In this tutorial we will look at each of these operations separately.
Lesson contentAdding Decimals
As we know, a decimal fraction consists of an integer and a fractional part. When adding decimals, integer and fractional parts are added separately.
For example, let's add the decimal fractions 3.2 and 5.3. It is more convenient to add decimal fractions in a column.
Let us first write these two fractions in a column, with the integer parts necessarily being under the integers, and the fractional parts under the fractional ones. At 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 add the fractional parts: 2 + 3 = 5. We write the five in the fractional part of our answer:
Now we add up the whole parts: 3 + 5 = 8. We write an eight 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 follow the rule "comma under comma" :
We received an answer of 8.5. This means that the expression 3.2 + 5.3 equals 8.5
3,2 + 5,3 = 8,5
In fact, not everything is as simple as it seems at first glance. There are also pitfalls here, which we will talk about now.
Places in decimals
Decimal fractions, like ordinary numbers, have their own digits. These are places of tenths, places of hundredths, places of thousandths. 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, and the third digit after the decimal point for the thousandths place.
Places in decimal fractions contain some useful information. Specifically, they tell you how many tenths, hundredths, and thousandths there are in a decimal.
For example, consider the decimal fraction 0.345
The position where the three 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 thousandth place
Let's look at this drawing. We see that there is a three in the tenths place. This means that there are three tenths in the decimal fraction 0.345.
If we add the fractions, we get the original decimal fraction 0.345
At first we got the answer, but we converted it to a decimal fraction and got 0.345.
When adding decimal fractions, the same rules apply as when adding ordinary numbers. The addition of decimal fractions occurs in digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.
Therefore, when adding decimal fractions, you must follow the rule "comma under comma". The comma under the comma provides the very 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 up the fractional parts 5 + 4 = 9. We write nine in the fractional part of our answer:
Now we add the integer parts 1 + 3 = 4. We write the four in the integer part of our answer:
Now we separate the whole part from the fractional part with a comma. To do this, we again follow the “comma under comma” rule:
We received an answer of 4.9. This means that 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 “comma under comma” rule.
First of all, we add up the fractional part, namely the hundredths of 1+2=3. We write a triple in the hundredth part of our answer:
Now add the tenths 5+2=7. We write a seven in the tenth part of our answer:
Now we add the whole parts 3+1=4. We write the four in the whole part of our answer:
We separate the whole part from the fractional part with a comma, observing the “comma under comma” rule:
The answer we received was 4.73. This means the value of the expression 3.51 + 1.22 is equal to 4.73
3,51 + 1,22 = 4,73
As with regular numbers, when adding decimals, . 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 the column:
Add the hundredths parts 5+7=12. The number 12 will not fit into the hundredth part of our answer. Therefore, in the hundredth part we write the number 2, and move the unit to the next digit:
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 we add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:
The answer we received was 5.92. This means the value of the expression 2.65 + 3.27 is equal to 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 the column
We add the fractional parts 5 + 8 = 13. The number 13 will not fit into the fractional part of our answer, so we first write down the number 3, and move 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 whole part from the fractional part with a comma:
We received the answer 12.3. This means that the value of the expression 9.5 + 2.8 is 12.3
9,5 + 2,8 = 12,3
When adding decimals, 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 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, but the fraction 1.7 has only one. This means that in the fraction 1.7 you need to add two zeros at the end. Then we get the fraction 1.700. Now you can write this expression in a column and start calculating:
Add the thousandths parts 5+0=5. We write the number 5 in the thousandth part of our answer:
Add the hundredths parts 2+0=2. We write the number 2 in the hundredth part of our answer:
Add the tenths 7+7=14. The number 14 will not fit into a tenth of our answer. Therefore, we first write down the number 4, and move the unit to the next digit:
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 whole part from the fractional part with a comma:
We received a response of 14,425. This means the value of the expression 12.725+1.700 is 14.425
12,725+ 1,700 = 14,425
Subtracting Decimals
When subtracting decimal fractions, you must follow the same rules as when adding: “comma under the decimal point” and “equal number of digits after the 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:
We calculate the integer part 2−2=0. We write zero in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
We received an answer of 0.3. This means 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
This expression has a different number of decimal places. The fraction 7.353 has three digits after the decimal point, but the fraction 3.1 has only one. This means that in the fraction 3.1 you need to add two zeros 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:
We received a response of 4,253. This means 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 will have to borrow one from an adjacent digit if subtraction becomes impossible.
Example 3. Find the value of the expression 3.46 − 2.39
Subtract hundredths of 6−9. You cannot subtract the number 9 from the number 6. Therefore, you need to borrow one from the adjacent digit. By borrowing one from the adjacent digit, the number 6 turns into the number 16. Now you can calculate the hundredths of 16−9=7. We write a seven in the hundredth part of our answer:
Now we subtract tenths. Since we took one unit in the tenths place, the figure that was located there decreased by one unit. In other words, in the tenths place there 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 we subtract the whole parts 3−2=1. We write one in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
We received an answer of 1.07. This means 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 a whole number. Let us write this expression in a column so that whole part the decimal fraction 1.23 turned out to be the number 3
Now let's make the number of digits after the decimal point the same. To do this, after the number 3 we put a comma and add one zero:
Now we subtract tenths: 0−2. You cannot subtract the number 2 from zero. Therefore, you need to borrow one from the adjacent digit. Having borrowed one from the neighboring digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write an eight in the tenth part of our answer:
Now we subtract the whole parts. Previously, the number 3 was located in the whole, but we took one unit from it. As a result, it turned into the number 2. Therefore, from 2 we subtract 1. 2−1=1. We write one in the integer part of our answer:
Separate the whole part from the fractional part with a comma:
The answer we received was 1.8. This means the value of the expression 3−1.2 is 1.8
Multiplying Decimals
Multiplying decimals is simple and even fun. To multiply decimals, you multiply them like regular numbers, ignoring the commas.
Having received the answer, 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 both fractions, then count the same number of digits from the right in the answer and put a comma.
Example 1. Find the value of the expression 2.5 × 1.5
Let's multiply these decimal fractions like ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:
We got 375. 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 fractions 2.5 and 1.5. The first fraction has one digit after the decimal point, and the second fraction also has one. Total two numbers.
We return to the number 375 and begin to move from right to left. We need to count two digits to the right and put a comma:
We received an answer of 3.75. So the value of the expression 2.5 × 1.5 is 3.75
2.5 × 1.5 = 3.75
Example 2. Find the value of the expression 12.85 × 2.7
Let's multiply these decimal fractions, 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 count the number of digits after the decimal point in the fractions 12.85 and 2.7. The fraction 12.85 has two digits after the decimal point, and the fraction 2.7 has 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:
We received a response of 34,695. So the value of the expression 12.85 × 2.7 is 34.695
12.85 × 2.7 = 34.695
Multiplying a decimal by a regular number
Sometimes situations arise when you need to multiply a decimal fraction by a regular number.
To multiply a decimal and a number, you multiply them without paying attention to the comma in the decimal. Having received the answer, 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 decimal fraction, then count the same number of digits from the right in the answer and put a comma.
For example, multiply 2.54 by 2
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 number 508 and begin to move from right to left. We need to count two digits to the right and put a comma:
We received an answer of 5.08. So the value of the expression 2.54 × 2 is 5.08
2.54 × 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. You need to perform the multiplication, not paying attention to the comma in the decimal fraction, then in the answer, separate the whole part from the fractional part, counting from the right the same number of digits as there were digits after the decimal point.
For example, multiply 2.88 by 10
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 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 2.88. We see that the fraction 2.88 has 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 to the right and put a comma:
We received an answer of 28.80. Let's drop the last zero and get 28.8. This means the value of the expression 2.88×10 is 28.8
2.88 × 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 moving the decimal point to the right by as many digits as there are zeros in the factor.
For example, let's solve the previous example 2.88×10 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 we move the decimal point to the right one digit, we get 28.8.
2.88 × 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 there are in it. We see that there are two zeros in it. Now in the fraction 2.88 we move the decimal point to the right two digits, we get 288
2.88 × 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 there are in it. We see that there are three zeros in it. Now in the fraction 2.88 we move the decimal point to the right by three digits. There is no third digit there, so we add another zero. As a result, we get 2880.
2.88 × 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 the 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:
We got 325. 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 fractions 3.25 and 0.1. The fraction 3.25 has two digits after the decimal point, and the fraction 0.1 has one digit. Total three numbers.
We return to the number 325 and begin to move from right to left. We need to count three digits from the right and put a comma. After counting down three digits, we find that the numbers have run out. In this case, you need to add one zero and add a comma:
We received an answer of 0.325. This means that the value of the expression 3.25 × 0.1 is 0.325
3.25 × 0.1 = 0.325
There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much simpler and more convenient. It consists in moving the decimal point to the left by as many digits as there are zeros in the factor.
For example, let's solve the previous example 3.25 × 0.1 this way. Without giving any calculations, we immediately look at the multiplier of 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 we move the decimal point to the left by one digit. By 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. The result is 0.325
3.25 × 0.1 = 0.325
Let's try multiplying 3.25 by 0.01. We immediately look at the multiplier of 0.01. 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 we move the decimal point to the left two digits, we get 0.0325
3.25 × 0.01 = 0.0325
Let's try multiplying 3.25 by 0.001. We immediately look at the multiplier of 0.001. 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 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 decimal fractions 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 decimal point is moved to the right by the same number of digits as there are zeros in the multiplier.
And when multiplying by 0.1, 0.01 and 0.001, the decimal point is moved to the left by the same number of digits as there are zeros in the multiplier.
If at first it is difficult to remember, you can use the first method, in which multiplication is performed as with ordinary numbers. In the answer, you will need to separate the whole part from the fractional part, counting the same number of digits on the right as there are digits after the decimal point in both fractions.
Dividing a smaller number by a larger number. Advanced level.
In one of the previous lessons, we said that when dividing a smaller number by a larger number, a fraction is obtained, the numerator of which is the dividend, and the denominator is the divisor.
For example, to divide one apple between two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. As a result, we get the fraction . This means each friend will get an apple. In other words, half an apple. The fraction is the answer to the problem “how to divide one apple into two”
It turns out that you can solve this problem further if you divide 1 by 2. After all, the fractional line in any fraction means division, and therefore this division is allowed in the fraction. But how? We are accustomed to the fact that the dividend is always greater than the divisor. But here, on the contrary, the dividend is less than the divisor.
Everything will become clear if we remember that a fraction means crushing, division, division. This means that the unit can be split into as many parts as desired, and not just into two parts.
When you divide a smaller number by a larger number, you get a decimal fraction in which the integer part is 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 completely divided into two. If you ask a question “how many twos are there 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 to get 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. Divide 10 by 2, we get 5. We write the five in the fractional part of our answer:
Now we take out the last remainder to complete the calculation. Multiply 5 by 2 to get 10
We received an answer of 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 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 there in a four? Not at all. We write 0 in the quotient and put a comma:
We multiply 0 by 5, we get 0. We write a 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, add a zero to the right of 4 and divide 40 by 5, we get 8. We write eight in the quotient.
We complete the example by multiplying 8 by 5 to get 40:
We received an answer of 0.8. This means the value of the expression 4:5 is 0.8
Example 3. Find the value of expression 5: 125
How many numbers are 125 in five? Not at all. We write 0 in the quotient and put a comma:
We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract 0 from five
Now let's start splitting (dividing) the five into 125 parts. To do this, we write a zero to the right of this five:
Divide 50 by 125. How many numbers are 125 in the number 50? Not at all. So in the quotient we write 0 again
Multiply 0 by 125, we get 0. Write this zero under 50. Immediately subtract 0 from 50
Now divide the number 50 into 125 parts. To do this, we write another zero to the right of 50:
Divide 500 by 125. How many numbers are 125 in the number 500? There are four numbers 125 in the number 500. Write the four in the quotient:
We complete the example by multiplying 4 by 125 to get 500
We received an answer of 0.04. This means the value of expression 5: 125 is 0.04
Dividing numbers without a remainder
So, let’s put a comma after the unit in the quotient, thereby indicating that the division of integer parts is over and we are proceeding to the fractional part:
Let's add zero to the remainder 4
Now divide 40 by 5, we get 8. We write eight in the quotient:
40−40=0. We got 0 left. This means that the division is completely completed. Dividing 9 by 5 gives the decimal fraction 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:
We got 16 in private and 4 more left. Now let's 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. We write the eight in the quotient after the decimal point:
and complete the example by checking whether 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, 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 normal 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 equals two. We write two in the quotient 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. Remainder equal to zero. We do not write down zero yet, since the solution is not completed. Next, 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:
We received an answer of 2.4. The value of the expression 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 2:
Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:
Divide 24 by 3, we get 8. We write eight in the quotient. Immediately multiply it by the divisor to find the remainder of the division:
24−24=0. The remainder is zero. We don’t write down zero yet. We take away the last three from the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:
The answer we received was 2.81. This means the value of the expression 8.43: 3 is 2.81
Dividing a decimal by a decimal
To divide a decimal fraction by a decimal fraction, you need to move the decimal point in the dividend and divisor to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by the usual number.
For example, divide 5.95 by 1.7
Let's write this expression with a corner
Now in the dividend and in the divisor we move the decimal point 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. This means that in the dividend and divisor we must move the decimal point to the right by one digit. We transfer:
After moving the decimal point to the right one digit, the decimal fraction 5.95 became the 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 a decimal fraction by a regular number. Further calculation is not difficult:
The comma is moved to the right to make division easier. This is allowed because 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 quotient 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 comes out of it:
(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 move 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 transformed into the fraction 59.1 and the fraction 1.7 was transformed into the usual number 17.
In fact, inside this process there was a multiplication by 10. This is what it looked like:
5.91 × 10 = 59.1
Therefore, the number of digits after the decimal point in the divisor determines 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 decimal point 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, divide 2.1 by 10. Solve this example using a corner:
But there is 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 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. This means that in the dividend of 2.1 you need to move the decimal point to the left by one digit. We move the comma to the left one digit and see that there are no more digits left. In this case, add another 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. This means that in the dividend 2.1 we 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
Dividing 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, you need to move the decimal point 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, let’s move the commas in the dividend and 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. This means we move the commas in the dividend and divisor to the right by one digit.
After moving the decimal point to the right one digit, the decimal fraction 6.3 becomes the usual number 63, and the decimal fraction 0.1 after moving the decimal point to the right one digit turns into one. And dividing 63 by 1 is very simple:
This means the value of the expression 6.3: 0.1 is 63
But there is a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the right by as many digits as there are zeros in the divisor.
Let's solve the previous example this way. 6.3: 0.1. Let's look at the divisor. 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 decimal point 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 of 0.01 has two zeros. This means that in the dividend 6.3 we need to move the decimal point to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, you need to add another zero at the end. As a result we get 630
Let's try to divide 6.3 by 0.001. The divisor of 0.001 has three zeros. This means that in the dividend 6.3 we need to move the decimal point to the right by three digits:
6,3: 0,001 = 6300
Tasks for independent solution
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Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
The purpose of the lesson:
- In a fun way, introduce to students the rule for multiplying a decimal fraction by a natural number, by a place value unit, and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge when solving examples and problems.
- Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their own work and the work of each other.
- Cultivate interest in mathematics, activity, mobility, and communication skills.
Equipment: interactive board, a poster with a cyphergram, posters with statements by mathematicians.
During the classes
- Organizing time.
- Oral arithmetic – generalization of previously studied material, preparation for studying new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of acquired knowledge in game form using a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not be teaching it alone, but with my friend. And my friend is also unusual, you will see him now. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, buddy? Komposha replies: “My name is Komposha.” Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is hung on the board with an oral calculation for adding and subtracting decimal fractions, as a result of which the children receive the following code 523914687. )
| 5 | 2 | 3 | 9 | 1 | 4 | 6 | 8 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Komposha helps decipher the received code. The result of decoding is the word MULTIPLICATION. Multiplication is keyword topics of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how to multiply natural numbers. Today we will look at multiplying decimal numbers by a natural number. Multiplying a decimal fraction by a natural number can be considered as a sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 ·3 = 5.21 + 5.21 + 5.21 = 15.63 This means 5.21·3 = 15.63. Representing 5.21 as common fraction by a natural number, we get
And in this case we got the same result: 15.63. Now, ignoring the comma, instead of the number 5.21, take the number 521 and multiply it by this natural number. Here we must remember that in one of the factors the comma has been moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now in this example we move the comma to the left two places. Thus, by how many times one of the factors was increased, by how many times the product was decreased. Based on the similarities of these methods, we will draw a conclusion.
To multiply a decimal fraction by a natural number, you need to:
1) without paying attention to the comma, multiply natural numbers;
2) in the resulting product, separate as many digits from the right with a comma as there are in the decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21·3 = 15.63 and 7.624·15 = 114.34. Then I show multiplication by a round number 12.6·50 = 630. Next, I move on to multiplying a decimal fraction by a place value unit. I show the following examples: 7.423 ·100 = 742.3 and 5.2·1000 = 5200. So, I introduce the rule for multiplying a decimal fraction by a digit unit:
To multiply a decimal fraction by digit units 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many places as there are zeros in the digit unit.
I finish my explanation by expressing the decimal fraction as a percentage. I introduce the rule:
To express a decimal fraction as a percentage, you must multiply it by 100 and add the % sign.
I’ll give an example on a computer: 0.5 100 = 50 or 0.5 = 50%.
4. At the end of the explanation I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, we are doing a mathematical physical education session together with Komposha to consolidate the topic. Everyone stands up, shows the solved examples to the class, and they must answer whether the example was solved correctly or incorrectly. If the example is solved correctly, then they raise their arms above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and stretch their fingers.
6. And now you have rested a little, you can solve the tasks. Open your textbook to page 205, № 1029. In this task you need to calculate the value of the expressions:
The tasks appear on the computer. As they are solved, a picture appears with the image of a boat that floats away when fully assembled.
No. 1031 Calculate:
By solving this task on a computer, the rocket gradually folds up; after solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year, spaceships take off from the Baikonur Cosmodrome from Kazakhstan’s soil to the stars. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.
No. 1035. Problem.
How far will a passenger car travel in 4 hours if the speed of the passenger car is 74.8 km/h.
This task is accompanied by sound design and a brief condition of the task displayed on the monitor. If the problem is solved, correctly, then the car begins to move forward until the finish flag.
№ 1033. Write the decimals as percentages.
0,2 = 20%; 0,5 = 50%; 0,75 = 75%; 0,92 = 92%; 1,24 =1 24%; 3,5 = 350%; 5,61= 561%.
By solving each example, when the answer appears, a letter appears, resulting in a word Well done.
The teacher asks Komposha why this word would appear? Komposha replies: “Well done, guys!” and says goodbye to everyone.
The teacher sums up the lesson and gives grades.
You already know that a * 10 = a + a + a + a + a + a + a + a + a + a. For example, 0.2 * 10 = 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 sum is equal to 2, i.e. 0.2 * 10 = 2.
Similarly, you can verify 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 decimal point in this fraction to the right by one digit.
How to multiply a decimal fraction by 100?
We have: a * 100 = a * 10 * 10. Then:
2,375 * 100 = 2,375 * 10 * 10 = 23,75 * 10 = 237,5 .
Reasoning similarly, we get that:
3,2 * 100 = 320 ;
28,431 * 100 = 2843,1 ;
0,57964 * 100 = 57,964 .
Multiply the fraction 7.1212 by the number 1,000.
We have: 7.1212 * 1,000 = 7.1212 * 100 * 10 = 712.12 * 10 = 7121.2.
These examples illustrate the following rule.
To multiply a decimal fraction by 10, 100, 1,000, etc., you need to move the decimal point in this fraction to the right by 1, 2, 3, etc., respectively. numbers.
So, if the comma is moved to the right by 1, 2, 3, etc. numbers, then the fraction will increase accordingly by 10, 100, 1,000, etc. once.
Hence, if the comma is moved to the left by 1, 2, 3, etc. numbers, then the fraction will decrease accordingly by 10, 100, 1,000, etc. once .
Let us show that the decimal form of writing fractions makes it possible to multiply them, guided by the rule of multiplication of 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 increased the product by 1,000 times.
Therefore, the product of the natural numbers 34 and 123 is 1,000 times greater than the desired product.
We have: 34 * 123 = 4182. Then to get the answer you need to reduce the number 4,182 by 1,000 times. Let's write: 4 182 = 4 182.0. Moving the decimal point in the number 4,182.0 three digits to the left, we get the number 4.182, which is 1,000 times smaller than the number 4,182. Therefore 3.4 * 1.23 = 4.182.
The same result can be obtained using the following rule.
To multiply two decimal fractions:
1) multiply them as integers, ignoring commas;
2) in the resulting product, separate with a comma on the right as many digits as there are after the commas in both factors together.
In cases where the product contains fewer digits than required to be separated by a comma, the required number of zeros are added to the left before the product, and then the comma is moved to the left by the required number of digits.
For example, 2 * 3 = 6, then 0.2 * 3 = 0.006; 25 * 33 = 825, then 0.025 * 0.33 = 0.00825.
In cases where one of the multipliers 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., you need to move the decimal point in this fraction to the left, respectively, to 1, 2, 3, etc. numbers.
For example, 1.58 * 0.1 = 0.158 ; 324.7 * 0.01 = 3.247.
The properties of multiplication of natural numbers also apply to fractional numbers:
ab = ba is the commutative property of multiplication,
(ab) с = a(b с) – associative property of multiplication,
a(b + c) = ab + ac is the distributive property of multiplication relative to addition.
Just like regular numbers.
2. We count the number of decimal places for the 1st decimal fraction and for the 2nd. We add up their numbers.
3. In the final result, count from right to left the same number of digits as in the paragraph above, and put a comma.
Rules for multiplying decimal fractions.
1. Multiply without paying attention to the comma.
2. In the product, we separate the same number of digits after the decimal point as there are after the decimal points in both factors together.
When multiplying a decimal fraction by a natural number, you need to:
1. Multiply numbers without paying attention to the comma;
2. As a result, we place the comma so that there are as many digits to the right of it as there are in the decimal fraction.
Multiplying decimal fractions by column.
Let's look at an example:
We write the decimal fractions in a column and multiply them as natural numbers, not paying attention to the commas. Those. We consider 3.11 as 311, and 0.01 as 1.
The result is 311. Next, we count the number of signs (digits) after the decimal point for both fractions. The first decimal has 2 digits and the 2nd has 2. Total number digits after decimal points:
2 + 2 = 4
We count from right to left four digits of the result. The final result contains fewer numbers than need to be separated by a comma. In this case, you need to add the missing number of zeros to the left.
In our case, the first digit is missing, so we add 1 zero to the left.
Note:
When multiplying any decimal fraction by 10, 100, 1000, and so on, the decimal point in the decimal fraction is moved to the right by as many places as there are zeros after the 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 decimal point in this fraction to the left by as many places as there are zeros before the one.
We count zero integers!
For example:
12 . 0,1 = 1,2
0,05 . 0,1 = 0,005
1,256 . 0,01 = 0,012 56
In the last lesson, we learned how to add and subtract decimals (see lesson “Adding and subtracting decimals”). At the same time, we assessed how much calculations are simplified compared to ordinary “two-story” fractions.
Unfortunately, this effect does not occur with multiplying and dividing decimals. In some cases, decimal notation even complicates these operations.
First, let's introduce a new definition. We'll see him quite often, and not just in this lesson.
The significant part of a number is everything between the first and last non-zero digit, including the ends. We are talking about numbers only, the decimal point is not taken into account.
The digits included in the significant part of a number are called significant digits. They can be repeated and even be equal to zero.
For example, consider several decimal fractions and write out the 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 (significant figure only one: 3).
Please note: the 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 lesson “ Decimals”).
This point is so important, and mistakes are made here so often, that in the near future I will publish a test on this topic. Be sure to practice! And we, armed with the concept of the significant part, will proceed, in fact, to the topic of the lesson.
Multiplying Decimals
The multiplication operation consists of three successive 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 obtain the significant part of the desired fraction;
- Find out where and by how many digits the decimal point in the original fractions is shifted to obtain the corresponding significant part. Perform reverse shifts for 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 down the significant parts for the numbers from this expression: 28 and 125;
- Their product: 28 · 125 = 3500;
- In the first factor the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second it is shifted by 1 more digit. In total, you need a shift to the left by three digits: 3500 → 3,500 = 3.5.
Now let's look at 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 digit, respectively. 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 trailing zeros.
We reached 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 moves to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We shift by 5 to the left: 45,050 → .45050 = 0.4505. The zero was removed at the end, and added at the front so as not to leave a “naked” decimal point.
The following expression is: 0.0108 · 1600.5.
- We write the 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 there are 1. The total is 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).
Note in the last example: since the decimal point moves in different directions, the total shift is found through the difference. This is a 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 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 potential savings.
Therefore, let's look at a universal algorithm, which is a little longer, but much more reliable:
- Convert all decimal fractions to ordinary 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 lesson “Multiplying and dividing numerical fractions");
- If possible, present the result again as a decimal fraction. This step is also quick, since the denominator is often already a power of ten.
Task. Find the meaning of the expression:
- 3,51: 3,9;
- 1,47: 2,1;
- 6,4: 25,6:
- 0,0425: 2,5;
- 0,25: 0,002.
Let's consider the first expression. First, let's convert fractions to decimals:
Let's do the same with the second expression. The numerator of the first fraction will again be factorized:
There is an important point in the third and fourth examples: after getting rid of decimal notation reducible fractions arise. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction contains a prime number. There’s simply nothing to factorize here, so we consider it straight ahead:
Sometimes division results in an 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 arise that cannot be converted to decimals. This distinguishes division from multiplication, where the results are always represented 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 do not intentionally shorten ordinary fractions, derived from decimals. Otherwise, this will complicate the inverse task - 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.