You know what to share natural number a by a natural number b means to find a natural number c that, when multiplied by b, gives the number a. This statement remains true if at least one of the numbers a, b, c is a decimal fraction.
Let's look at a few examples in which the divisor is a natural number.
1.2: 4 = 0.3, since 0.3 * 4 = 1.2;
2.5: 5 = 0.5, since 0.5 * 5 = 2.5;
1: 2 = 0.5, since 0.5 * 2 = 1.
But what to do in cases where division cannot be performed orally?
For example, how do you divide 43.52 by 17?
By increasing the dividend 43.52 by 100 times, we get the number 4,352. Then the value of the expression 4,352: 17 is 100 times greater than the value of the expression 43.52: 17. By dividing with a corner, you can easily establish that 4,352: 17 = 256. Here the dividend is increased by 100 times. So, 43.52: 17 = 2.56. Note that 2.56 * 17 = 43.52, which confirms that the division was performed correctly.
The quotient 2.56 can be obtained differently. We will divide 4352 by 17 with a corner, ignoring the comma. In this case, the comma in the quotient should be placed immediately before the first digit after the decimal point in the dividend is used:
If the dividend is less than the divisor, then the integer part of the quotient is zero. For example:
Let's look at another example. Let's find the quotient 3.1:5. We have:
We stopped the division process because the digits of the dividend ran out and we didn’t get a zero as a remainder. You know that a decimal fraction will not change if any number of zeros are added to it on the right. Then it becomes clear that the numbers of the dividend cannot end. We have:
Now we can find the quotient of two natural numbers when the dividend is not evenly divisible by the divisor. For example, let's find the quotient 31:5. Obviously, the number 31 is not divisible by 5:
We stopped the division process because we ran out of dividend digits. However, if you represent the dividend as a decimal fraction, then the division can be continued.
We have: 31:5 = 31.0:5. Next, let's do the division with a corner:
Therefore, 31:5 = 6.2.
In the previous paragraph, we found out that if the comma is moved to the right by 1, 2, 3, etc. digits, then the fraction will increase by 10, 100, 1,000, etc. times, respectively, and if the comma is moved to the left by 1, 2, 3, etc. digits, then the fraction will decrease by 10, 100, 1,000, etc., respectively etc. times.
Therefore, in cases where the divisor is 10, 100, 1,000, etc., use the following rule.
To divide a decimal fraction by 10, 100, 1,000, etc., you need to move the decimal point in this fraction to the left by 1, 2, 3, etc. digits.
For example: 4.23: 10 = 0.423; 2: 100 = 0.02; 58.63: 1,000 = 0.05863.
So, we learned how to divide a decimal fraction by a natural number.
Let's show how division by a decimal fraction can be reduced to division by a natural number.
$\frac(2)(5) km = 400 m$
,$\frac(20)(50) km = 400 m$
,$\frac(200)(500) km = 400 m$
.We get that
$\frac(2)(5) = \frac(20)(50) = \frac(200)(500)$
Those. 2:5 = 20:50 = 200:500.
This example illustrates the following: if the dividend and divisor are increased simultaneously by 10, 100, 1,000, etc. times, then the quotient will not change .
Let's find the quotient 43.52: 1.7.
Let's increase both the dividend and the divisor by 10 times. We have:
43,52 : 1,7 = 435,2 : 17 .
Let's increase both the dividend and the divisor by 10 times. We have: 43.52: 1.7 = 25.6.
To divide a decimal fraction by a decimal:
1) move the commas in the dividend and divisor to the right by as many digits as there are after the decimal point in the divisor;
2) divide by a natural number.
Example 1 . Vanya collected 140 kg of apples and pears, of which 0.24 were pears. How many kilograms of pears did Vanya collect?
Solution. We have:
$0.24=\frac(24)(100)$
.1) 140: 100 = 1.4 (kg) - is
Apples and pears.
2) 1.4 * 24 = 33.6 (kg) - pears were collected.
Answer: 33.6 kg.
Example 2 . For breakfast, Winnie the Pooh ate 0.7 barrels of honey. How many kilograms of honey were in the barrel if Winnie the Pooh ate 4.2 kg?
Solution. We have:
$0.7=\frac(7)(10)$
.1) 4.2: 7 = 0.6 (kg) - is
Just honey.
2) 0.6 * 10 = 6 (kg) - there was honey in the barrel.
Answer: 6 kg.
Division by a decimal fraction is reduced to division by a natural number.
The rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are in the divisor after the decimal point. After this, divide by a natural number.
Examples.
Divide by decimal fraction:
To divide by a decimal, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are after the decimal point in the divisor, that is, by one digit. We get: 35.1: 1.8 = 351: 18. Now we perform the division with a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To divide decimal fractions, in both the dividend and the divisor we move the decimal point to the right one place: 14.76: 3.6 = 147.6: 36. Now we perform a natural number. Result: 14.76: 3.6 = 4.1.
To divide a natural number by a decimal fraction, you need to move both the dividend and the divisor to the right as many places as there are in the divisor after the decimal point. Since a comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 = 7000: 175. Divide the resulting natural numbers with a corner: 70: 1.75 = 7000: 175 = 40.
4) 0,1218: 0,058
To divide one decimal fraction by another, we move the decimal point to the right in both the dividend and the divisor by as many digits as there are in the divisor after the decimal point, that is, by three decimal places. Thus, 0.1218: 0.058 = 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8
I. To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.
Examples.
Perform division: 1) 96,25: 5; 2) 4,78: 4; 3) 183,06: 45.
Solution.
Example 1) 96,25: 5.
We divide with a “corner” in the same way as natural numbers are divided. After we take down the number 2 (the number of tenths is the first digit after the decimal point in the dividend 96, 2 5), in the quotient we put a comma and continue the division.
Answer: 19,25.
Example 2) 4,78: 4.
We divide as natural numbers are divided. In the quotient we will put a comma as soon as we remove it 7
— the first digit after the decimal point in the dividend 4, 7
8. We continue the division further. When subtracting 38-36 we get 2, but the division is not completed. How do we proceed? We know that zeros can be added to the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 20 by 4. We get 5 - the division is over.
Answer: 1,195.
Example 3) 183,06: 45.
Divide as 18306 by 45. In the quotient we put a comma as soon as we remove the number 0
— the first digit after the decimal point in the dividend 183, 0
6. Just as in example 2), we had to assign zero to the number 36 - the difference between the numbers 306 and 270.
Answer: 4,068.
Conclusion: when dividing a decimal fraction by a natural number in private we put a comma immediately after we take down the figure in the tenths place of the dividend. Please note: all highlighted numbers in red in these three examples belong to the category tenths of the dividend.
II. To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.
Examples.
Perform division: 1) 41,56: 10; 2) 123,45: 100; 3) 0,47: 100; 4) 8,5: 1000; 5) 631,2: 10000.
Solution.
Moving the decimal point to the left depends on how many zeros after the one are in the divisor. So, when dividing a decimal fraction by 10
we will carry over in the dividend comma to the left one digit; when divided by 100
- move the comma left two digits; when divided by 1000
convert to this decimal fraction comma three digits to the left.
Adding decimals is the same as adding whole numbers. Let's see this with examples.
1) 0.132 + 2.354. Let's label the terms one below the other.
Here, adding 2 thousandths to 4 thousandths resulted in 6 thousandths;
from adding 3 hundredths with 5 hundredths the result is 8 hundredths;
from adding 1 tenth with 3 tenths -4 tenths and
from adding 0 integers with 2 integers - 2 integers.
2) 5,065 + 7,83.
There are no thousandths in the second term, so it is important not to make mistakes when labeling terms one after another.
3) 1,2357 + 0,469 + 2,08 + 3,90701.
Here, when adding thousandths, the result is 21 thousandths; we wrote 1 under the thousandths, and added 2 to the hundredths, so in the hundredths place we got the following terms: 2 + 3 + 6 + 8 + 0; in total they give 19 hundredths, we signed 9 under hundredths, and 1 counted as tenths, etc.
Thus, when adding decimal fractions, the following order must be observed: sign the fractions one below the other so that in all terms the same digits are located under each other and all commas are in the same vertical column; To the right of the decimal places of some terms, such a number of zeros are added, at least mentally, so that all terms after the decimal point have the same number of digits. Then perform addition by digits, starting from right side, and in the resulting sum a comma is placed in the same vertical column in which it is located in these terms.
§ 108. Subtraction of decimal fractions.
Subtracting decimals works the same way as subtracting whole numbers. Let's show this with examples.
1) 9.87 - 7.32. Let's sign the subtrahend under the minuend so that units of the same digit are under each other:
2) 16.29 - 4.75. Let's sign the subtrahend under the minuend, as in the first example:
To subtract tenths, you had to take one whole unit from 6 and split it into tenths.
3) 14.0213- 5.350712. Let's sign the subtrahend under the minuend:
The subtraction was performed as follows: since we cannot subtract 2 millionths from 0, we should turn to the nearest digit on the left, i.e., hundred thousandths, but in place of hundred thousandths there is also zero, so we take 1 ten thousandth from 3 ten thousandths and We break it up into hundred thousandths, we get 10 hundred thousandths, of which we leave 9 hundred thousandths in the hundred thousandths category, and we break 1 hundred thousandth into millionths, we get 10 millionths. Thus, in the last three digits we got: millionths 10, hundred thousandths 9, ten thousandths 2. For greater clarity and convenience (so as not to forget), these numbers are written above the corresponding fractional digits of the minuend. Now you can start subtracting. From 10 millionths we subtract 2 millionths, we get 8 millionths; from 9 hundred thousandths we subtract 1 hundred thousandth, we get 8 hundred thousandths, etc.
Thus, when subtracting decimal fractions, the following order is observed: sign the subtrahend under the minuend so that the same digits are located under each other and all commas are in the same vertical column; on the right they add, at least mentally, so many zeros in the minuend or subtrahend so that they have the same number of digits, then they subtract by digits, starting from the right side, and in the resulting difference they put a comma in the same vertical column in which it is located in minuend and subtract.
§ 109. Multiplication of decimal fractions.
Let's look at some examples of multiplying decimal fractions.
To find the product of these numbers, we can reason as follows: if the factor is increased by 10 times, then both factors will be integers and we can then multiply them according to the rules for multiplying integers. But we know that when one of the factors increases several times, the product increases by the same amount. This means that the number that is obtained from multiplying the integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to obtain the true product, the found product must be reduced by 10 times. Therefore, here you will have to multiply by 10 once and divide by 10 once, but multiplying and dividing by 10 is done by moving the decimal point to the right and left by one place. Therefore, you need to do this: in the factor, move the comma to the right one place, this will make it equal to 23, then you need to multiply the resulting integers:
This product is 10 times larger than the true product. Therefore, it must be reduced by 10 times, for which we move the comma one place to the left. Thus, we get
28 2,3 = 64,4.
For verification purposes, you can write a decimal fraction with a denominator and perform the action according to the rule for multiplying ordinary fractions, i.e.
2) 12,27 0,021.
The difference between this example and the previous one is that here both factors are represented as decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, i.e. we will temporarily increase the multiplicand by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1,227 by 21, we get:
1 227 21 = 25 767.
Considering that the resulting product is 100,000 times larger than the true product, we must now reduce it by 100,000 times by properly placing a comma in it, then we get:
32,27 0,021 = 0,25767.
Let's check:
Thus, in order to multiply two decimal fractions, it is enough, without paying attention to commas, to multiply them as whole numbers and in the product to separate as many decimal places with a comma on the right side as there were in the multiplicand and in the multiplier together.
The last example resulted in a product with five decimal places. If such great precision is not required, then the decimal fraction is rounded. When rounding, you should use the same rule as was indicated for integers.
§ 110. Multiplication using tables.
Multiplying decimals can sometimes be done using tables. For this purpose, you can, for example, use those multiplication tables for two-digit numbers, the description of which was given earlier.
1) Multiply 53 by 1.5.
We will multiply 53 by 15. In the table, this product is equal to 795. We found the product 53 by 15, but our second factor was 10 times smaller, which means the product must be reduced by 10 times, i.e.
53 1,5 = 79,5.
2) Multiply 5.3 by 4.7.
First, we find in the table the product of 53 by 47, it will be 2,491. But since we increased the multiplicand and the multiplier by a total of 100 times, the resulting product is 100 times larger than it should be; so we must reduce this product by 100 times:
5,3 4,7 = 24,91.
3) Multiply 0.53 by 7.4.
First, we find in the table the product 53 by 74; it will be 3,922. But since we increased the multiplicand by 100 times, and the multiplier by 10 times, the product increased by 1,000 times; so we now have to reduce it by 1,000 times:
0,53 7,4 = 3,922.
§ 111. Division of decimal fractions.
We will look at dividing decimal fractions in this order:
1. Dividing a decimal fraction by integer,
1. Divide a decimal fraction by a whole number.
1) Divide 2.46 by 2.
We divided by 2 first whole, then tenths and finally hundredths.
2) Divide 32.46 by 3.
32,46: 3 = 10,82.
We divided 3 tens by 3, then began to divide 2 units by 3; since the number of units of the dividend (2) is less than the divisor (3), we had to put 0 in the quotient; further, to the remainder we took 4 tenths and divided 24 tenths by 3; received 8 tenths in the quotient and finally divided 6 hundredths.
3) Divide 1.2345 by 5.
1,2345: 5 = 0,2469.
Here in the quotient the first place is zero integers, since one integer is not divisible by 5.
4) Divide 13.58 by 4.
The peculiarity of this example is that when we received 9 hundredths in the quotient, we discovered a remainder equal to 2 hundredths, we split this remainder into thousandths, got 20 thousandths and completed the division.
Rule. Dividing a decimal fraction by an integer is performed in the same way as dividing integers, and the resulting remainders are converted into decimal fractions, smaller and smaller; Division continues until the remainder is zero.
2. Divide a decimal by a decimal.
1) Divide 2.46 by 0.2.
We already know how to divide a decimal fraction by a whole number. Let's think, is it possible to reduce this new case of division to the previous one? At one time, we considered the remarkable property of a quotient, which consists in the fact that it remains unchanged when the dividend and divisor simultaneously increase or decrease by the same number of times. We could easily divide the numbers given to us if the divisor were an integer. To do this, it is enough to increase it by 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same amount, i.e., 10 times. Then the division of these numbers will be replaced by the division of the following numbers:
Moreover, there will no longer be any need to make any amendments to the particulars.
Let's do this division:
So 2.46: 0.2 = 12.3.
2) Divide 1.25 by 1.6.
We increase the divisor (1.6) by 10 times; so that the quotient does not change, we increase the dividend by 10 times; 12 integers are not divisible by 16, so we write 0 in the quotient and divide 125 tenths by 16, we get 7 tenths in the quotient and the remainder 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Please note to the following:
a) when there are no integers in a particular, then zero integers are written in their place;
b) when, after adding the digit of the dividend to the remainder, a number is obtained that is not divisible by the divisor, then zero is written in the quotient;
c) when, after removing the last digit of the dividend, the division does not end, then, adding zeros to the remainder, the division continues;
d) if the dividend is an integer, then when dividing it by a decimal fraction, it is increased by adding zeros to it.
Thus, to divide a number by a decimal fraction, you need to discard the comma in the divisor, and then increase the dividend by as many times as the divisor increased when discarding the comma in it, and then perform the division according to the rule for dividing a decimal fraction by a whole number.
§ 112. Approximate quotients.
In the previous paragraph, we looked at the division of decimal fractions, and in all the examples we solved the division was completed, i.e., an exact quotient was obtained. However, in most cases, an exact quotient cannot be obtained, no matter how far we continue the division. Here is one such case: divide 53 by 101.
We have already received five digits in the quotient, but the division has not yet ended and there is no hope that it will ever end, since in the remainder we begin to have numbers that have already been encountered before. In the quotient, numbers will also be repeated: it is obvious that after the number 7 the number 5 will appear, then 2, etc. endlessly. In such cases, the division is interrupted and limited to the first few digits of the quotient. This quotient is called close ones. We will show with examples how to perform division.
Let it be necessary to divide 25 by 3. Obviously, an exact quotient, expressed as an integer or a decimal fraction, cannot be obtained from such a division. Therefore, we will look for an approximate quotient:
25: 3 = 8 and remainder 1
The approximate quotient is 8; it is, of course, less than the exact quotient, because there is a remainder 1. To obtain the exact quotient, you need to add the fraction that is obtained by dividing the remainder equal to 1 by 3 to the found approximate quotient, i.e., to 8; this will be a fraction 1/3. This means that the exact quotient will be expressed as a mixed number 8 1/3. Since 1/3 represents correct fraction, i.e. fraction, less than one, then, discarding it, we will allow error, which less than one. The quotient 8 will be approximate quotient up to unity with a disadvantage. If instead of 8 we take 9 in the quotient, then we will also allow an error that is less than one, since we will not add the whole unit, but 2/3. Such a private will approximate quotient to within one with excess.
Let's now take another example. Let’s say we need to divide 27 by 8. Since here we won’t get an exact quotient expressed as an integer, we will look for an approximate quotient:
27: 8 = 3 and remainder 3.
Here the error is equal to 3/8, it is less than one, which means that the approximate quotient (3) was found accurate to one with a disadvantage. Let's continue the division: split the remainder 3 into tenths, we get 30 tenths; divide them by 8.
We got 3 in the quotient in place of tenths and 6 tenths in the remainder. If we limit ourselves to the number 3.3 and discard the remainder 6, then we will allow an error of less than one tenth. Why? Because the exact quotient would be obtained when we added to 3.3 the result of dividing 6 tenths by 8; this division would yield 6/80, which is less than one tenth. (Check!) Thus, if in the quotient we limit ourselves to tenths, then we can say that we have found the quotient accurate to one tenth(with a disadvantage).
Let's continue division to find another decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; divide them by 8.
In the quotient in third place it turned out to be 7 and the remainder 4 hundredths; if we discard them, we will allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases they say that the quotient has been found accurate to one hundredth(with a disadvantage).
In the example we are now looking at, we can get the exact quotient expressed as a decimal fraction. To do this, it is enough to split the last remainder, 4 hundredths, into thousandths and divide by 8.
However, in the vast majority of cases it is impossible to obtain an exact quotient and one has to limit oneself to its approximate values. We will now look at this example:
40: 7 = 5,71428571...
The dots placed at the end of the number indicate that the division is not completed, i.e. the equality is approximate. Usually the approximate equality is written as follows:
40: 7 = 5,71428571.
We took the quotient with eight decimal places. But if such great accuracy is not required, you can limit yourself to only the whole part of the quotient, i.e., the number 5 (more precisely 6); for greater accuracy, one could take into account tenths and take the quotient equal to 5.7; if for some reason this accuracy is insufficient, then you can stop at hundredths and take 5.71, etc. Let’s write out the individual quotients and name them.
The first approximate quotient accurate to one 6.
Second » » » to one tenth 5.7.
Third » » » to one hundredth 5.71.
Fourth » » » to one thousandth 5.714.
Thus, in order to find an approximate quotient accurate to some, for example, 3rd decimal place (i.e., up to one thousandth), stop division as soon as this sign is found. In this case, you need to remember the rule set out in § 40.
§ 113. The simplest problems involving percentages.
After learning about decimals, we'll do some more percent problems.
These problems are similar to those we solved in the fractions department; but now we will write hundredths in the form of decimal fractions, that is, without an explicitly designated denominator.
First of all, you need to be able to easily move from an ordinary fraction to a decimal with a denominator of 100. To do this, you need to divide the numerator by the denominator:
The table below shows how a number with a % (percentage) symbol is replaced by a decimal fraction with a denominator of 100:
Let us now consider several problems.
1. Finding the percentage of a given number.
Task 1. Only 1,600 people live in one village. Number of children school age makes up 25% of the total number of residents. How many school-age children are there in this village?
In this problem you need to find 25%, or 0.25, of 1,600. The problem is solved by multiplying:
1,600 0.25 = 400 (children).
Therefore, 25% of 1,600 is 400.
To clearly understand this task, it is useful to recall that for every hundred of the population there are 25 school-age children. Therefore, to find the number of all school-age children, you can first find out how many hundreds there are in the number 1,600 (16), and then multiply 25 by the number of hundreds (25 x 16 = 400). This way you can check the validity of the solution.
Task 2. Savings banks provide depositors with a 2% return annually. How much income will a depositor receive in a year if he puts in the cash register: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rub.?
In all four cases, to solve the problem you will need to calculate 0.02 of the indicated amounts, i.e. each of these numbers will have to be multiplied by 0.02. Let's do it:
a) 200 0.02 = 4 (rub.),
b) 500 0.02 = 10 (rub.),
c) 750 0.02 = 15 (rub.),
d) 1,000 0.02 = 20 (rub.).
Each of these cases can be verified by the following considerations. Savings banks give investors 2% income, i.e. 0.02 of the amount deposited in savings. If the amount was 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the investor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds there are in a given number, and multiply 2 rubles by this number of hundreds. In example a) there are 2 hundreds, which means
2 2 = 4 (rub.).
In example d) there are 10 hundreds, which means
2 10 = 20 (rub.).
2. Finding a number by its percentage.
Task 1. The school graduated 54 students in the spring, representing 6% of its total enrollment. How many students were there in the school last year? academic year?
Let us first clarify the meaning of this task. The school graduated 54 students, which is 6% of the total number of students, or, in other words, 6 hundredths (0.06) of all students at the school. This means that we know the part of the students expressed by the number (54) and the fraction (0.06), and from this fraction we must find the entire number. Thus, we have before us an ordinary task of finding a number from its fraction (§90, paragraph 6). Problems of this type are solved by division:
This means that there were only 900 students in the school.
It is useful to check such problems by solving the inverse problem, i.e. after solving the problem, you should, at least in your head, solve a problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage of it indicated in the solved problem , namely:
900 0,06 = 54.
Task 2. The family spends 780 rubles on food during the month, which is 65% of the father’s monthly earnings. Determine his monthly salary.
This task has the same meaning as the previous one. It gives part of the monthly earnings, expressed in rubles (780 rubles), and indicates that this part is 65%, or 0.65, of the total earnings. And what you are looking for is all the earnings:
780: 0,65 = 1 200.
Therefore, the required income is 1200 rubles.
3. Finding the percentage of numbers.
Task 1. There are only 6,000 books in the school library. Among them are 1,200 books on mathematics. What percentage of math books make up the total number of books in the library?
We have already considered (§97) problems of this kind and came to the conclusion that to calculate the percentage of two numbers, you need to find the ratio of these numbers and multiply it by 100.
In our problem we need to find the percentage ratio of the numbers 1,200 and 6,000.
Let's first find their ratio, and then multiply it by 100:
Thus, the percentage of the numbers 1,200 and 6,000 is 20. In other words, math books make up 20% of the total number of all books.
To check, let’s solve the inverse problem: find 20% of 6,000:
6 000 0,2 = 1 200.
Task 2. The plant should receive 200 tons of coal. 80 tons have already been delivered. What percentage of coal has been delivered to the plant?
This problem asks what percentage one number (80) is of another (200). The ratio of these numbers will be 80/200. Let's multiply it by 100:
This means that 40% of the coal has been delivered.
Sooner or later, all children at school begin to learn fractions: their addition, division, multiplication and all the possible operations that can be performed with fractions. In order to provide proper assistance to the child, parents themselves should not forget how to divide integers into fractions, otherwise you will not be able to help him in any way, but will only confuse him. If you need to remember this action, but you just can’t put all the information in your head into a single rule, then this article will help you: you will learn to divide a number by a fraction and see clear examples.
How to divide a number into a fraction
Write your example down as a rough draft so you can make notes and erasures. Remember that the integer number is written between the cells, right at their intersection, and fractional numbers are written each in its own cell.
- IN this method you need to turn the fraction upside down, that is, write the denominator into the numerator, and the numerator into the denominator.
- The division sign must be changed to multiplication.
- Now all you have to do is perform the multiplication according to the rules you have already learned: the numerator is multiplied by an integer, but you do not touch the denominator.
Of course, as a result of such an action you will get very big number in the numerator. You cannot leave a fraction in this state - the teacher simply will not accept this answer. Reduce the fraction by dividing the numerator by the denominator. Write the resulting integer to the left of the fraction in the middle of the cells, and the remainder will be the new numerator. The denominator remains unchanged.
This algorithm is quite simple, even for a child. After completing it five or six times, the child will remember the procedure and will be able to apply it to any fractions.
How to divide a number by a decimal
There are other types of fractions - decimals. The division into them occurs according to a completely different algorithm. If you encounter such an example, then follow the instructions:
- First, convert both numbers to decimals. This is easy to do: your divisor is already represented as a fraction, and you separate the natural number being divided with a comma, getting a decimal fraction. That is, if the dividend was 5, you get the fraction 5.0. You need to separate a number by as many digits as there are after the decimal point and divisor.
- After this, you must make both decimal fractions natural numbers. It may seem a little confusing at first, but it's the most quick way division, which will take you seconds after a few practices. The fraction 5.0 will become the number 50, the fraction 6.23 will become 623.
- Do the division. If the numbers are large, or the division will occur with a remainder, do it in a column. This way you can clearly see all the actions of this example. You don't need to put a comma on purpose, as it will appear on its own during the long division process.
This type of division initially seems too confusing, since you need to turn the dividend and divisor into a fraction, and then back into natural numbers. But after a short practice, you will immediately begin to see those numbers that you simply need to divide by each other.
Remember that the ability to correctly divide fractions and whole numbers by them can come in handy many times in life, therefore, know these rules and simple principles the child needs ideally so that in higher grades they do not become a stumbling block, due to which the child cannot solve more complex problems.
