Multiplying decimal fractions by an integer column. Multiplication of decimal fractions: rules, examples, solutions. Division of a decimal by a regular number
In this tutorial, we will look at each of these operations separately.
Lesson contentAdding Decimals
As we know, the decimal fraction has a whole and a fractional part. When adding decimal fractions, whole and fractional parts are added separately.
For example, add the decimal fractions 3.2 and 5.3. It is more convenient to add decimal fractions in a column.
Let's first write these two fractions in a column, while the whole parts must necessarily be under the whole, and the fractional part under the fractional. At school this requirement is called Comma under comma.
Let's write fractions in a column so that the comma is below the comma:
We begin to add the fractional parts: 2 + 3 \u003d 5. We write the five in the fractional part of our answer:
Now we add the whole parts: 3 + 5 \u003d 8. We write the eight in the whole part of our answer:
Now we separate the whole part from the fractional part with a comma. To do this, again, we follow the rule Comma under comma:
The answer was 8.5. So expressions 3.2 + 5.3 is 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.
Decimal places
Decimal fractions, like ordinary numbers, have their places. These are tenths, hundredths, thousandths. In this case, the digits begin after the decimal point.
The first digit after the decimal point is responsible for the tenth place, the second digit after the decimal point for the hundredth place, the third digit after the decimal point for the thousandth place.
The decimal places hold some useful information. In particular, they report how many tenths, hundredths and thousandths are in decimal fraction.
For example, consider the decimal 0.345
The position where the triplet is located is called in tenths
The position where the four is located is called hundredths
The position where the five is located is called thousandths
Let's take a look at this figure. We see that in the tenth place there is a three. This suggests that the decimal 0.345 contains three tenths.
If we add the fractions, and we get the original decimal fraction 0.345
It can be seen that at first we received 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. Decimal fractions are added in digits: tenths are added with tenths, hundredths with hundredths, thousandths with thousandths.
Therefore, when adding decimal fractions, you must observe the rule Comma under comma... The comma below the 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, add the fractional parts 5 + 4 \u003d 9. Write the nine in the fractional part of our answer:
Now we add the whole parts 1 + 3 \u003d 4. We write the four in the whole part of our answer:
Now we separate the whole part from the fractional part with a comma. To do this, we again observe the “comma under the comma” rule:
The answer was 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 down this expression in a column, observing the "comma under the comma" rule
First of all, add the fractional part, namely the hundredths 1 + 2 \u003d 3. We write the three in the hundredth part of our answer:
Now add the tenths 5 + 2 \u003d 7. We write the seven in the tenth part of our answer:
Now add the whole parts 3 + 1 \u003d 4. We write a four in the whole part of our answer:
Separate the whole part from the fractional part with a comma, observing the “comma under the comma” rule:
The answer was 4.73. So the value of the expression 3.51 + 1.22 is 4.73
3,51 + 1,22 = 4,73
As with normal numbers, adding decimal fractions can occur. 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 5 + 7 \u003d 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 we transfer the unit to the next digit:
Now we add the tenths 6 + 2 \u003d 8 plus the one that came from the previous operation, we get 9. We write the number 9 in the tenth part of our answer:
Now add the whole parts 2 + 3 \u003d 5. We write the number 5 in the whole part of our answer:
The answer was 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
We write this expression in a column
We add the fractional parts 5 + 8 \u003d 13. The number 13 will not fit into the fractional part of our answer, so first we write down the number 3, and we transfer the unit to the next digit, or rather we transfer it to the whole part:
Now we add the whole parts 9 + 2 \u003d 11 plus the one that came from the previous operation, we get 12. We write the number 12 in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
The answer was 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 numbers, 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 down this expression in a column, let's make the number of digits after the decimal point in both fractions the same. There are three digits in the decimal fraction 12.725 after the decimal point, and in the fraction 1.7 there is 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 down this expression in a column and start calculating:
Add the thousandths 5 + 0 \u003d 5. We write down the number 5 in the thousandth part of our answer:
Add the hundredths 2 + 0 \u003d 2. We write down the number 2 in the hundredth part of our answer:
Add tenths 7 + 7 \u003d 14. The number 14 will not fit in a tenth of our answer. Therefore, first we write down the number 4, and transfer the unit to the next digit:
Now we add the whole parts 12 + 1 \u003d 13 plus the one that got from the previous operation, we get 14. We write the number 14 in the whole part of our answer:
Separate the whole part from the fractional part with a comma:
The answer was 14.425. So the value of the expression 12.725 + 1.700 is equal to 14.425
12,725+ 1,700 = 14,425
Subtracting decimal fractions
When subtracting decimal fractions, you must follow the same rules as when adding: "comma under the comma" and "equal number of digits after the decimal point."
Example 1. Find the value of expression 2.5 - 2.2
We write down this expression in a column, observing the "comma under the comma" rule:
Evaluate the fractional part 5−2 \u003d 3. We write the number 3 in the tenth part of our answer:
Evaluate the integer part 2−2 \u003d 0. We write zero in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
The answer was 0.3. So the value of the expression 2.5 - 2.2 is 0.3
2,5 − 2,2 = 0,3
Example 2. Find the value of the expression 7.353 - 3.1
This expression has a different number of digits after the decimal point. There are three digits after the decimal point in the fraction 7.353, and in the fraction 3.1 there is only one. This means that in the fraction 3.1 at the end, you need to add two zeros 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:
The answer was 4.253. So the value of the expression 7.353 - 3.1 is equal to 4.253
7,353 — 3,1 = 4,253
As with ordinary numbers, sometimes you have to occupy one from the adjacent digit if subtraction becomes impossible.
Example 3. Find the value of expression 3.46 - 2.39
Subtract the hundredths of 6-9. From the number 6, do not subtract the number 9. Therefore, you need to take one from the adjacent bit. Having taken one from the adjacent bit, the number 6 turns into the number 16. Now you can calculate the hundredths of 16-9 \u003d 7. We write the seven in the hundredth part of our answer:
Now let's subtract tenths. Since we occupied one unit in the tenth place, the figure that was located there decreased by one unit. In other words, in the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3 \u003d 0. We write zero in the tenth part of our answer:
Now we subtract the whole parts 3−2 \u003d 1. We write one in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
The answer was 1.07. So the value of expression 3.46−2.39 is 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. We write this expression in a column so that the integer part of the decimal fraction 1.23 is 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 we subtract the tenths: 0−2. You cannot subtract the number 2 from zero. Therefore, you need to take one from the adjacent bit. Taking one from the adjacent bit, 0 becomes 10. Now we can calculate the tenths of 10−2 \u003d 8. We write the eight in the tenth part of our answer:
Now we subtract whole parts. Previously, the integer contained the number 3, but we borrowed one unit from it. As a result, it turned into number 2. Therefore, subtract 1.2 from 2. 2−1 \u003d 1. We write one in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
The answer was 1.8. So the value of the expression 3−1.2 is 1.8
Decimal multiplication
Decimal multiplication is easy and even fun. To multiply decimal fractions, you multiply them like ordinary numbers, ignoring the commas.
Having received the answer, 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 both fractions, then in the answer, count the same number of digits from the right and put a comma.
Example 1. Find the value of the expression 2.5 × 1.5
Let's multiply these decimal fractions as usual numbers, ignoring the commas. In order not to pay attention to the commas, you can imagine for a while that they are absent at all:
Received 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 2.5 and 1.5. In the first fraction after the decimal point there is one digit, in the second fraction there is also one. There are two digits in total.
We return to the number 375 and start moving from right to left. We need to count two digits to the right and put a comma:
The answer was 3.75. So the value of the expression 2.5 × 1.5 is 3.75
2.5 x 1.5 \u003d 3.75
Example 2. Find the value of the expression 12.85 × 2.7
Let's multiply these decimal fractions, ignoring the commas:
Received 34695. 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 fractions 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 start moving from right to left. We need to count three digits to the right and put a comma:
The answer was 34.695. So the value of the expression 12.85 × 2.7 is 34.695
12.85 × 2.7 \u003d 34.695
Decimal multiplication by an ordinary number
Sometimes situations arise when you need to multiply a decimal fraction by an ordinary number.
To multiply a decimal fraction and an ordinary number, you need to multiply them, ignoring the comma in the decimal fraction. Having received the answer, 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 the decimal fraction, then in the answer, count the same number of digits on 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 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 2.54. There are two digits after the decimal point in the fraction 2.54.
We return to the number 508 and start moving from right to left. We need to count two digits to the right and put a comma:
The answer was 5.08. So the value of the expression 2.54 × 2 is 5.08
2.54 x 2 \u003d 5.08
Decimal multiplication by 10, 100, 1000
Multiplying decimal fractions by 10, 100, or 1000 is done in the same way as multiplying decimal fractions by regular numbers. You need to perform multiplication, not paying attention to the comma in the decimal fraction, then in the answer separate the whole part from the fractional part, counting as many digits to the right as there were digits after the decimal point in the decimal fraction.
For example, multiply 2.88 by 10
Multiply the decimal 2.88 by 10, ignoring the decimal point:
Received 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 there are two digits after the decimal point in the fraction 2.88.
We return to the number 2880 and start moving from right to left. We need to count two digits to the right and put a comma:
The answer was 28.80. If we drop the last zero, we get 28.8. So the value of the expression 2.88 × 10 is 28.8
2.88 x 10 \u003d 28.8
There is also a second way to multiply decimal fractions by 10, 100, 1000. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction is moved to the right by as many digits as there are zeros in the factor.
For example, let's solve the previous 2.88 × 10 example this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros there are in it. We see that there is one zero in it. Now in the fraction 2.88, move the comma to the right one digit, we get 28.8.
2.88 x 10 \u003d 28.8
Let's try to multiply 2.88 by 100. Immediately we look at the factor 100. We are interested in how many zeros it contains. We see that there are two zeros in it. Now in the fraction 2.88, move the comma to the right by two digits, we get 288
2.88 × 100 \u003d 288
Let's try to multiply 2.88 by 1000. Immediately look at the multiplier 1000. We are interested in how many zeros there are. We see that there are three zeros in it. Now in the fraction 2.88, move the comma 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 × 1000 \u003d 2880
Decimal multiplication by 0.1 0.01 and 0.001
Multiplying decimal fractions by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal fraction by a decimal fraction. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits to 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:
Received 325. 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 fractions 3.25 and 0.1. There are two digits after the decimal point in the fraction 3.25, in the fraction 0.1 there is one digit. There are three numbers in total.
We return to the number 325 and start moving from right to left. We need to count three digits to the right and put a comma. After counting three digits, we find that the digits are over. In this case, you need to add one zero and put a comma:
The answer was 0.325. So the value of the expression 3.25 × 0.1 is equal to 0.325
3.25 × 0.1 \u003d 0.325
There is also a second way to multiply decimal fractions 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 is moved to the left by as many digits as there are zeros in the multiplier.
For example, let's solve the previous 3.25 × 0.1 example this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros there are in it. We see that there is one zero in it. Now in the fraction 3.25, move the comma to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits in front of the three. In this case, add one zero and add a comma. As a result, we get 0.325
3.25 × 0.1 \u003d 0.325
Let's try to multiply 3.25 by 0.01. Immediately look at the 0.01 multiplier. We are interested in how many zeros there are in it. We see that there are two zeros in it. Now in the fraction 3.25, move the comma to the left by two digits, we get 0.0325
3.25 × 0.01 \u003d 0.0325
Let's try to multiply 3.25 by 0.001. Immediately look at the 0.001 multiplier. We are interested in how many zeros there are in it. We see that there are three zeros in it. Now in the fraction 3.25, move the comma to the left by three digits, we get 0.00325
3.25 × 0.001 \u003d 0.00325
Multiplying decimal fractions by 0.1, 0.001 and 0.001 should not be confused with multiplying by 10, 100, 1000. This is a typical mistake most people make.
When multiplying by 10, 100, 1000, the comma is shifted to the right by the same number of digits as there are zeros in the factor.
And when multiplied by 0.1, 0.01 and 0.001, the comma is transferred to the left by the same number of digits as 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, counting as many digits from the right as the 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 you divide a smaller number by a larger one, you get a fraction, in the numerator of which is the dividend and in the denominator - the divisor.
For example, to divide one apple by two, you need to write 1 (one apple) in the numerator and 2 (two friends) in the denominator. As a result, we get a fraction. So each friend will get an apple. In other words, half an apple. Fraction is the answer to the problem "How to split one apple for 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, and therefore this division is 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 fraction means division, division, division. This means that a 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, you get a decimal fraction, in which the integer part will be 0 (zero). The fractional part can be any.
So, let's divide 1 by 2. Let's solve this example with a corner:
One cannot simply be divided into two completely. If you ask a question "How many twos are in one" , then the answer will be 0. Therefore, in the quotient we write 0 and put a comma:
Now, as usual, we multiply the quotient by the divisor in order 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 resulting one:
We got 10. We divide 10 by 2, we get 5. We write the five in the fractional part of our answer:
Now we pull out the last remainder to complete the calculation. Multiply 5 by 2 to get 10
The answer was 0.5. So the fraction is 0.5
Half an apple can also be written using a decimal fraction of 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 you 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 the expression 4: 5
How many fives are in the four? Not at all. We write 0 in the private and put a comma:
Multiply 0 by 5, we get 0. Write zero under the four. We 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, add zero and divide 40 by 5, we get 8. We write the eight in the quotient.
Finishing the example by multiplying 8 by 5 to get 40:
The answer was 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 the five? Not at all. We write 0 in the quotient and put a comma:
Multiply 0 by 5, we get 0. Write 0 under the five. Immediately subtract 0 from the five
Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write down zero:
Divide 50 by 125. How many numbers 125 are there in 50? Not at all. So in the quotient we again write 0
Multiply 0 by 125, we get 0. Write this zero under 50. Immediately subtract 0 from 50
Now we divide the number 50 by 125 parts. To do this, to the right of 50, write another zero:
Divide 500 by 125. How many numbers 125 are in the number 500. There are four numbers 125 in the number 500. Write the four in the quotient:
Finish the example by multiplying 4 by 125 to get 500
The answer was 0.04. So the value of the expression 5: 125 is 0.04
Division of numbers without remainder
So, we put a comma in the quotient after the one, thereby indicating that the division of the whole parts is over and we proceed to the fractional part:
Add zero to remainder 4
Now we divide 40 by 5, we get 8. Write the eight in the quotient:
40-40 \u003d 0. Got 0 in the remainder. This means that the division is completely completed. Dividing 9 by 5 gives the decimal 1.8:
9: 5 = 1,8
Example 2... Divide 84 by 5 without a remainder
First, divide 84 by 5 as usual with a remainder:
Received in private 16 and 4 more in the remainder. Now divide this remainder by 5. Put a comma in the quotient, and add 0 to the remainder 4
Now we divide 40 by 5, we get 8. Write the eight in the quotient after the decimal point:
and end the example by checking if there is still a remainder:
Division of a decimal by a regular number
The decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by an ordinary number, you first need to:
- divide the whole part of the decimal fraction by this number;
- after the whole part is divided, you need to immediately put a comma in the quotient and continue the calculation, as in ordinary division.
For example, divide 4.8 by 2
Let's write this example in a corner:
Now let's divide the whole part by 2. Four divided by two is two. We write down the two in the quotient and immediately put a comma:
Now we multiply the quotient by the divisor and see if there is a remainder of the division:
4−4 \u003d 0. The remainder is zero. We do not write down zero yet, since the solution is not complete. Then we continue to calculate as in ordinary division. Take down 8 and divide it by 2
8: 2 \u003d 4. We write the four in the quotient and immediately multiply it by the divisor:
The answer was 2.4. The expression value 4.8: 2 is 2.4
Example 2. Find the value of the expression 8.43: 3
Divide 8 by 3, we get 2. Immediately put a comma after the two:
Now we multiply the quotient by the divisor 2 × 3 \u003d 6. Write the six under the eight and find the remainder:
Divide 24 by 3, we get 8. We write the eight in the quotient. Immediately multiply it by the divisor to find the remainder of the division:
24-24 \u003d 0. The remainder is zero. We do not write down zero yet. Dividing the last 3 from the dividend and dividing by 3, we get 1. Immediately multiply 1 by 3 to complete this example:
The answer was 2.81. So the value of the expression 8.43: 3 is 2.81
Division of a decimal by a decimal
To divide a decimal fraction by a decimal fraction, it is necessary in the dividend and in the divisor to 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 an ordinary number.
For example, divide 5.95 by 1.7
Let's write this expression in a corner
Now, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the comma in the divisor. There is one digit after the decimal point. So we have to move the comma to the right by one digit in the dividend and in the divisor. We transfer:
After moving the comma to the right one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7 after transferring the comma 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 wrapped 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 the interesting features of division. It is called the property of the quotient. Consider the expression 9: 3 \u003d 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) \u003d 18: 6 \u003d 3
As you can see from the example, the quotient has not changed.
The same happens when we carry the comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma in the dividend and divisor one digit to the right. After moving the decimal point, the fraction 5.91 was converted to a fraction of 59.1 and the fraction 1.7 was converted to the usual number 17.
In fact, inside this process, there was a multiplication by 10. This is how it looked:
5.91 x 10 \u003d 59.1
Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and the 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.
Dividing a decimal 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 is lighter. The essence of this method is that the comma in the dividend is shifted 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 divisor. We are interested in how many zeros there are in it. We see that there is one zero. So in the dividend 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 numbers left. In this case, 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 100. So in the dividend 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 1000. This means that in the dividend 2,1 you need to move the comma to the left by three digits:
2,1: 1000 = 0,0021
Division of a decimal by 0.1, 0.01, and 0.001
Dividing a decimal fraction by 0.1, 0.01, and 0.001 is done in the same way as. In the dividend and in the divisor, the comma must be moved to the right by as many digits as there are after the comma in the divisor.
For example, divide 6.3 by 0.1. First of all, move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the comma in the divisor. There is one digit after the decimal point. So we transfer commas in the dividend and in the divisor to the right by one digit.
After moving the comma 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 comma to the right one digit turns into one. And dividing 63 by 1 is very simple:
So the value of the expression 6.3: 0.1 is 63
But there is also a second way. It is lighter. The essence of this method is that the comma in the dividend is shifted 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. We look at the divider. We are interested in how many zeros there are in it. We see that there is one zero. This means that in the dividend of 6.3, you need to move the comma to the right by one digit. Move the comma to the right one digit and get 63
Let's try to divide 6.3 by 0.01. The divisor 0.01 has two zeros. This means that in the dividend 6,3 it is necessary to move the comma to the right by two digits. But there is only one digit after the comma in the dividend In this case, one more zero must be added at the end. As a result, we get 630
Let's try to divide 6.3 by 0.001. The divisor 0.001 has three zeros. This means that in the dividend 6.3, you need to move the comma to the right by three digits:
6,3: 0,001 = 6300
Self-help assignments
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Like regular numbers.
2. We count the number of decimal places in the 1st decimal fraction and in the 2nd. We add up their number.
3. In the final result, count from right to left as many digits as you get in the paragraph above, and put a comma.
Decimal multiplication rules.
1. Multiply without paying attention to the comma.
2. In the product, separate as many digits after the comma as there are after the commas in both factors together.
Multiplying a decimal fraction by a natural number, you need:
1. Multiply numbers without paying attention to the comma;
2. As a result, put the comma so that there are as many digits to the right of it as there are in decimal fraction.
Column multiplication of decimal fractions.
Let's take an example:
We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. Those. We regard 3.11 as 311, and 0.01 as 1.
The result is 311. Then we count the number of decimal places for both fractions. In the 1st decimal there are 2 digits and in the 2nd - 2. The total number of digits after the commas:
2 + 2 = 4
We count from right to left four characters in the result. In the final result, there are fewer numbers than you need to separate with a comma. In this case, it is necessary to add the missing number of zeros to the left.
In our case, the 1st digit is missing, so we add 1 zero to the left.
Note:
Multiplying any decimal fraction by 10, 100, 1000 and so on, the decimal point is moved to the right by as many digits as there are zeros after one.
for example:
70,1 . 10 = 701
0,023 . 100 = 2,3
5,6 . 1 000 = 5 600
Note:
To multiply a decimal by 0.1; 0.01; 0.001; and so on, you need to move the comma to the left in this fraction by as many digits as there are zeros in front of the unit.
We count zero integers!
For example:
12 . 0,1 = 1,2
0,05 . 0,1 = 0,005
1,256 . 0,01 = 0,012 56
§ 1 Application of the rule of multiplication of decimal fractions
In this lesson, you will get acquainted and learn how to apply the rule for multiplying decimal fractions and the rule for multiplying a decimal fraction by a digit unit, such as 0.1, 0.01, etc. In addition, we will look at the properties of multiplication when finding the values \u200b\u200bof expressions containing decimal fractions.
Let's solve the problem:
The vehicle travels at a speed of 59.8 km / h.
Which way will the car cover in 1.3 hours?
As you know, to find a path, you need to multiply the speed by time, i.e. 59.8 times 1.3.
Let's write down the numbers in a column and start multiplying them, not noticing the commas: 8 multiplied by 3, it will be 24, 4 we write 2 in the mind, 3 multiplied by 9 is 27, plus 2, we get 29, we write 9, 2 in the mind. Now we multiply 3 by 5, it will be 15 and add 2 more, we get 17.
We pass to the second line: 1 multiplied by 8, it will be 8, 1 multiplied by 9, we get 9, 1 multiplied by 5, we get 5, add these two lines, we get 4, 9 + 8 equals 17, 7 write 1 in our mind, 7 +9 is 16 and 1 more, it will be 17, 7 we write 1 in our mind, 1 + 5 and 1 more we get 7.
Now let's see how many decimal places are there in both decimal fractions! In the first fraction there is one digit after the decimal point and in the second fraction there is one digit after the decimal point, only two digits. This means that on the right side of the result you need to count two digits and put a comma, i.e. will be 77.74. So, when multiplying 59.8 by 1.3, we got 77.74. So the answer in the problem is 77.74 km.
Thus, to multiply two decimal fractions, you need:
First: do the multiplication, ignoring the commas
Second: in the resulting product, separate as many digits on the right with a comma as there are after the comma in both factors together.
If there are less digits in the resulting product than must be separated by a comma, then one or more zeros must be added in front.
For example: 0.145 multiplied by 0.03, we get 435 in the product, and we need to separate 5 digits from the right with a comma, so we add 2 more zeros before the number 4, put a comma and add one more zero. We get the answer 0.00435.
§ 2 Properties of multiplication of decimal fractions
When multiplying decimal fractions, all the same properties of multiplication are preserved as for natural numbers. Let's do a few tasks.
Task number 1:
Let's solve this example by applying the distribution property of multiplication to addition.
We put 5.7 (common factor) outside the parenthesis, in parentheses there will be 3.4 plus 0.6. The value of this sum is 4, and now 4 must be multiplied by 5.7, we get 22.8.
Task number 2:
Let's apply the transposition property of multiplication.
First we multiply 2.5 by 4, we get 10 integers, and now we need to multiply 10 by 32.9 and we get 329.
In addition, when multiplying decimal fractions, you can notice the following:
When multiplying a number by an incorrect decimal, i.e. greater than or equal to 1, it increases or does not change, for example:
When multiplying a number by a correct decimal fraction, i.e. less than 1, it decreases, for example:
Let's solve an example:
23.45 times 0.1.
We have to multiply 2345 by 1 and separate the three decimal places on the right, we get 2.345.
Now let's solve another example: 23.45 divided by 10, we have to move the comma to the left one digit, because 1 is a zero in a bit, we get 2.345.
From these two examples, we can conclude that multiplying a decimal fraction by 0.1, 0.01, 0.001, etc. means dividing the number by 10, 100, 1000, etc., i.e. it is necessary to move the comma to the left in the decimal fraction by as many digits as there are zeros before 1 in the multiplier.
Using the resulting rule, we find the values \u200b\u200bof the products:
13.45 times 0.01
there are 2 zeros in front of the number 1, so we move the comma to the left by 2 digits, we get 0.1345.
0.02 times 0.001
before the number 1 there are 3 zeros, which means we move the comma three characters to the left, we get 0.00002.
Thus, in this lesson you learned how to multiply decimal fractions. To do this, you just need to perform multiplication, ignoring the commas, and in the resulting product, separate as many digits on the right with a comma as there are after the comma in both factors together. In addition, we got acquainted with the rule for multiplying a decimal fraction by 0.1, 0.01, etc., and also considered the properties of multiplying decimal fractions.
List of used literature:
- Mathematics grade 5. Vilenkin N.Ya., Zhokhov V.I. et al. 31st ed., erased. - M: 2013.
- Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
- We calculate without errors. Works with self-test in mathematics 5-6 grades. Author - Minaeva S.S. - year 2014
- Didactic materials in mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
- Control and independent work in mathematics, grade 5. Authors - Popov M.A. - year 2012
- Maths. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Erased. - M .: Mnemosina, 2009
To understand how to multiply decimal fractions, let's look at specific examples.
Decimal multiplication rule
1) Multiply, ignoring the comma.
2) As a result, we separate as many digits after the comma as there are after the commas in both factors together.
Examples.
Find the product of decimal fractions:
To multiply decimal fractions, we multiply without paying attention to the commas. That is, we are not multiplying 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the comma as there are after the commas in both factors together. The first multiplier after the decimal point has one digit, the second - also one. In total, we separate two digits after the decimal point. Thus, we got the final answer: 6.8 ∙ 3.4 \u003d 23.12.
Multiply decimals without taking into account the comma. That is, in fact, instead of multiplying 36.85 by 1.14, we multiply 3685 by 14. We get 51590. Now, in this result, we need to separate as many digits with a comma as there are in both factors together. The first number after the decimal point has two digits, the second - one. In total, we separate three digits with a comma. Since there is a zero at the end of the entry after the comma, we do not write it in response: 36.85 ∙ 1.4 \u003d 51.59.
To multiply these decimal fractions, we multiply the numbers, ignoring the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, you need to separate four digits after the decimal point - as many as there are in both factors together (two in each). The final answer: 23.15 ∙ 0.07 \u003d 1.6205.
Multiplication of a decimal fraction by a natural number is performed in the same way. We multiply the numbers, not paying attention to the comma, that is, we multiply 75 by 16. In the result, after the comma, there should be as many digits as there are in both factors together - one. Thus, 75 ∙ 1.6 \u003d 120.0 \u003d 120.
We start multiplying decimal fractions by multiplying natural numbers, since we don't pay attention to commas. After that, we separate after the decimal point as many digits as there are in both factors together. The first number has two decimal places, the second has two. In total, as a result, there should be four digits after the decimal point: 4.72 ∙ 5.04 \u003d 23.7888.
You already know that a * 10 \u003d a + a + a + a + a + a + a + a + a + a.For example, 0.2 * 10 \u003d 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2. It is easy to guess that this amount is equal to 2, i.e. 0.2 * 10 \u003d 2.
Similarly, you can make sure that:
5,2 * 10 = 52 ;
0,27 * 10 = 2,7 ;
1,253 * 10 = 12,53 ;
64,95 * 10 = 649,5 .
You probably guessed that when multiplying a decimal fraction by 10, you need to move the comma to the right by one digit in this fraction.
How to multiply a decimal by 100?
We have: a * 100 \u003d a * 10 * 10. Then:
2,375 * 100 = 2,375 * 10 * 10 = 23,75 * 10 = 237,5 .
Arguing similarly, we obtain that:
3,2 * 100 = 320 ;
28,431 * 100 = 2843,1 ;
0,57964 * 100 = 57,964 .
Multiply the fraction 7.1212 by 1000.
We have: 7.1212 * 1000 \u003d 7.1212 * 100 * 10 \u003d 712.12 * 10 \u003d 7121.2.
These examples illustrate the following rule.
To multiply a decimal fraction by 10, 100, 1,000, etc., you need to move the comma to the right in this fraction by 1, 2, 3, etc., respectively. figures.
So, if you move the comma to the right by 1, 2, 3, etc. digits, then the fraction will increase respectively by 10, 100, 1,000, etc. time.
Hence, 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. time .
Let us show that the decimal form of writing fractions gives the opportunity to multiply them, guided by the rule of multiplying natural numbers.
Let's find, for example, the product 3.4 * 1.23. Let's increase the first factor by 10 times, and the second by 100 times. This means that we have enlarged the work 1000 times.
Therefore, the product of natural numbers 34 and 123 is 1,000 times larger than the desired product.
We have: 34 * 123 \u003d 4182. Then, to get the answer, the number 4 182 must be reduced by 1000 times. We write: 4 182 \u003d 4 182.0. Moving the comma in the number 4 182.0 three digits to the left, we get the number 4.182, which is 1,000 times less than the number 4 182. Therefore 3.4 * 1.23 \u003d 4.182.
The same result can be obtained using the following rule.
To multiply two decimal fractions, you need:
1) multiply them as natural numbers, not paying attention to commas;
2) in the resulting product, separate as many digits with a comma on the right as they are after the commas in both factors together.
In cases where the product contains fewer digits than it is required to separate with a comma, the required number of zeros is added to the left before this, and then the comma is moved to the left by the required number of digits.
For example, 2 * 3 \u003d 6, then 0.2 * 3 \u003d 0.006; 25 * 33 \u003d 825, then 0.025 * 0.33 \u003d 0.00825.
In cases where one of the factors is 0.1; 0.01; 0.001, etc., it is convenient to use the following rule.
To multiply a decimal by 0.1; 0.01; 0.001, etc., it is necessary in this fraction to move the comma to the left, respectively, 1, 2, 3, etc. figures.
For example, 1.58 * 0.1 \u003d 0.158; 324.7 * 0.01 \u003d 3.247.
The properties of multiplication of natural numbers are also fulfilled for fractional numbers:
ab \u003d ba is the displacement property of multiplication,
(ab) c \u003d a (b c) is the combination property of multiplication,
a (b + c) \u003d ab + ac - the distribution property of multiplication with respect to addition.
