Addition of mixed fractions with different denominators. How to subtract fractions with different denominators
Fractional expressions are difficult for a child to understand. Most people have difficulties with . When studying the topic "addition of fractions with integers", the child falls into a stupor, finding it difficult to solve the task. In many examples, a series of calculations must be performed before an action can be performed. For example, convert fractions or translate improper fraction to the correct one.
Explain to the child clearly. Take three apples, two of which will be whole, and the third will be cut into 4 parts. Separate one slice from the cut apple, and put the remaining three next to two whole fruits. We get ¼ apples on one side and 2 ¾ on the other. If we combine them, we get three whole apples. Let's try to reduce 2 ¾ apples by ¼, that is, remove one more slice, we get 2 2/4 apples.
Let's take a closer look at actions with fractions, which include integers:
First, let's recall the calculation rule for fractional expressions with a common denominator:
At first glance, everything is easy and simple. But this applies only to expressions that do not require conversion.
How to find the value of an expression where the denominators are different
In some tasks, it is necessary to find the value of an expression where the denominators are different. Consider a specific case:
3 2/7+6 1/3
Find the value of this expression, for this we find a common denominator for two fractions.
For numbers 7 and 3, this is 21. We leave the integer parts the same, and reduce the fractional parts to 21, for this we multiply the first fraction by 3, the second by 7, we get:
6/21+7/21, do not forget that whole parts are not subject to conversion. As a result, we get two fractions with one denominator and calculate their sum:
3 6/21+6 7/21=9 15/21
What if the result of addition is an improper fraction that already has an integer part:
2 1/3+3 2/3
In this case, we add the integer parts and fractional parts, we get:
5 3/3, as you know, 3/3 is one, so 2 1/3+3 2/3=5 3/3=5+1=6
With finding the sum, everything is clear, let's analyze the subtraction:
From all that has been said, the rule of operations on mixed numbers follows, which sounds like this:
- If it is necessary to subtract an integer from a fractional expression, it is not necessary to represent the second number as a fraction, it is enough to operate only on integer parts.
Let's try to calculate the value of expressions on our own:
Let's take a look more example under the letter "m":
4 5/11-2 8/11, the numerator of the first fraction is less than the second. To do this, we take one integer from the first fraction, we get,
3 5/11+11/11=3 whole 16/11, subtract the second from the first fraction:
3 16/11-2 8/11=1 whole 8/11
- Be careful when completing the task, do not forget to convert improper fractions to mixed ones, highlighting the whole part. To do this, it is necessary to divide the value of the numerator by the value of the denominator, then what happened takes the place of the integer part, the remainder will be the numerator, for example:
19/4=4 ¾, check: 4*4+3=19, in the denominator 4 remains unchanged.
Summarize:
Before proceeding with the task related to fractions, it is necessary to analyze what kind of expression it is, what transformations need to be performed on the fraction in order for the solution to be correct. Look for more rational solutions. Don't go the hard way. Plan all the actions, decide first in a draft version, then transfer to a school notebook.
To avoid confusion when solving fractional expressions, it is necessary to follow the sequence rule. Decide everything carefully, without rushing.
Adding and subtracting fractions with the same denominators
Adding and subtracting fractions with different denominators
The concept of the NOC
Bringing fractions to the same denominator
How to add a whole number and a fraction
1 Adding and subtracting fractions with the same denominators
To add fractions with the same denominators, you need to add their numerators, and leave the denominator the same, for example:
To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:
To add mixed fractions, you must separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,
If, when adding the fractional parts, an improper fraction is obtained, we select the integer part from it and add it to the integer part, for example:
2 Adding and subtracting fractions with different denominators
In order to add or subtract fractions with different denominators, you must first bring them to the same denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the LCM (least common multiple). For the numerator of each of the fractions, additional factors are found by dividing the LCM by the denominator of this fraction. We'll look at an example later, after we figure out what an LCM is.
3 Least common multiple (LCM)
The least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both of these numbers without a remainder. Sometimes NOCs can be picked up orally, but more often, especially when working with big numbers, you have to find the LCM in writing, using the following algorithm:
In order to find the LCM of several numbers, you need:
- Decompose these numbers into prime factors
- Take the largest expansion, and write these numbers as a product
- Select in other expansions the numbers that do not occur in the largest expansion (or occur in it a smaller number of times), and add them to the product.
- Multiply all the numbers in the product, this will be the LCM.
For example, let's find the LCM of numbers 28 and 21:
4Reducing fractions to the same denominator
Let's go back to adding fractions with different denominators.
When we reduce fractions to the same denominator, equal to the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:
Thus, in order to bring fractions to one indicator, you must first find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then put additional factors on the numerators of the fractions. You can find them by dividing the common denominator (LCD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.
5How to add a whole number and a fraction
In order to add a whole number and a fraction, you just need to add this number in front of the fraction, and you get a mixed fraction, for example.
§ 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. Addition of fractions with different denominators.
3. Addition of mixed numbers.
1. Addition of fractions with the same denominators.
Consider an example: 1 / 5 + 2 / 5 .
Take the segment AB (Fig. 17), take it as a unit and divide 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.
It can be seen from the drawing that if we take the segment AD, then it will be equal to 3/5 AB; but segment AD is precisely the sum of segments AC and CD. So, we can write:
1 / 5 + 2 / 5 = 3 / 5
Considering these terms and the resulting amount, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.
From this we get the following rule: To add fractions with the same denominators, you must add their numerators and leave the same denominator.
Consider an example:
2. Addition of fractions with different denominators.
Let's add fractions: 3/4 + 3/8 First they need to be reduced to the lowest common denominator:
The intermediate link 6/8 + 3/8 could not have been written; we have written it here for greater clarity.
Thus, to add fractions with different denominators, you must first bring them to the lowest common denominator, add their numerators and sign the common denominator.
Consider an example (we will write additional factors over the corresponding fractions):
3. Addition of mixed numbers.
Let's add the numbers: 2 3 / 8 + 3 5 / 6.
Let us first bring the fractional parts of our numbers to a common denominator and rewrite them again:
Now add the integer and fractional parts in sequence:
§ 88. Subtraction of fractions.
Subtraction of fractions is defined in the same way as subtraction of whole numbers. This is an action by which, given the sum of two terms and one of them, another term is found. Let's consider three cases in turn:
1. Subtraction of fractions with the same denominators.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.
1. Subtraction of fractions with the same denominators.
Consider an example:
13 / 15 - 4 / 15
Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then the AC part of this segment will be 1/15 of AB, and the AD part of the same segment will correspond to 13/15 AB. Let's set aside another segment ED, equal to 4/15 AB.
We need to subtract 4/15 from 13/15. In the drawing, this means that the segment ED must be subtracted from the segment AD. As a result, segment AE will remain, which is 9/15 of segment AB. So we can write:
The example we made shows that the numerator of the difference was obtained by subtracting the numerators, and the denominator remained the same.
Therefore, in order to subtract fractions with the same denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.
2. Subtraction of fractions with different denominators.
Example. 3/4 - 5/8
First, let's reduce these fractions to the smallest common denominator:
The intermediate link 6 / 8 - 5 / 8 is written here for clarity, but it can be skipped in the future.
Thus, in order to subtract a fraction from a fraction, you must first bring them to the smallest common denominator, then subtract the numerator of the subtrahend from the numerator of the minuend and sign the common denominator under their difference.
Consider an example:
3. Subtraction of mixed numbers.
Example. 10 3 / 4 - 7 2 / 3 .
Let's bring the fractional parts of the minuend and the subtrahend to the lowest common denominator:
We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the integer part of the reduced, split it into those parts in which the fractional part is expressed, and add to the fractional part of the reduced. And then the subtraction will be performed in the same way as in the previous example:
§ 89. Multiplication of fractions.
When studying the multiplication of fractions, we will consider the following questions:
1. Multiplying a fraction by an integer.
2. Finding a fraction of a given number.
3. Multiplication of a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding percentages of a given number. Let's consider them sequentially.
1. Multiplying a fraction by an integer.
Multiplying a fraction by an integer has the same meaning as multiplying an integer by an integer. Multiplying a fraction (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. Consequently,
Consideration of this action shows that multiplying a fraction by an integer is equivalent to increasing this fraction as many times as there are units in the integer. And since the increase in the fraction is achieved either by increasing its numerator
or by decreasing its denominator 
, then we can either multiply the numerator by the integer, or divide the denominator by it, if such a division is possible.
From here we get the rule:
To multiply a fraction by an integer, you need to multiply the numerator by this integer and leave the denominator the same, or, if possible, divide the denominator by this number, leaving the numerator unchanged.
When multiplying, abbreviations are possible, for example:
2. Finding a fraction of a given number. There are many problems in which you have to find, or calculate, a part of a given number. The difference between these tasks and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce the method of solving them.
Task 1. I had 60 rubles; 1 / 3 of this money I spent on the purchase of books. How much did the books cost?
Task 2. The train must cover the distance between cities A and B, equal to 300 km. He has already covered 2/3 of that distance. How many kilometers is this?
Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there?
Here are some of the many problems that we have to deal with to find a fraction of a given number. They are usually called problems for finding a fraction of a given number.
Solution of problem 1. From 60 rubles. I spent 1 / 3 on books; So, to find the cost of books, you need to divide the number 60 by 3:
Problem 2 solution. The meaning of the problem is that you need to find 2 / 3 of 300 km. Calculate first 1/3 of 300; this is achieved by dividing 300 km by 3:
300: 3 = 100 (that's 1/3 of 300).
To find two-thirds of 300, you need to double the resulting quotient, that is, multiply by 2:
100 x 2 = 200 (that's 2/3 of 300).
Solution of problem 3. Here you need to determine the number of brick houses, which are 3/4 of 400. Let's first find 1/4 of 400,
400: 4 = 100 (that's 1/4 of 400).
To calculate three quarters from 400, the resulting quotient must be tripled, i.e., multiplied by 3:
100 x 3 = 300 (that's 3/4 of 400).
Based on the solution of these problems, we can derive the following rule:
To find the value of a fraction of a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.
3. Multiplication of a whole number by a fraction.
Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 \u003d 5 + 5 + 5 + 5 \u003d 20). In this paragraph (paragraph 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.
In both cases, the multiplication consisted in finding the sum of identical terms.
Now we move on to multiplying a whole number by a fraction. Here we will meet with such, for example, multiplication: 9 2 / 3. It is quite obvious that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.
Because of this, we will have to give a new definition of multiplication, i.e., in other words, to answer the question of what should be understood by multiplication by a fraction, how this action should be understood.
The meaning of multiplying an integer by a fraction is clear from the following definition: to multiply an integer (multiplier) by a fraction (multiplier) means to find this fraction of the multiplier.
Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it's easy to figure out that we end up with 6.
But now an interesting and important question arises: why such seemingly different actions as finding the sum of equal numbers and finding the fraction of a number are called the same word “multiplication” in arithmetic?
This happens because the previous action (repeating the number with terms several times) and the new action (finding the fraction of a number) give an answer to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by one and the same action.
To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?
This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).
Let's take the same problem, but in it the amount of cloth will be expressed as a fractional number: “1 m of cloth costs 50 rubles. How much will 3/4 m of such 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 .
Consequently.
Consider another example: 12 5 / 8 = ?
1/8 of 12 is 12/8,
5/8 of the number 12 is .
Consequently,
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:
In this way,
Another example: 5/8 times 4/9.
1/9 of 5/8 is ,
4/9 numbers 5/8 are .
In this way, 
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 total number children who were 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 problem statement.
We have solved three problems. Despite the fact that these tasks were about different things (delivery of firewood for the school, the number of children of different ages, the expenses of the worker), they were solved in the same way. This happened because in all tasks it was necessary to find a few percent of the given numbers.
§ 90. Division of fractions.
When studying the division of fractions, we will consider the following questions:
1. Divide an integer by an integer.
2. Division of a fraction by an integer
3. Division of an integer by a fraction.
4. Division of a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number given its fraction.
7. Finding a number by its percentage.
Let's consider them sequentially.
1. Divide an integer by an integer.
As it was indicated in the section of integers, division is the action consisting in the fact that, given the product of two factors (the dividend) and one of these factors (the divisor), another factor is found.
The division of an integer by an integer we considered in the department of integers. We met there two cases of division: division without a remainder, or "entirely" (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 in the remainder). We can therefore say that in the realm of integers, exact division is not always possible, because the dividend is not always the product of the divisor and the integer. After the introduction of multiplication by a fraction, we can consider any case of division of integers as possible (only division by zero is excluded).
For example, dividing 7 by 12 means finding a number whose product times 12 would be 7. This number is the fraction 7/12 because 7/12 12 = 7. Another example: 14: 25 = 14/25 because 14/25 25 = 14.
Thus, to divide an integer by an integer, you need to make a fraction, the numerator of which is equal to the dividend, and the denominator is the divisor.
2. Division of a fraction by an integer.
Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find such a second factor, which from multiplication by 3 would give this work 6/7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6 / 7 by 3 times.
We already know that the reduction of a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore, you can write:
In this case, the numerator 6 is divisible by 3, so the numerator should be reduced by 3 times.
Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:
Based on this, we can state the rule: To divide a fraction by an integer, you need to divide the numerator of the fraction by that integer(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.
3. Division of an integer by a fraction.
Let it be required to divide 5 by 1 / 2, i.e. find a number that, after multiplying by 1 / 2, will give the product 5. Obviously, this number must be greater than 5, since 1 / 2 is a proper fraction, and when multiplying a number by a proper fraction, the product must be less than the multiplicand. To make it clearer, let's write our actions as follows: 5: 1 / 2 = X , so x 1 / 2 \u003d 5.
We must find such a number X , which, when multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is 5, and the whole number X twice as much, i.e. 5 2 \u003d 10.
So 5: 1 / 2 = 5 2 = 10
Let's check: 
Let's consider one more example. Let it be required to divide 6 by 2 / 3 . Let's first try to find the desired result using the drawing (Fig. 19).
Fig.19
Draw a segment AB, equal to 6 of some units, and divide each unit into 3 equal parts. In each unit, three-thirds (3 / 3) in the entire segment AB is 6 times larger, i.e. e. 18/3. We connect with the help of small brackets 18 obtained segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in b units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 integer units. Consequently,
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 have done the following actions:
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 .
Consequently,
Thus, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second and make the first product the numerator and the second the denominator.
Let's write the rule using letters:
When dividing, abbreviations are possible, for example:
5. Division of mixed numbers.
When dividing mixed numbers, they must first be converted into improper fractions, and then the resulting fractions should be divided according to the rules for dividing fractional numbers. Consider an example:
Convert mixed numbers to improper fractions:
Now let's split:
Thus, to divide mixed numbers, you need to convert them to improper fractions and then divide according to the rule for dividing fractions.
6. Finding a number given its fraction.
Among various tasks 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 to the problem of finding a fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number is given and it is required to find this number itself. This idea will become even clearer if we turn to the solution of this type of problem.
Task 1. On the first day, glaziers glazed 50 windows, which is 1 / 3 of all windows of the built house. How many windows are in this house?
Solution. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means that there are 3 times more windows in total, i.e.
The house had 150 windows.
Task 2. The shop sold 1,500 kg of flour, which is 3/8 of the total stock of flour in the shop. What was the store's initial supply of flour?
Solution. It can be seen from the condition of the problem that the sold 1,500 kg of flour make up 3/8 of the total stock; this means that 1/8 of this stock will be 3 times less, i.e., to calculate it, you need to reduce 1500 by 3 times:
1,500: 3 = 500 (that's 1/8 of the stock).
Obviously, the entire stock will be 8 times larger. Consequently,
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. Therefore, the reciprocal of 7 will be 1 / 7, because 7 \u003d 7 / 1; for the number 10 the reverse is 1 / 10 since 10 = 10 / 1
This idea can be expressed in another way: the reciprocal of a given number is obtained by dividing one by the given number. This statement is true not only for integers, but also for fractions. Indeed, if you want to write a number that is the reciprocal of 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. 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.
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. That is:
- 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 choose the same denominator in an easier way, then you just need to perform the steps by multiplying all the denominators in series with 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.
It is impossible to turn an improper fraction into a proper one, 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 integer resulting from 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, we divide the number "nine" by "eight", as a result we get a mixed fraction with an integer and a 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 - this is a value that is equal to the sum of a whole number and a proper ordinary 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 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.
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How to add decimals
It is more convenient to add decimal fractions in a column. To perform addition decimal fractions you have to follow one simple rule:
- The digit must be under the digit, comma under the comma.
As you can see in the example, whole units are under each other, tenths and hundredths are under each other. Now we add the numbers, ignoring the comma. What to do with a comma? The comma is transferred to the place where it stood in the discharge of integers.
Adding fractions with equal denominators
To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction, which will be the total amount.
Adding fractions with different denominators by finding a common multiple
The first thing to pay attention to is the denominators. The denominators are different, are they not divisible by one another, are they prime numbers. First you need to bring to one common denominator, there are several ways to do this:
- 1/3 + 3/4 = 13/12, to solve this example, we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b - LCM (a; b). In this example LCM (3;4)=12. Check: 12:3=4; 12:4=3.
- We multiply the factors and perform the addition of the resulting numbers, we get 13/12 - an improper fraction.
- In order to convert an improper fraction to a proper one, we divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.
Adding fractions using cross multiplication
For adding fractions with different denominators, there is another way according to the “cross by cross” formula. This is a guaranteed way to equalize the denominators, for this you need to multiply the numerators with the denominator of one fraction and vice versa. If you are only on initial stage learning fractions, then this method is the easiest and most accurate, how to get the right result when adding fractions with different denominators.
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