multiply several fractions, it is necessary to multiply their numerators among themselves and the denominators among themselves and make the first product the numerator and the second the denominator of the product.
Comment. This rule can also be applied to such products in which some factors of the number are integer or mixed, if only we consider the whole number as a fraction whose denominator is one, and we turn mixed numbers into improper fractions. For instance:
§ 147. Basic properties of multiplication. Those properties of multiplication that we have indicated for integers (§ 56, 57, 59) also belong to the multiplication of fractional numbers. Let's specify these properties.
1) The product does not change from changing the places of the factors.
For instance:
Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their members differ only in the order of the integer factors, and the product of integers does not change when the factors change places.
2) The product will not change if any group of factors is replaced by their product.
For instance:
The results are the same.
From this property of multiplication, one can deduce the following conclusion:
to multiply some number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, and so on.
For instance:
3) The distributive law of multiplication (with respect to addition). To multiply the sum by some number, you can multiply each term by this number separately and add the results.
This law has been explained by us (§ 59) as applied to whole numbers. It remains true without any changes for fractional numbers.
Let us show, in fact, that the equality
(a + b + c + .)m = am + bm + cm + .
(the distributive law of multiplication with respect to addition) remains true even when the letters mean fractional numbers. Let's consider three cases.
1) Suppose first that the factor m is an integer, for example m = 3 (a, b, c are any numbers). According to the definition of multiplication by an integer, one can write (limited for simplicity to three terms):
(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).
On the basis of the associative law of addition, we can omit all brackets on the right side; applying the commutative law of addition, and then again the combination law, we can obviously rewrite the right-hand side as follows:
(a + a + a) + (b + b + b) + (c + c + c).
(a + b + c) * 3 = a * 3 + b * 3 + c * 3.
Hence, the distributive law in this case is confirmed.
Multiplication and division of fractions
Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.
To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the "inverted" second.
From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what exactly will not happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.
By definition we have:
Multiplication of fractions with an integer part and negative fractions
If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:
- We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.
Task. Find the value of the expression:
We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:
Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the value of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what is left of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.
There is simply no other reason to reduce fractions, so the right decision the previous task looks like this:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
Multiplication of fractions.
To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.
Multiplying a fraction by a fraction.
To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.
Let's consider an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.
Multiplying a fraction by a number.
Let's start with the rule any number can be represented as a fraction \(\bf n = \frac \) .
Let's use this rule for multiplication.
The improper fraction \(\frac = \frac = \frac + \frac = 2 + \frac = 2\frac \\\) was converted to a mixed fraction.
In other words, When multiplying a number by a fraction, multiply the number by the numerator and leave the denominator unchanged. Example:
Multiplication of mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied with the denominator.
Multiplication of reciprocal fractions and numbers.
Related questions:
How to multiply a fraction by a fraction?
Answer: the product of ordinary fractions is the multiplication of the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter if the denominators of fractions are the same or different, multiplication occurs according to the rule for finding the product of the numerator with the numerator, the denominator with the denominator.
How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction to an improper fraction and then find the product according to the rules of multiplication.
How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac \times \frac \) b) \(\frac \times \frac \)
Example #2:
Calculate the product of a number and a fraction: a) \(3 \times \frac \) b) \(\frac \times 11\)
Example #3:
Write the reciprocal of the fraction \(\frac \)?
Answer: \(\frac = 3\)
Example #4:
Calculate the product of two reciprocals: a) \(\frac \times \frac \)
Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?
Solution:
a) Let's use an example to answer the first question. The fraction \(\frac \) is correct, its reciprocal will be equal to \(\frac \) - an improper fraction. The answer is no.
b) in almost all enumerations of fractions, this condition is not met, but there are some numbers that fulfill the condition of being an improper fraction at the same time. For example, the improper fraction is \(\frac \) , its reciprocal is \(\frac \). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.
c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac \), then its reciprocal will be \(\frac \). The fraction \(\frac \) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac = \frac = 1\). The number 1 is a natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.
Example #6:
Perform the product of mixed fractions: a) \(4 \times 2\frac \) b) \(1\frac \times 3\frac \)
Solution:
a) \(4 \times 2\frac = \frac \times \frac = \frac = 11\frac \\\\ \)
b) \(1\frac \times 3\frac = \frac \times \frac = \frac = 4\frac \)
Example #7:
Can two reciprocal numbers be simultaneously mixed numbers?
Let's look at an example. Let's take a mixed fraction \(1\frac \), find its reciprocal, for this we translate it into an improper fraction \(1\frac = \frac \) . Its reciprocal will be equal to \(\frac \) . The fraction \(\frac \) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.
Multiplying a decimal by a natural number
Lesson presentation
Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested this work please download the full version.
- In a fun way, introduce students to the rule of multiplying a decimal fraction by a natural number, by a bit unit and the rule of expressing a decimal fraction as a percentage. Develop the ability to apply the acquired knowledge in solving examples and problems.
- Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their work and the work of each other.
- To cultivate interest in mathematics, activity, mobility, ability to communicate.
Equipment: interactive board, a poster with a cyphergram, posters with mathematicians' statements.
- Organizing time.
- Oral counting is a generalization of previously studied material, preparation for the study of new material.
- Explanation of new material.
- Home assignment.
- Mathematical physical education.
- Generalization and systematization of the acquired knowledge in a playful way with the help of a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not spend it alone, but with my friend. And my friend is also unusual, now you will see him. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, friend? Komposha replies: "My name is Komposha." Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is posted on the board with an oral account for adding and subtracting decimal fractions, as a result of which the guys get the following code 523914687. )
Komposha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is keyword topics of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how multiplication is done natural numbers. Today we are going to look at multiplication. decimal numbers to a natural number. The multiplication of a decimal fraction by a natural number can be considered as the sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 3 = 5.21 + 5, 21 + 5.21 = 15.63 So 5.21 3 = 15.63. Representing 5.21 as an ordinary fraction of a natural number, we get
And in this case, we got the same result of 15.63. Now, ignoring the comma, let's take the number 521 instead of the number 5.21 and multiply by the given natural number. Here we must remember that in one of the factors the comma is moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now, in this example, we will move the comma to the left by two digits. Thus, by how many times one of the factors was increased, the product was reduced by so many times. Based on the similar points of these methods, we draw a conclusion.
To multiply a decimal by a natural number, you need:
1) ignoring the comma, perform the multiplication of natural numbers;
2) in the resulting product, separate with a comma on the right as many characters as there are in a decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 3 = 15.63 and 7.624 15 = 114.34. After I show multiplication by a round number 12.6 50 \u003d 630. Next, I turn to the multiplication of a decimal fraction by a bit unit. I show the following examples: 7.423 100 \u003d 742.3 and 5.2 1000 \u003d 5200. So, I introduce the rule for multiplying a decimal fraction by a bit unit:
To multiply a decimal fraction by bit units 10, 100, 1000, etc., it is necessary to move the comma to the right in this fraction by as many digits as there are zeros in the bit unit record.
I end the explanation with the expression of a decimal fraction as a percentage. I enter the rule:
To express a decimal as a percentage, multiply it by 100 and add the % sign.
I give an example on a computer 0.5 100 = 50 or 0.5 = 50%.
4. At the end of the explanation, I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, to consolidate the topic, we do a mathematical physical education session together with Komposha. Everyone stands up, shows the class the solved examples and they must answer whether the example is correct or incorrect. If the example is solved correctly, then they raise their hands above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and knead their fingers.
6. And now you have a little rest, you can solve the tasks. Open your textbook to page 205, № 1029. in this task it is necessary to calculate the value of expressions:
Tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when fully assembled, sails away.
Solving this task on a computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the Kazakh land from the Baikonur cosmodrome take off to the stars spaceships. Near Baikonur, Kazakhstan is building its new Baiterek cosmodrome.
How far will a car travel in 4 hours if the speed of the car is 74.8 km/h.
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Multiplication and division of fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:
For instance:
Everything is extremely simple. And please don't look for a common denominator! Don't need it here...
To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:
For instance:
If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For instance:
In high school, you often have to deal with three-story (or even four-story!) fractions. For instance:
How to bring this fraction to a decent form? It's very simple! Use division through two points:
But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:
then divide-multiply in order, left to right!
And very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:
The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.
That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Note practical advice, and they (errors) will be less!
Practical Tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.
2. In examples with different types of fractions - go to ordinary fractions.
3. We reduce all fractions to the stop.
4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.
Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.
So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. Only Then look at the answers.
Calculate:
Did you decide?
Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.
0; 17/22; 3/4; 2/5; 1; 25.
And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.
If you like this site ...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.
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