In the last lesson, we learned how to add and subtract decimal fractions (see the lesson " Adding and subtracting decimal fractions"). At the same time, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.
Unfortunately, with multiplication and division decimal fractions no such effect occurs. In some cases, decimal notation even complicates these operations.
First, let's introduce a new definition. We will meet him quite often, and not only in this lesson.
The significant part of a number is everything between the first and last non-zero digit, including the trailers. We are only talking about numbers, the decimal point is not taken into account.
The digits included in the significant part of the number are called significant digits. They can be repeated and even be equal to zero.
For example, consider several decimal fractions and write out their corresponding significant parts:
- 91.25 → 9125 (significant figures: 9; 1; 2; 5);
- 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
- 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
- 0.0304 → 304 (significant figures: 3; 0; 4);
- 3000 → 3 (there is only one significant figure: 3).
Please note: zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see the lesson “ Decimal Fractions”).
This point is so important, and errors are made here so often that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of a significant part, will proceed, in fact, to the topic of the lesson.
Decimal multiplication
The multiplication operation consists of three consecutive steps:
- For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
- Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We get the significant part of the desired fraction;
- Find out where and by how many digits the decimal point is shifted in the original fractions to obtain the corresponding significant part. Perform reverse shifts on the significant part obtained in the previous step.
Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.
- 0.28 12.5;
- 6.3 1.08;
- 132.5 0.0034;
- 0.0108 1600.5;
- 5.25 10,000.
We work with the first expression: 0.28 12.5.
- Let's write out the significant parts for the numbers from this expression: 28 and 125;
- Their product: 28 125 = 3500;
- In the first multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.
Now let's deal with the expression 6.3 1.08.
- Let's write out the significant parts: 63 and 108;
- Their product: 63 108 = 6804;
- Again, two shifts to the right: by 2 and 1 digits, respectively. In total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no zeros at the end.
We got to the third expression: 132.5 0.0034.
- Significant parts: 1325 and 34;
- Their product: 1325 34 = 45,050;
- In the first fraction, the decimal point goes to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We perform a shift by 5 to the left: 45050 → .45050 = 0.4505. Zero was removed at the end, and added to the front so as not to leave a “bare” decimal point.
The following expression: 0.0108 1600.5.
- We write significant parts: 108 and 16 005;
- We multiply them: 108 16 005 = 1 728 540;
- We count the numbers after the decimal point: in the first number there are 4, in the second - 1. In total - again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.
Finally, the last expression: 5.25 10,000.
- Significant parts: 525 and 1;
- We multiply them: 525 1 = 525;
- The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52 500 (we had to add zeros).
Pay attention to the last example: since the decimal point moves in different directions, the total shift is through the difference. This is very important point! Here's another example:
Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12 500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 digits to the left. As a result, we stepped 2 − 1 = 1 digit to the left.
Decimal division
Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case, there are many subtleties that negate the potential savings.
So let's look at a generic algorithm that is a little longer, but much more reliable:
- Convert all decimals to common fractions. With a little practice, this step will take you a matter of seconds;
- Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the "inverted" second (see the lesson " Multiplication and division of numerical fractions");
- If possible, return the result as a decimal. This step is also fast, because often the denominator already has a power of ten.
Task. Find the value of the expression:
- 3,51: 3,9;
- 1,47: 2,1;
- 6,4: 25,6:
- 0,0425: 2,5;
- 0,25: 0,002.
We consider the first expression. First, let's convert obi fractions to decimals:
We do the same with the second expression. The numerator of the first fraction is again decomposed into factors:
There is an important point in the third and fourth examples: after getting rid of the decimal notation, cancellable fractions appear. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction is a prime number. There is simply nothing to factorize here, so we consider it “blank through”:
Sometimes the result of division is integer(I'm talking about the last example). In this case, the third step is not performed at all.
In addition, when dividing, “ugly” fractions often appear that cannot be converted to decimals. This is where division differs from multiplication, where the results are always expressed in decimal form. Of course, in this case, the last step is again not performed.
Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce ordinary fractions obtained from decimals. Otherwise, it will complicate the inverse problem - representing the final answer again in decimal form.
Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.
In this tutorial, we'll look at each of these operations one by one.
Lesson contentAdding decimals
As we know, a decimal has an integer part and a fractional part. When adding decimals, the integer and fractional parts are added separately.
For example, let's add the decimals 3.2 and 5.3. It is more convenient to add decimal fractions in a column.
First, we write these two fractions in a column, while the integer parts must be under the integer parts, and the fractional ones under the fractional parts. In school, this requirement is called "comma under comma".
Let's write the fractions in a column so that the comma is under the comma:
We begin to add the fractional parts: 2 + 3 \u003d 5. We write down the five in the fractional part of our answer:
Now we add up the integer parts: 3 + 5 = 8. We write the eight in the integer part of our answer:
Now we separate the integer part from the fractional part with a comma. To do this, we again follow the rule "comma under comma":
Got the answer 8.5. So the expression 3.2 + 5.3 is equal to 8.5
In fact, not everything is as simple as it seems at first glance. Here, too, there are pitfalls, which we will now talk about.
Places in decimals
Decimals, like ordinary numbers, have their own digits. These are tenth places, hundredth places, thousandth places. In this case, the digits begin after the decimal point.
The first digit after the decimal point is responsible for the tenths place, the second digit after the decimal point for the hundredths place, the third digit after the decimal point for the thousandths place.
The digits in decimal fractions store some useful information. In particular, they report how many tenths, hundredths, and thousandths are in a decimal.
For example, consider the decimal 0.345
The position where the triple is located is called tenth place
The position where the four is located is called hundredths place
The position where the five is located is called thousandths
Let's look at this figure. We see that in the category of tenths there is a three. This suggests that there are three tenths in the decimal fraction 0.345.
If we add the fractions, and then we get the original decimal fraction 0.345
It can be seen that at first we got the answer, but converted it to a decimal fraction and got 0.345.
When adding decimal fractions, the same principles and rules are followed as when adding ordinary numbers. The addition of decimal fractions occurs by digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.
Therefore, when adding decimal fractions, it is required to follow the rule "comma under comma". A comma under a comma provides the same order in which tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.
Example 1 Find the value of the expression 1.5 + 3.4
First of all, we add the fractional parts 5 + 4 = 9. We write the nine in the fractional part of our answer:
Now we add up the integer parts 1 + 3 = 4. We write down the four in the integer part of our answer:
Now we separate the integer part from the fractional part with a comma. To do this, we again observe the rule "comma under a comma":
Got the answer 4.9. So the value of the expression 1.5 + 3.4 is 4.9
Example 2 Find the value of the expression: 3.51 + 1.22
We write this expression in a column, observing the rule "comma under a comma"
First of all, add the fractional part, namely the hundredths 1+2=3. We write the triple in the hundredth part of our answer:
Now add tenths of 5+2=7. We write down the seven in the tenth part of our answer:
Now add the whole parts 3+1=4. We write down the four in the whole part of our answer:
We separate the integer part from the fractional part with a comma, observing the “comma under the comma” rule:
Got the answer 4.73. So the value of the expression 3.51 + 1.22 is 4.73
3,51 + 1,22 = 4,73
As with ordinary numbers, when adding decimal fractions, . In this case, one digit is written in the answer, and the rest are transferred to the next digit.
Example 3 Find the value of the expression 2.65 + 3.27
We write this expression in a column:
Add hundredths of 5+7=12. The number 12 will not fit in the hundredth part of our answer. Therefore, in the hundredth part, we write the number 2, and transfer the unit to the next bit:
Now we add the tenths of 6+2=8 plus the unit that we got from the previous operation, we get 9. We write the number 9 in the tenth of our answer:
Now add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:
Got the answer 5.92. So the value of the expression 2.65 + 3.27 is 5.92
2,65 + 3,27 = 5,92
Example 4 Find the value of the expression 9.5 + 2.8
Write this expression in a column
We add the fractional parts 5 + 8 = 13. The number 13 will not fit in the fractional part of our answer, so we first write down the number 3, and transfer the unit to the next digit, or rather transfer it to the integer part:
Now we add the integer parts 9+2=11 plus the unit that we got from the previous operation, we get 12. We write the number 12 in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 12.3. So the value of the expression 9.5 + 2.8 is 12.3
9,5 + 2,8 = 12,3
When adding decimal fractions, the number of digits after the decimal point in both fractions must be the same. If there are not enough digits, then these places in the fractional part are filled with zeros.
Example 5. Find the value of the expression: 12.725 + 1.7
Before writing this expression in a column, let's make the number of digits after the decimal point in both fractions the same. The decimal fraction 12.725 has three digits after the decimal point, while the fraction 1.7 has only one. So in the fraction 1.7 at the end you need to add two zeros. Then we get the fraction 1,700. Now you can write this expression in a column and start calculating:
Add thousandths of 5+0=5. We write the number 5 in the thousandth part of our answer:
Add hundredths of 2+0=2. We write the number 2 in the hundredth part of our answer:
Add tenths of 7+7=14. The number 14 will not fit in a tenth of our answer. Therefore, we first write down the number 4, and transfer the unit to the next bit:
Now we add the integer parts 12+1=13 plus the unit that we got from the previous operation, we get 14. We write the number 14 in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 14,425. So the value of the expression 12.725+1.700 is 14.425
12,725+ 1,700 = 14,425
Subtraction of decimals
When subtracting decimal fractions, you must follow the same rules as when adding: “a comma under a comma” and “an equal number of digits after a decimal point”.
Example 1 Find the value of the expression 2.5 − 2.2
We write this expression in a column, observing the “comma under comma” rule:
We calculate the fractional part 5−2=3. We write the number 3 in the tenth part of our answer:
Calculate the integer part 2−2=0. We write zero in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
We got the answer 0.3. So the value of the expression 2.5 − 2.2 is equal to 0.3
2,5 − 2,2 = 0,3
Example 2 Find the value of the expression 7.353 - 3.1
In this expression different amount digits after the decimal point. In the fraction 7.353 there are three digits after the decimal point, and in the fraction 3.1 there is only one. This means that in the fraction 3.1, two zeros must be added at the end to make the number of digits in both fractions the same. Then we get 3,100.
Now you can write this expression in a column and calculate it:
Got the answer 4,253. So the value of the expression 7.353 − 3.1 is 4.253
7,353 — 3,1 = 4,253
As with ordinary numbers, sometimes you will have to borrow one from the adjacent bit if subtraction becomes impossible.
Example 3 Find the value of the expression 3.46 − 2.39
Subtract hundredths of 6−9. From the number 6 do not subtract the number 9. Therefore, you need to take a unit from the adjacent digit. Having borrowed one from the neighboring digit, the number 6 turns into the number 16. Now we can calculate the hundredths of 16−9=7. We write down the seven in the hundredth part of our answer:
Now subtract tenths. Since we took one unit in the category of tenths, the figure that was located there decreased by one unit. In other words, the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3=0. We write zero in the tenth part of our answer:
Now subtract the integer parts 3−2=1. We write the unit in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 1.07. So the value of the expression 3.46−2.39 is equal to 1.07
3,46−2,39=1,07
Example 4. Find the value of the expression 3−1.2
This example subtracts a decimal from an integer. Let's write this expression in a column so that whole part decimal fraction 1.23 was under the number 3
Now let's make the number of digits after the decimal point the same. To do this, after the number 3, put a comma and add one zero:
Now subtract tenths: 0−2. Do not subtract the number 2 from zero. Therefore, you need to take a unit from the adjacent digit. By borrowing one from the adjacent digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write down the eight in the tenth part of our answer:
Now subtract the whole parts. Previously, the number 3 was located in the integer, but we borrowed one unit from it. As a result, it turned into the number 2. Therefore, we subtract 1 from 2. 2−1=1. We write the unit in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 1.8. So the value of the expression 3−1.2 is 1.8
Decimal multiplication
Multiplying decimals is easy and even fun. To multiply decimals, you need to multiply them like regular numbers, ignoring the commas.
Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in both fractions, then count the same number of digits on the right in the answer and put a comma.
Example 1 Find the value of the expression 2.5 × 1.5
We multiply these decimal fractions as ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:
We got 375. In this number, it is necessary to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in fractions of 2.5 and 1.5. In the first fraction there is one digit after the decimal point, in the second fraction there is also one. A total of two numbers.
We return to the number 375 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 3.75. So the value of the expression 2.5 × 1.5 is 3.75
2.5 x 1.5 = 3.75
Example 2 Find the value of the expression 12.85 × 2.7
Let's multiply these decimals, ignoring the commas:
We got 34695. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 12.85 and 2.7. In the fraction 12.85 there are two digits after the decimal point, in the fraction 2.7 there is one digit - a total of three digits.
We return to the number 34695 and begin to move from right to left. We need to count three digits from the right and put a comma:
Got the answer 34,695. So the value of the expression 12.85 × 2.7 is 34.695
12.85 x 2.7 = 34.695
Multiplying a decimal by a regular number
Sometimes there are situations when you need to multiply a decimal by common number.
To multiply a decimal and an ordinary number, you need to multiply them, regardless of the comma in the decimal. Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then in the answer, count the same number of digits to the right and put a comma.
For example, multiply 2.54 by 2
We multiply the decimal fraction 2.54 by the usual number 2, ignoring the comma:
We got the number 508. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. The fraction 2.54 has two digits after the decimal point.
We return to the number 508 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 5.08. So the value of the expression 2.54 × 2 is 5.08
2.54 x 2 = 5.08
Multiplying decimals by 10, 100, 1000
Multiplying decimals by 10, 100, or 1000 is done in the same way as multiplying decimals by regular numbers. It is necessary to perform the multiplication, ignoring the comma in the decimal fraction, then in the answer, separate the integer part from the fractional part, counting the same number of digits on the right as there were digits after the decimal point in the decimal fraction.
For example, multiply 2.88 by 10
Let's multiply the decimal fraction 2.88 by 10, ignoring the comma in the decimal fraction:
We got 2880. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.88. We see that in the fraction 2.88 there are two digits after the decimal point.
We return to the number 2880 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 28.80. We discard the last zero - we get 28.8. So the value of the expression 2.88 × 10 is 28.8
2.88 x 10 = 28.8
There is a second way to multiply decimal fractions by 10, 100, 1000. This method is much simpler and more convenient. It consists in the fact that the comma in the decimal fraction moves to the right by as many digits as there are zeros in the multiplier.
For example, let's solve the previous example 2.88×10 in this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 2.88 we move the decimal point to the right by one digit, we get 28.8.
2.88 x 10 = 28.8
Let's try to multiply 2.88 by 100. We immediately look at the factor 100. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 2.88 we move the decimal point to the right by two digits, we get 288
2.88 x 100 = 288
Let's try to multiply 2.88 by 1000. We immediately look at the factor 1000. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 2.88 we move the decimal point to the right by three digits. The third digit is not there, so we add another zero. As a result, we get 2880.
2.88 x 1000 = 2880
Multiplying decimals by 0.1 0.01 and 0.001
Multiplying decimals by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal by a decimal. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits on the right as there are digits after the decimal point in both fractions.
For example, multiply 3.25 by 0.1
We multiply these fractions like ordinary numbers, ignoring the commas:
We got 325. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 3.25 and 0.1. In the fraction 3.25 there are two digits after the decimal point, in the fraction 0.1 there is one digit. A total of three numbers.
We return to the number 325 and begin to move from right to left. We need to count three digits on the right and put a comma. After counting three digits, we find that the numbers are over. In this case, you need to add one zero and put a comma:
We got the answer 0.325. So the value of the expression 3.25 × 0.1 is 0.325
3.25 x 0.1 = 0.325
There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction moves to the left by as many digits as there are zeros in the multiplier.
For example, let's solve the previous example 3.25 × 0.1 in this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 3.25 we move the decimal point to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits before the three. In this case, add one zero and put a comma. As a result, we get 0.325
3.25 x 0.1 = 0.325
Let's try multiplying 3.25 by 0.01. Immediately look at the multiplier of 0.01. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 3.25 we move the comma to the left by two digits, we get 0.0325
3.25 x 0.01 = 0.0325
Let's try multiplying 3.25 by 0.001. Immediately look at the multiplier of 0.001. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 3.25 we move the decimal point to the left by three digits, we get 0.00325
3.25 × 0.001 = 0.00325
Do not confuse multiplying decimals by 0.1, 0.001 and 0.001 with multiplying by 10, 100, 1000. Common Mistake most people.
When multiplying by 10, 100, 1000, the comma is moved to the right by as many digits as there are zeros in the multiplier.
And when multiplying by 0.1, 0.01 and 0.001, the comma is moved to the left by as many digits as there are zeros in the multiplier.
If at first it is difficult to remember, you can use the first method, in which the multiplication is performed as with ordinary numbers. In the answer, you will need to separate the integer part from the fractional part by counting as many digits on the right as there are digits after the decimal point in both fractions.
Dividing a smaller number by a larger one. Advanced level.
In one of the previous lessons, we said that when dividing a smaller number by a larger one, a fraction is obtained, in the numerator of which is the dividend, and in the denominator is the divisor.
For example, to divide one apple into two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. The result is a fraction. So each friend will get an apple. In other words, half an apple. A fraction is the answer to a problem how to split one apple between two
It turns out that you can solve this problem further if you divide 1 by 2. After all, a fractional bar in any fraction means division, which means that this division is also allowed in a fraction. But how? We are used to the fact that the dividend is always greater than the divisor. And here, on the contrary, the dividend is less than the divisor.
Everything will become clear if we remember that a fraction means crushing, dividing, dividing. This means that the unit can be split into as many parts as you like, and not just into two parts.
When dividing a smaller number by a larger one, a decimal fraction is obtained, in which the integer part will be 0 (zero). The fractional part can be anything.
So, let's divide 1 by 2. Let's solve this example with a corner:
One cannot be divided into two just like that. If you ask a question "how many twos are in one" , then the answer will be 0. Therefore, in private we write 0 and put a comma:
Now, as usual, we multiply the quotient by the divisor to pull out the remainder:
The moment has come when the unit can be split into two parts. To do this, add another zero to the right of the received one:
We got 10. We divide 10 by 2, we get 5. We write down the five in the fractional part of our answer:
Now we take out the last remainder to complete the calculation. Multiply 5 by 2, we get 10
We got the answer 0.5. So the fraction is 0.5
Half an apple can also be written using the decimal fraction 0.5. If we add these two halves (0.5 and 0.5), we again get the original one whole apple: 
This point can also be understood if we imagine how 1 cm is divided into two parts. If you divide 1 centimeter into 2 parts, you get 0.5 cm
Example 2 Find the value of expression 4:5
How many fives are in four? Not at all. We write in private 0 and put a comma:
We multiply 0 by 5, we get 0. We write zero under the four. Immediately subtract this zero from the dividend:
Now let's start splitting (dividing) the four into 5 parts. To do this, to the right of 4, we add zero and divide 40 by 5, we get 8. We write the eight in private.
We complete the example by multiplying 8 by 5, and get 40:
We got the answer 0.8. So the value of the expression 4: 5 is 0.8
Example 3 Find the value of expression 5: 125
How many numbers 125 are in five? Not at all. We write 0 in private and put a comma:
We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract from the five 0
Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write zero:
Divide 50 by 125. How many numbers 125 are in 50? Not at all. So in the quotient we again write 0
We multiply 0 by 125, we get 0. We write this zero under 50. Immediately subtract 0 from 50
Now we divide the number 50 into 125 parts. To do this, to the right of 50, we write another zero:
Divide 500 by 125. How many numbers are 125 in the number 500. In the number 500 there are four numbers 125. We write the four in private:
We complete the example by multiplying 4 by 125, and get 500
We got the answer 0.04. So the value of the expression 5: 125 is 0.04
Division of numbers without a remainder
So, let's put a comma in the quotient after the unit, thereby indicating that the division of integer parts is over and we proceed to the fractional part:
Add zero to the remainder 4
Now we divide 40 by 5, we get 8. We write the eight in private:
40−40=0. Received 0 in the remainder. So the division is completely completed. Dividing 9 by 5 results in a decimal of 1.8:
9: 5 = 1,8
Example 2. Divide 84 by 5 without a remainder
First we divide 84 by 5 as usual with a remainder:
Received in private 16 and 4 more in the balance. Now we divide this remainder by 5. We put a comma in the private, and add 0 to the remainder 4
Now we divide 40 by 5, we get 8. We write the figure eight in the quotient after the decimal point:
and complete the example by checking if there is still a remainder:
Dividing a decimal by a regular number
A decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by a regular number, first of all you need:
- divide the integer part of the decimal fraction by this number;
- after the integer part is divided, you need to immediately put a comma in the private part and continue the calculation, as in ordinary division.
For example, let's divide 4.8 by 2
Let's write this example as a corner:
Now let's divide the whole part by 2. Four divided by two is two. We write the deuce in private and immediately put a comma:
Now we multiply the quotient by the divisor and see if there is a remainder from the division:
4−4=0. The remainder is zero. We do not write zero yet, since the solution is not completed. Then we continue to calculate, as in ordinary division. Take down 8 and divide it by 2
8: 2 = 4. We write the four in the quotient and immediately multiply it by the divisor:
Got the answer 2.4. Expression value 4.8: 2 equals 2.4
Example 2 Find the value of the expression 8.43:3
We divide 8 by 3, we get 2. Immediately put a comma after the two:
Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:
We divide 24 by 3, we get 8. We write the eight in private. We immediately multiply it by the divisor to find the remainder of the division:
24−24=0. The remainder is zero. Zero is not recorded yet. Take the last three of the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:
Got the answer 2.81. So the value of the expression 8.43: 3 is equal to 2.81
Dividing a decimal by a decimal
To divide a decimal fraction into a decimal fraction, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by a regular number.
For example, divide 5.95 by 1.7
Let's write this expression as a corner
Now, in the dividend and in the divisor, we move the comma to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we must move the comma to the right by one digit in the dividend and in the divisor. Transferring:
After moving the decimal point to the right by one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7, after moving the decimal point to the right by one digit, turned into the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculation is not difficult:
The comma is moved to the right to facilitate division. This is allowed due to the fact that when multiplying or dividing the dividend and the divisor by the same number, the quotient does not change. What does it mean?
This is one of interesting features division. It is called the private property. Consider expression 9: 3 = 3. If in this expression the dividend and the divisor are multiplied or divided by the same number, then the quotient 3 will not change.
Let's multiply the dividend and divisor by 2 and see what happens:
(9 × 2) : (3 × 2) = 18: 6 = 3
As can be seen from the example, the quotient has not changed.
The same thing happens when we carry a comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma one digit to the right in the dividend and divisor. After moving the comma, the fraction 5.91 was converted to the fraction 59.1 and the fraction 1.7 was converted to the usual number 17.
In fact, inside this process, multiplication by 10 took place. Here's what it looked like:
5.91 × 10 = 59.1
Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and divisor will be multiplied by. In other words, the number of digits after the decimal point in the divisor will determine how many digits in the dividend and in the divisor the comma will be moved to the right.
Decimal division by 10, 100, 1000
Dividing a decimal by 10, 100, or 1000 is done in the same way as . For example, let's divide 2.1 by 10. Let's solve this example with a corner:
But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the left by as many digits as there are zeros in the divisor.
Let's solve the previous example in this way. 2.1: 10. We look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 2.1, you need to move the comma to the left by one digit. We move the comma to the left by one digit and see that there are no more digits left. In this case, we add one more zero before the number. As a result, we get 0.21
Let's try to divide 2.1 by 100. There are two zeros in the number 100. So in the divisible 2.1, you need to move the comma to the left by two digits:
2,1: 100 = 0,021
Let's try to divide 2.1 by 1000. There are three zeros in the number 1000. So in the divisible 2.1, you need to move the comma to the left by three digits:
2,1: 1000 = 0,0021
Decimal division by 0.1, 0.01 and 0.001
Dividing a decimal by 0.1, 0.01, and 0.001 is done in the same way as . In the dividend and in the divisor, you need to move the comma to the right by as many digits as there are after the decimal point in the divisor.
For example, let's divide 6.3 by 0.1. First of all, we move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we move the commas in the dividend and in the divisor to the right by one digit.
After moving the decimal point to the right by one digit, the decimal fraction 6.3 turns into the usual number 63, and the decimal fraction 0.1, after moving the decimal point to the right by one digit, turns into one. And dividing 63 by 1 is very simple:
So the value of the expression 6.3: 0.1 is equal to 63
But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is transferred to the right by as many digits as there are zeros in the divisor.
Let's solve the previous example in this way. 6.3:0.1. Let's look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 6.3, you need to move the comma to the right by one digit. We move the comma to the right by one digit and get 63
Let's try to divide 6.3 by 0.01. Divisor 0.01 has two zeros. So in the divisible 6.3, you need to move the comma to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, one more zero must be added at the end. As a result, we get 630
Let's try dividing 6.3 by 0.001. The divisor of 0.001 has three zeros. So in the divisible 6.3, you need to move the comma to the right by three digits:
6,3: 0,001 = 6300
Tasks for independent solution
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Division by a decimal is reduced to division by natural number.
Rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, it is necessary both in the dividend and in the divisor to move the comma as many digits to the right as there are in the divisor after the decimal point. After that, divide by a natural number.
Examples.
Perform division by decimal:
To divide by a decimal fraction, you need to move the comma as many digits to the right in both the dividend and the divisor as there are after the decimal point in the divisor, that is, by one sign. We get: 35.1: 1.8 \u003d 351: 18. Now we perform division by a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To perform the division of decimal fractions, both in the dividend and in the divisor, move the comma to the right by one sign: 14.76: 3.6 \u003d 147.6: 36. Now we perform on a natural number. Result: 14.76: 3.6 = 4.1.
To perform division by a decimal fraction of a natural number, it is necessary both in the dividend and in the divisor to move as many characters to the right as there are in the divisor after the decimal point. Since the comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 \u003d 7000: 175. We divide the resulting natural numbers with a corner: 70: 1.75 \u003d 7000: 175 \u003d 40.
4) 0,1218: 0,058
To divide one decimal fraction into another, we move the comma to the right both in the dividend and in the divisor by as many digits as there are in the divisor after the decimal point, that is, by three digits. Thus, 0.1218: 0.058 \u003d 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
Adding decimals is done in the same way as adding whole numbers. Let's see this with examples.
1) 0.132 + 2.354. Let's sign the terms one under the other.
Here, from the addition of 2 thousandths with 4 thousandths, 6 thousandths were obtained;
from the addition of 3 hundredths with 5 hundredths, it turned out 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 signing the terms under each other.
3) 1,2357 + 0,469 + 2,08 + 3,90701.
Here, when adding thousandths, we get 21 thousandths; we wrote 1 under the thousandths, and 2 added to the hundredths, so in the hundredth place we got the following terms: 2 + 3 + 6 + 8 + 0; in sum, they give 19 hundredths, we signed 9 under hundredths, and 1 was counted as tenths, etc.
Thus, when adding decimal fractions, the following order must be observed: fractions are signed one under the other so that in all terms the same digits are under each other and all commas are in the same vertical column; to the right of the decimal places of some terms, they attribute, at least mentally, such a number of zeros that all terms after the decimal point have the same number of digits. Then, addition is performed by digits, starting from the right side, and in the resulting amount a comma is placed in the same vertical column as it is in these terms.
§ 108. Subtraction of decimal fractions.
Subtracting decimals is done in 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 the 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, one 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 refer to the nearest digit to the left, i.e., to hundred-thousandths, but there is also zero in place of hundred-thousandths, so we take 1 ten-thousandth from 3 ten-thousandths and we split it into hundred-thousandths, we get 10 hundred-thousandths, of which 9 hundred-thousandths are left in the category of hundred-thousandths, and 1 hundred-thousandth is crushed 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 (not to forget), these numbers are written on top of the corresponding fractional digits of the reduced. Now we can start subtracting. We subtract 2 millionths from 10 millionths, we get 8 millionths; subtract 1 hundred-thousandth from 9 hundred-thousandths, we get 8 hundred-thousandths, etc.
Thus, when subtracting decimal fractions, the following order is observed: the subtracted is signed under the reduced so that the same digits are one under the other and all the commas are in the same vertical column; on the right, they attribute, at least mentally, in the reduced or subtracted so many zeros so that they have the same number of digits, then subtract by digits, starting from the right side, and in the resulting difference put a comma in the same vertical column in which it is in reduced and subtracted.
§ 109. Multiplication of decimal fractions.
Consider a few 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 is increased several times, the product increases by the same amount. This means that the number that will be obtained from multiplying integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to get true work, you need to reduce the found product by 10 times. Therefore, here you have to perform a multiplication by 10 once and a division by 10 once, but multiplication and division by 10 is performed by moving the comma to the right and left by one sign. Therefore, you need to do this: in the multiplier, move the comma to the right by one sign, from this it will be equal to 23, then you need to multiply the resulting integers:
This product is 10 times larger than the true one. Therefore, it must be reduced by 10 times, for which we move the comma one character 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 an 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 by decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, that is, we will temporarily increase the multiplier by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1227 by 21, we get:
1 227 21 = 25 767.
Taking into account that the resulting product is 100,000 times greater than the true one, 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 integers and in the product to separate with a comma on the right side as many decimal places as there were in the multiplicand and in the factor together.
In the last example, the result is a product with five decimal places. If such greater accuracy is not required, then rounding of the decimal fraction is done. When rounding, you should use the same rule that 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 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 of 53 by 15, but our second factor was 10 times less, which means that the product must be reduced by 10 times, i.e.
53 1,5 = 79,5.
2) Multiply 5.3 by 4.7.
First, let's find the product of 53 by 47 in the table, it will be 2491. But since we increased the multiplicand and the multiplier by a total of 100 times, then the resulting product is 100 times larger than it should be; so we have to reduce this product by a factor of 100:
5,3 4,7 = 24,91.
3) Multiply 0.53 by 7.4.
First we find in the table the product of 53 by 74; this will be 3,922. But since we have increased the multiplier by 100 times, and the multiplier by 10 times, the product has increased by 1,000 times; so we now have to reduce it by a factor of 1,000:
0,53 7,4 = 3,922.
§ 111. Division of decimals.
We will look at decimal division in this order:
1. Division of a decimal fraction by an integer,
1. Division of a decimal fraction by an integer.
1) Divide 2.46 by 2.
We divided by 2 first integers, then tenths and finally hundredths.
2) Divide 32.46 by 3.
32,46: 3 = 10,82.
We divided 3 tens by 3, then we 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 demolished 4 tenths and divided 24 tenths by 3; received in private 8 tenths and finally divided 6 hundredths.
3) Divide 1.2345 by 5.
1,2345: 5 = 0,2469.
Here, in the quotient in the first place, zero integers turned out, since one integer is not divisible by 5.
4) Divide 13.58 by 4.
The peculiarity of this example is that when we got 9 hundredths in private, then a remainder equal to 2 hundredths was found, we split this remainder into thousandths, got 20 thousandths and brought the division to the end.
Rule. The division of a decimal fraction by an integer is carried out in the same way as the division of integers, and the resulting remainders are converted into decimal fractions, more and more small; division continues until the remainder is zero.
2. Division of a decimal fraction by a decimal fraction.
1) Divide 2.46 by 0.2.
We already know how to divide a decimal fraction by an integer. Let's think about whether this new case of division can also be reduced to the previous one? At one time, we considered a remarkable property of the quotient, which consists in the fact that it remains unchanged while increasing or decreasing the dividend and divisor by the same number of times. We would easily perform the division of the numbers offered to us if the divisor were an integer. To do this, it is enough to increase it 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same number of times, that is, 10 times. Then the division of these numbers will be replaced by the division of such numbers:
and there is no need to make any amendments in private.
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 in quotient 0 and divide 125 tenths by 16, we get 7 tenths in quotient and the remainder is 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Pay attention to the following:
a) when integers are not obtained in the quotient, then zero integers are written in their place;
b) when, after taking 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 the last digit of the dividend has been removed, the division does not end, then, by assigning zeros to the remainders, the division continues;
d) if the dividend is an integer, then when dividing it by a decimal fraction, its increase is carried out by assigning zeros to it.
Thus, in order to divide a number by a decimal fraction, you need to discard a comma in the divisor, and then increase the dividend as many times as the divisor increased when the comma was dropped in it, and then perform the division according to the rule of dividing the decimal fraction by an integer.
§ 112. Approximate quotient.
In the previous paragraph, we considered the division of decimal fractions, and in all the examples we solved, the division was brought to the end, i.e., an exact quotient was obtained. However, in most cases the exact quotient cannot be obtained, no matter how far we extend 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 the numbers that we have met before begin to appear in the remainder. Numbers will also be repeated in the quotient: obviously, after the number 7, the number 5 will appear, then 2, and so on without end. In such cases, division is interrupted and limited to the first few digits of the quotient. This private is called approximate. How to perform division in this case, we will show with examples.
Let it be required to divide 25 by 3. It is obvious that the exact quotient, expressed as an integer or 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 of 1. To get the exact quotient, you need to add to the found approximate quotient, that is, to 8, the fraction that results from dividing the remainder, equal to 1, by 3; it 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 is a proper fraction, i.e. a fraction, less than one, then, discarding it, we assume error, which less than one. Private 8 will approximate quotient up to one with a drawback. If we take 9 instead of 8, then we also allow an error that is less than one, since we will add not a whole unit, but 2 / 3. Such a private will approximate quotient up to one with an excess.
Let's take another example now. Let it be required to divide 27 by 8. Since here we will not 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 3 / 8 , it is less than one, which means that the approximate quotient (3) is found up to one with a drawback. We continue the division: we split the remainder of 3 into tenths, we get 30 tenths; Let's divide them by 8.
We got in private on the spot tenths 3 and in the remainder b tenths. If we confine ourselves to the number 3.3 in particular, and discard the remainder 6, then we will allow an error 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; from this division would be 6/80, which is less than one tenth. (Check!) Thus, if we limit ourselves to tenths in the quotient, then we can say that we have found the quotient accurate to one tenth(with disadvantage).
Let's continue the division to find one more decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; Let's divide them by 8.
In private in third place it turned out 7 and in the remainder 4 hundredths; if we discard them, then we allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases, the quotient is said to be found. accurate to one hundredth(with disadvantage).
In the example that we are now considering, you 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 consider such an example:
40: 7 = 5,71428571...
The dots at the end of the number indicate that the division is not completed, that is, the equality is approximate. Usually approximate equality is written like this:
40: 7 = 5,71428571.
We took the quotient with eight decimal places. But if such great precision is not required, one can confine oneself to the whole part of the quotient, i.e., the number 5 (more precisely, 6); for greater accuracy, tenths could be taken into account and the quotient taken equal to 5.7; if for some reason this accuracy is insufficient, then we can stop at hundredths and take 5.71, etc. Let's write out the individual quotients and name them.
The first approximate quotient up to one 6.
The second » » » to one tenth 5.7.
Third » » » up to one hundredth 5.71.
Fourth » » » up to one thousandth of 5.714.
Thus, in order to find an approximate quotient up to some, for example, the 3rd decimal place (i.e., up to one thousandth), division is stopped as soon as this sign is found. In this case, one must remember the rule set forth in § 40.
§ 113. The simplest problems for interest.
After studying decimal fractions, we will solve a few more percentage problems.
These problems are similar to those we solved in the department of ordinary fractions; 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 switch from an ordinary fraction to a decimal fraction 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 with a denominator of 100:
Let's now consider a few problems.
1. Finding percentages of a given number.
Task 1. Only 1,600 people live in one village. Number of children school age is 25% of total number residents. How many school-age children are 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.
For a clear understanding of 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 are in the number 1600 (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 give depositors 2% of income annually. How much income per year will be received by a depositor who has deposited: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rubles?
In all four cases, to solve the problem, it will be necessary 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 (rubles),
b) 500 0.02 = 10 (rubles),
c) 750 0.02 = 15 (rubles),
d) 1,000 0.02 = 20 (rubles).
Each of these cases can be verified by the following considerations. Savings banks give depositors 2% of income, that is, 0.02 of the amount put into savings. If the amount were 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the depositor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds are in a given number, and multiply 2 rubles by this number of hundreds. In example a) hundreds of 2, so
2 2 \u003d 4 (rubles).
In example d) hundreds are 10, which means
2 10 \u003d 20 (rubles).
2. Finding a number by its percentage.
Task 1. In the spring, the school graduated 54 students, which is 6% of the total number of students. How many students were in the school in the past academic year?
Let us first clarify the meaning of this problem. 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 in 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 whole number. Thus, before us is an ordinary problem of finding a number by its fraction (§ 90 p. 6). Problems of this type are solved by division:
This means that there were 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 mind, solve the problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage indicated in the solved problem from it , 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 income. Determine his monthly income.
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 the desired is the entire earnings:
780: 0,65 = 1 200.
Therefore, the desired earnings is 1200 rubles.
3. Finding the percentage of numbers.
Task 1. The school library has a total of 6,000 books. 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) this kind of problem 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 task, we need to find the percentage of the numbers 1,200 and 6,000.
We 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, we 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.
At school, these actions are studied from simple to complex. Therefore, it is absolutely necessary to master well the algorithm for performing these operations on simple examples. So that later there will be no difficulties with dividing decimal fractions into a column. After all, this is the most difficult version of such tasks.
This subject requires consistent study. Gaps in knowledge are unacceptable here. This principle should be learned by every student already in the first grade. Therefore, if you skip several lessons in a row, you will have to master the material yourself. Otherwise, later there will be problems not only with mathematics, but also with other subjects related to it.
The second prerequisite for a successful study of mathematics is to move on to examples of division in a column only after addition, subtraction and multiplication have been mastered.
It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to learn it from the Pythagorean table. There is nothing superfluous, and multiplication is easier to digest in this case.
How are natural numbers multiplied in a column?
If there is a difficulty in solving examples in a column for division and multiplication, then it is necessary to start solving the problem with multiplication. Because division is the inverse of multiplication:
- Before multiplying two numbers, you need to look at them carefully. Choose the one with more digits (longer), write it down first. Place the second one under it. Moreover, the numbers of the corresponding category should be under the same category. That is, the rightmost digit of the first number must be above the rightmost digit of the second.
- Multiply the rightmost digit of the bottom number by each digit of the top number, starting from the right. Write the answer under the line so that its last digit is under the one by which it was multiplied.
- Repeat the same with the other digit of the bottom number. But the result of the multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.
Continue this multiplication in a column until the numbers in the second multiplier run out. Now they need to be folded. This will be the desired answer.
Algorithm for multiplying into a column of decimal fractions
First, it is supposed to imagine that not decimal fractions are given, but natural ones. That is, remove commas from them and then proceed as described in the previous case.
The difference begins when the answer is written. At this point, it is necessary to count all the numbers that are after the decimal points in both fractions. That is how many of them you need to count from the end of the answer and put a comma there.
It is convenient to illustrate this algorithm with an example: 0.25 x 0.33:
How to start learning to divide?
Before solving examples for division in a column, it is supposed to remember the names of the numbers that are in the example for division. The first of them (the one that divides) is the divisible. The second (divided by it) is a divisor. The answer is private.
After that, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it is easy to divide them equally between mom and dad. But what if you need to distribute them to your parents and brother?
After that, you can get acquainted with the rules of division and master them on concrete examples. Simple ones at first, and then moving on to more and more complex ones.
Algorithm for dividing numbers into a column
First, we present the procedure for natural numbers that are divisible by a single-digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then it is supposed to make small changes, but more on that later:
- Before doing division in a column, you need to find out where the dividend and divisor are.
- Write down the dividend. To the right of it is a divider.
- Draw a corner on the left and bottom near the last corner.
- Determine the incomplete dividend, that is, the number that will be the minimum for division. Usually it consists of one digit, maximum of two.
- Choose the number that will be written first in the answer. It must be the number of times the divisor fits in the dividend.
- Write down the result of multiplying this number by a divisor.
- Write it under an incomplete divisor. Perform subtraction.
- Carry to the remainder the first digit after the part that has already been divided.
- Pick up the answer again.
- Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: demolish the number, pick up the number, multiply, subtract.
How to solve long division if there is more than one digit in the divisor?
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then it is supposed to work with the first three digits.
There is another nuance in this division. The fact is that the remainder and the figure carried to it are sometimes not divisible by a divisor. Then it is supposed to attribute one more figure in order. But at the same time, the answer must be zero. If division is made three-digit numbers in a column, you may need to demolish more than two digits. Then the rule is introduced: zeros in the answer should be one less than the number of digits taken down.
You can consider such a division using the example - 12082: 863.
- The incomplete divisible in it is the number 1208. The number 863 is placed in it only once. Therefore, in response, it is supposed to put 1, and write 863 under 1208.
- After subtraction, the remainder is 345.
- To him you need to demolish the number 2.
- In the number 3452, 863 fits four times.
- Four must be written in response. Moreover, when multiplied by 4, this number is obtained.
- The remainder after subtraction is zero. That is, the division is completed.
The answer in the example is 14.
What if the dividend ends in zero?
Or a few zeros? In this case, a zero remainder is obtained, and there are still zeros in the dividend. Do not despair, everything is easier than it might seem. It is enough just to attribute to the answer all the zeros that remained undivided.
For example, you need to divide 400 by 5. The incomplete dividend is 40. Five is placed in it 8 times. This means that the answer is supposed to be written 8. When subtracting, there is no remainder. That is, the division is over, but zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 gives 80.
What if you need to divide a decimal?
Again, this number looks like a natural number, if not for the comma separating the integer part from the fractional part. This suggests that the division of decimal fractions into a column is similar to the one described above.
The only difference will be the semicolon. It is supposed to be answered immediately, as soon as the first digit from the fractional part is taken down. In another way, it can be said like this: the division of the integer part has ended - put a comma and continue the solution further.
When solving examples for dividing into a column with decimal fractions, you need to remember that any number of zeros can be assigned to the part after the decimal point. Sometimes this is necessary in order to complete the numbers to the end.
Division of two decimals
It may seem complicated. But only at the beginning. After all, how to perform division in a column of fractions by a natural number is already clear. So, we need to reduce this example to the already familiar form.
Make it easy. You need to multiply both fractions by 10, 100, 1,000, or 10,000, or maybe a million if the task requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, as a result, it turns out that you will have to divide a fraction by a natural number.
And it will be in the worst case. After all, it may turn out that the dividend from this operation becomes an integer. Then the solution of the example with division into a column of fractions will be reduced to the simplest option: operations with natural numbers.
As an example: 28.4 divided by 3.2:
- First, they must be multiplied by 10, since in the second number there is only one digit after the decimal point. Multiplying will give 284 and 32.
- They are supposed to be divided. And at once the whole number is 284 by 32.
- The first matched number for the answer is 8. Multiplying it gives 256. The remainder is 28.
- The division of the integer part is over, and a comma is supposed to be put in the answer.
- Demolish to remainder 0.
- Take 8 again.
- Remainder: 24. Add another 0 to it.
- Now you need to take 7.
- The result of the multiplication is 224, the remainder is 16.
- Demolish another 0. Take 5 and get exactly 160. The remainder is 0.
Division completed. The result of the 28.4:3.2 example is 8.875.
What if the divisor is 10, 100, 0.1, or 0.01?
As with multiplication, long division is not needed here. It is enough just to move the comma in the right direction for a certain number of digits. Moreover, according to this principle, you can solve examples with both integers and decimal fractions.
So, if you need to divide by 10, 100 or 1000, then the comma is moved to the left by as many digits as there are zeros in the divisor. That is, when a number is divisible by 100, the comma should move to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at the end of it.
This action produces the same result as if the number were to be multiplied by 0.1, 0.01, or 0.001. In these examples, the comma is also moved to the left by the number of digits, equal to the length fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the comma should move to the right by one digit (or two, three, depending on the number of zeros or the length of the fractional part).
It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be assigned to the left (in the integer part) or to the right (after the decimal point).
Division of periodic fractions
In this case, you will not be able to get the exact answer when dividing into a column. How to solve an example if a fraction with a period is encountered? Here it is necessary to move on to ordinary fractions. And then perform their division according to the previously studied rules.
For example, you need to divide 0, (3) by 0.6. The first fraction is periodic. It is converted to the fraction 3/9, which after reduction will give 1/3. The second fraction is the final decimal. It is even easier to write down an ordinary one: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions prescribes to replace division with multiplication and the divisor with the reciprocal of a number. That is, the example boils down to multiplying 1/3 by 5/3. The answer is 5/9.
If the example has different fractions...
Then there are several possible solutions. Firstly, common fraction You can try to convert to decimal. Then divide already two decimals according to the above algorithm.
Secondly, every final decimal fraction can be written as a common fraction. It's just not always convenient. Most often, such fractions turn out to be huge. Yes, and the answers are cumbersome. Therefore, the first approach is considered more preferable.
