Comma 3 0. Most likely: to put commas or not to put? Multiplying a decimal by a regular number
The expression “most likely” causes many difficulties with punctuation, since it may or may not require commas depending on its role in the sentence (context). However, learning to determine whether separation is necessary in a given situation is not a difficult matter.
Introductory construction
To correctly place punctuation marks, you need to determine whether the expression “most likely” is an introductory phrase.
What does it mean?
An introductory word (or a stable combination of words) is a construction that is not a member of the sentence and is not syntactically related to any of its members. It is impossible to ask a question to her either from the subject, or from the predicate, or from the secondary members, and it is also impossible to ask a question from her to other members.
Introductory words can, for example, convey the emotional coloring of a sentence (“fortunately,” “unfortunately”), express confidence (“of course,” “of course”) or uncertainty (“probably,” “maybe”) of the author, or indicate a reference to someone's opinion (“in my opinion”, “they say”).
“Most likely” is highlighted with commas if this is an introductory phrase with the meaning of uncertainty, since an introductory word or expression always requires isolation.
How to determine this?
- The introductory phrase can be rearranged into any part of the sentence without losing its meaning. If “most likely” is at the beginning of a sentence, then it can be used at the end or middle, and the essence of the sentence will remain unchanged.
- The introductory phrase can be replaced by any other synonymous introductory construction. You should try to replace the introductory expression “most likely” with the introductory word “probably” or the construction “maybe”. If “most likely” is an introductory word, then the degree of confidence will change, but the meaning of the statement will not disappear.
- The introductory turnover can be excluded. The sentence must remain grammatically correct.
If the conditions are met, “most likely” is separated by commas.
A phrase consisting of an adjective and a pronoun
The word "more likely" can be a comparative adjective and part of the predicate. Then “total” is a dependent word, also part of the predicate, and is a attributive pronoun.
How to determine this?
It is enough to check the same three conditions.
If the conditions are not met, that is, when discarded, moved to another part of the sentence or replaced with introductory constructions “maybe”, “probably” the sentence loses its meaning or becomes grammatically incorrect, “most likely” is not separated by commas.
Examples
Consider two similar proposals:
This behavior was most likely predicted in advance.
This behavior was most likely.
In the first case, to understand whether commas are needed, we move them to the beginning of the sentence “most likely”:
Most likely, this behavior was predicted in advance.
Replace the phrase with “probably”:
This behavior was probably predicted in advance.
Now let's try to discard the phrase in question:
This behavior was predicted in advance.
In all three cases, the sentence retained its meaning and remained grammatically correct. We can conclude that in this sentence “most likely” is an introductory construction. Separate with commas on both sides. Of course, except at the very beginning or end of a sentence, when a comma on one side is enough.
Let's move on to the second sentence.
Let's move "most likely" to the beginning of the sentence.
Most likely this was the behavior.
As you can see, the result is a phrase that is extremely inconvenient to understand. But to be sure, let's check the other two signs.
Let's replace it with "probably":
This kind of behavior probably happened.
The meaning is completely lost.
If we discard “most likely”, we are left with:
There was such behavior.
In this case, too, the meaning is completely lost.
Conclusion: in the considered sentence, “most likely” is not an introductory word. This means we don’t separate “most likely” with commas.
A comma between independent clauses combined into one complex and between subordinate clauses relating to the same main clause
A comma is placed between sentences that are combined into one difficult sentence through repeated conjunctions and...and, neither...nor, or...or, etc.
P., for example: And everything feels nauseous, and my head is spinning, and there are bloody boys in my eyes... (Pushkin).
A comma is placed between sentences that are combined into one complex sentence using the conjunctions and, yes (meaning “and”), yes and, or, or, as well as the conjunctions a and yes (meaning “but”), for example: The sea murmured dully, and the waves beat against the shore madly and angrily (M. Gorky).
Note. A comma is not placed before conjunctions and, yes (in the meaning of “and”), or, or if the sentences they connect have a common secondary member or a common subordinate clause. The presence of a common minor member or a common subordinate clause closely connects such sentences into one whole, for example: Trucks were moving along the streets and cars were racing. Every morning a boat or boat departed from the pier. The stars were already beginning to fade and the sky was turning grey, when the carriage drove up to the wing of the house in Vasilievsky (Turgenev).
A comma is placed between independent sentences that are combined into one complex without the help of conjunctions or through conjunctions but, however, nevertheless, nevertheless, only in cases where such sentences are closely related to each other in meaning, for example: The horses started moving, the bell rang, the wagon flew away (Pushkin).
A comma is placed between subordinate clauses that relate to the same main clause. My father eagerly and in detail told me how many birds and fish there are, how many different berries there are, how many lakes there are, what wonderful forests grow (S. Aksakov).
If such subordinate clauses are connected through single conjunctions and, yes (in the meaning of “and”), then no punctuation mark is placed between them, for example: She dreamed out loud about how she would live in Dubechnya and what it would be like interesting life(Chekhov).
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, the whole 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 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
In this expression different quantities numbers after the decimal point. 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 received 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 to 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 that 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 making 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 by 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 previous lessons we said that when a smaller number is divided by a larger number, a fraction is obtained, the numerator of which contains the dividend, and the denominator contains 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 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, 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 interesting features 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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Dividing by a decimal fraction is reduced to dividing by natural number.
The rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, you need to move the comma in both the dividend and the divisor to the right as many digits as there are in the divisor after the decimal point. After this, divide by a natural number.
Examples.
Divide by decimal fraction:
To divide by a decimal, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are after the decimal point in the divisor, that is, by one digit. We get: 35.1: 1.8 = 351: 18. Now we perform the division with a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To divide decimal fractions, in both the dividend and the divisor we move the decimal point to the right one place: 14.76: 3.6 = 147.6: 36. Now we perform a natural number. Result: 14.76: 3.6 = 4.1.
To divide a natural number by a decimal fraction, you need to move both the dividend and the divisor to the right as many places as there are in the divisor after the decimal point. Since a comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 = 7000: 175. Divide the resulting natural numbers with a corner: 70: 1.75 = 7000: 175 = 40.
4) 0,1218: 0,058
To divide one decimal fraction by another, we move the decimal point to the right in both the dividend and the divisor by as many digits as there are in the divisor after the decimal point, that is, by three decimal places. Thus, 0.1218: 0.058 = 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8
A comma usually indicates that the thought is incomplete. It is placed only inside a sentence and serves to separate simple phrases inside complex ones. As linguists point out, the comma often corresponds in importance to other punctuation marks that are placed only within phrases. In the case of opposition or comparison, the comma is replaced by a dash, colon or semicolon.
The comma is significant in its ambiguity and may require different intonation in different cases. A comma requires a raised voice before it on a stressed word.
For example:
“Days after days passed, | and there was no end in sight to the disputes between the crucian carp and the ruff. (Saltykov-Shchedrin “Crucian carp the idealist”)
The stressed word preceding the comma may not necessarily be immediately before the comma, but the rise in voice occurs precisely at the stressed word.
For example:
“The early willow blossomed, | and a bee flew to her, | and the bumblebee buzzed, | and the first butterfly folded its wings.” (M. Prishvin “In the Land of Grandfather Mazai”)
Stanislavsky writes that with a comma, you want to “bend the sound upward” and leave “the top note hanging in the air for a while. With this bend, the sound is transferred from bottom to top, like an object from a lower shelf to a higher one... the most remarkable thing in the nature of the comma is that, as if a hand was raised in warning, it forces listeners to patiently wait for the continuation of an unfinished phrase.”
When enumerating, the comma requires repeated, almost identical increases in voice on each of the listed words, and on the last one the voice drops to a point.
For example:
“There is no shortage of nuts, | lingonberry | and blueberries." (A. Pushkin “History of the village of Goryukhin”)
When a comma is not “readable”. The comma is not separated by a pause in oral speech.
1) Before or after the introductory word.
Without pauses the following is pronounced in oral speech: introductory words, as “of course”, “probably”, “perhaps”, “probably”, “it seems”, “maybe”, “however”, “what good”, “in my opinion”, “unfortunately”, “finally” and so on.
For example:
“And yet, Claudia was probably removed from the team leader ten times, and even now she is officially listed as “acting”.” (F. Abramov “Around and Around”)
“The heroine of this novel, | it goes without saying | there was Masha." (L. Tolstoy “Adolescence”)
2) Between the conjunction “and” and the participial phrase.
For example:
“The Chechen looked at him and, slowly turning away, began to look at the other shore.” (L. Tolstoy “Cossacks”)
3) Before the participial phrase, if it comes after the word being defined.
For example:
“A person (,) loving animals, | - poet." (Yu. Olesha “Not a day without a line”)
In the example given, the definition is a unity with the word being defined: not just “man,” but “a person who loves animals.”
But depending on the context, this rule may be broken.
4) Before comparative turnover.
For example:
"Herman | trembled (,) like a tiger, waiting for the appointed time.” (A. Pushkin “The Queen of Spades”)
5) The comma is often “unreadable” in complex sentences, when the connection between the main part and the subordinate part is carried out by conjunctions: “who”, “what”, “which”; in complex words: “because”, “so that”, “in order to”; relations: “all that”, “that which”.
For example:
“Borya felt | how his back and crown grow cold |, realizing (,) that she, | Lucy, | Even now he sees something terrible. (Astafiev “The Shepherd and the Shepherdess”)
“It is true (,) that we have books, | but this is not at all the same (,) as live conversation and society.” (A. Chekhov “Ward No. 6”)
“I invited you (,) gentlemen, | in order (,) to inform you | very unpleasant news." (N.V. Gogol “The Inspector General”)
“As a child | the whole world | belongs to the child, | and Akim | everything (,) that I saw, | turned into his own experience, | thought to myself as about a tree, | about an ant, | about the wind, | to guess | why do they live, | and what makes them feel good.” (Platonov “The Light of Books”)
6) Before an address in the middle or at the end of a sentence.
For example:
“But there is a lot of happiness, so much (,) guy, | that there would be enough for the whole district, | let not a single soul see him!” (A. Chekhov “Happiness”)
“I don’t blame you (,) Alexey Nikolaevich.” (I. Turgenev “A Month in the Village”)