In this article we will look at such an important operation with decimals as division. First let's formulate general principles, then we’ll figure out how to do division correctly decimals column for both other fractions and natural numbers. Next, we will analyze the division of ordinary fractions into decimals and vice versa, and at the end we will look at how to correctly divide fractions ending in 0, 1, 0, 01, 100, 10, etc.
Here we will take only cases with positive fractions. If there is a minus in front of the fraction, then to operate with it you need to study material about dividing rational and real numbers.
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All decimal fractions, both finite and periodic, are just a special form of writing ordinary fractions. Consequently, they are subject to the same principles as their corresponding ordinary fractions. Thus, we reduce the entire process of dividing decimal fractions to replacing them with ordinary ones, followed by calculation using methods already known to us. Let's take a specific example.
Example 1
Divide 1.2 by 0.48.
Solution
Let's write decimal fractions as ordinary fractions. We will get:
1 , 2 = 12 10 = 6 5
0 , 48 = 48 100 = 12 25 .
Thus, we need to divide 6 5 by 12 25. We count:
1, 2: 0, 48 = 6 2: 12 25 = 6 5 25 12 = 6 25 5 12 = 5 2
From the resulting improper fraction you can select the whole part and get mixed number 2 1 2, or you can represent it as a decimal fraction so that it corresponds to the original numbers: 5 2 = 2, 5. We have already written about how to do this earlier.
Answer: 1 , 2: 0 , 48 = 2 , 5 .
Example 2
Calculate how much 0 , (504) 0 , 56 will be.
Solution
First, we need to convert a periodic decimal fraction into a common fraction.
0 , (504) = 0 , 504 1 - 0 , 001 = 0 , 504 0 , 999 = 504 999 = 56 111
After this, we will also convert the final decimal fraction into another form: 0, 56 = 56,100. Now we have two numbers with which it will be easy for us to carry out the necessary calculations:
0 , (504) : 1 , 11 = 56 111: 56 100 = 56 111 100 56 = 100 111
We have a result that we can also convert to decimal form. To do this, divide the numerator by the denominator using the column method:
Answer: 0 , (504) : 0 , 56 = 0 , (900) .
If in the division example we encountered non-periodic decimal fractions, then we will act a little differently. We cannot reduce them to the usual ordinary fractions, so when dividing we have to first round them to a certain digit. This action must be performed with both the dividend and the divisor: we will also round the existing finite or periodic fraction in the interests of accuracy.
Example 3
Find how much 0.779... / 1.5602 is.
Solution
First, we round both fractions to the nearest hundredth. This is how we move from infinite non-periodic fractions to finite decimal ones:
0 , 779 … ≈ 0 , 78
1 , 5602 ≈ 1 , 56
We can continue the calculations and get an approximate result: 0, 779 ...: 1, 5602 ≈ 0, 78: 1, 56 = 78,100: 156,100 = 78,100 100,156 = 78,156 = 1 2 = 0, 5.
The accuracy of the result will depend on the degree of rounding.
Answer: 0 , 779 … : 1 , 5602 ≈ 0 , 5 .
How to divide a natural number by a decimal and vice versa
The approach to division in this case is almost the same: we replace finite and periodic fractions with ordinary ones, and round off infinite non-periodic ones. Let's start with the example of division with a natural number and a decimal fraction.
Example 4
Divide 2.5 by 45.
Solution
Let's reduce 2, 5 to the form of an ordinary fraction: 255 10 = 51 2. Next we just need to divide it by natural number. We already know how to do this:
25, 5: 45 = 51 2: 45 = 51 2 1 45 = 17 30
If we convert the result to decimal notation, we get 0.5 (6).
Answer: 25 , 5: 45 = 0 , 5 (6) .
The long division method is good not only for natural numbers. By analogy, we can use it for fractions. Below we indicate the sequence of actions that need to be carried out for this.
Definition 1
To divide a column of decimal fractions by natural numbers you need:
1. Add a few zeros to the decimal fraction on the right (for division we can add any number of them that we need).
2. Divide a decimal fraction by a natural number using an algorithm. When the division of the whole part of the fraction comes to an end, we put a comma in the resulting quotient and count further.
The result of such division can be either a finite or an infinite periodic decimal fraction. It depends on the remainder: if it is zero, then the result will be finite, and if the remainders begin to repeat, then the answer will be a periodic fraction.
Let's take several problems as an example and try to perform these steps with specific numbers.
Example 5
Calculate how much 65, 14 4 will be.
Solution
We use the column method. To do this, add two zeros to the fraction and get the decimal fraction 65, 1400, which will be equal to the original one. Now we write a column for dividing by 4:
The resulting number will be the result we need from dividing the integer part. We put a comma, separating it, and continue:
We have reached zero remainder, therefore the division process is complete.
Answer: 65 , 14: 4 = 16 , 285 .
Example 6
Divide 164.5 by 27.
Solution
We first divide the fractional part and get:
Separate the resulting number with a comma and continue dividing:
We see that the remainders began to repeat periodically, and in the quotient the numbers nine, two and five began to alternate. We will stop here and write the answer in the form of a periodic fraction 6, 0 (925).
Answer: 164 , 5: 27 = 6 , 0 (925) .
This division can be reduced to the process of finding the quotient of a decimal fraction and a natural number, already described above. To do this, we need to multiply the dividend and divisor by 10, 100, etc. so that the divisor turns into a natural number. Next we carry out the sequence of actions described above. This approach is possible due to the properties of division and multiplication. We wrote them down like this:
a: b = (a · 10) : (b · 10) , a: b = (a · 100) : (b · 100) and so on.
Let's formulate a rule:
Definition 2
To divide one final decimal fraction by another:
1. Move the comma in the dividend and divisor to the right by the number of digits necessary to turn the divisor into a natural number. If there are not enough signs in the dividend, we add zeros to it on the right side.
2. After this, divide the fraction by a column by the resulting natural number.
Let's look at a specific problem.
Example 7
Divide 7.287 by 2.1.
Solution: To make the divisor a natural number, we need to move the decimal place one place to the right. So we moved on to dividing the decimal fraction 72, 87 by 21. Let's write the resulting numbers in a column and calculate
Answer: 7 , 287: 2 , 1 = 3 , 47
Example 8
Calculate 16.30.021.
Solution
We will have to move the comma three places. There are not enough digits in the divisor for this, which means you need to use additional zeros. We think the result will be:
We see periodic repetition of residues 4, 19, 1, 10, 16, 13. In the quotient, 1, 9, 0, 4, 7 and 5 are repeated. Then our result is the periodic decimal fraction 776, (190476).
Answer: 16 , 3: 0 , 021 = 776 , (190476)
The method we described allows you to do the opposite, that is, divide a natural number by the final decimal fraction. Let's see how it's done.
Example 9
Calculate how much 3 5, 4 is.
Solution
Obviously, we will have to move the comma to the right one place. After this we can proceed to divide 30, 0 by 54. Let's write the data in a column and calculate the result:
Repeating the remainder gives us the final number 0, (5), which is a periodic decimal fraction.
Answer: 3: 5 , 4 = 0 , (5) .
How to divide decimals by 1000, 100, 10, etc.
According to the already studied rules for dividing ordinary fractions, dividing a fraction by tens, hundreds, thousands is similar to multiplying it by 1/1000, 1/100, 1/10, etc. It turns out that to perform the division, in this case it is enough to simply move the decimal point to the required amount numbers If there are not enough values in the number to transfer, you need to add the required number of zeros.
Example 10
So, 56, 21: 10 = 5, 621, and 0, 32: 100,000 = 0, 0000032.
In the case of infinite decimal fractions, we do the same.
Example 11
For example, 3, (56): 1,000 = 0, 003 (56) and 593, 374...: 100 = 5, 93374....
How to divide decimals by 0.001, 0.01, 0.1, etc.
Using the same rule, we can also divide fractions into the indicated values. This action will be similar to multiplying by 1000, 100, 10, respectively. To do this, we move the comma to one, two or three digits, depending on the conditions of the problem, and add zeros if there are not enough digits in the number.
Example 12
For example, 5.739: 0.1 = 57.39 and 0.21: 0.00001 = 21,000.
This rule also applies to infinite decimal fractions. We only advise you to be careful with the period of the fraction that appears in the answer.
So, 7, 5 (716) : 0, 01 = 757, (167) because after we moved the comma in the decimal fraction 7, 5716716716... two places to the right, we got 757, 167167....
If we have non-periodic fractions in the example, then everything is simpler: 394, 38283...: 0, 001 = 394382, 83....
How to divide a mixed number or fraction by a decimal and vice versa
We also reduce this action to operations with ordinary fractions. To do this you need to replace decimal numbers corresponding ordinary fractions, and write the mixed number as an improper fraction.
If we divide a non-periodic fraction by an ordinary or mixed number, we need to do the opposite, replacing common fraction or a mixed number with its corresponding decimal fraction.
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The rule for dividing decimal fractions by natural numbers.
Four identical toys cost a total of 921 rubles 20 kopecks. How much does one toy cost (see Fig. 1)?
Rice. 1. Illustration for the problem
Solution
To find the cost of one toy, you need to divide this amount by four. Let's convert the amount into kopecks:
Answer: the cost of one toy is 23,030 kopecks, that is, 230 rubles 30 kopecks, or 230.3 rubles.
You can solve this problem without converting rubles to kopecks, that is, divide the decimal fraction by a natural number: .
To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.
We divide into a column in the same way as natural numbers are divided. After we remove the number 2 (the number of tenths is the first digit after the decimal point in the dividend 921.20), we put a comma in the quotient and continue the division:
Answer: 230.3 rubles.
We divide into a column in the same way as natural numbers are divided. After we remove the number 6 (the number of tenths is the number after the decimal point in the notation of the dividend 437.6), we put a comma in the quotient and continue the division:
If the dividend is less than the divisor, then the quotient will start from zero.
1 is not divisible by 19, so we put zero in the quotient. The division of the whole part is completed, we put a comma in the quotient. We take down 7. 17 is not divisible by 19, in the quotient we write zero. We take down 6 and continue division:
We divide as natural numbers are divided. In the quotient, we put a comma as soon as we remove 8 - the first digit after the decimal point in the dividend 74.8. We continue the division further. When subtracting, we get 8, but the division is not completed. We know that zeros can be added to the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 80 by 10. We get 8 - the division is over.
To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in this fraction as many digits to the left as there are zeros after the one in the divisor.
In this lesson we learned how to divide a decimal fraction by a natural number. We considered the option with an ordinary natural number, as well as the option in which division by a digit unit occurs (10, 100, 1000, etc.).
Solve the equations:
To find unknown divisor, it is necessary to divide the dividend by the quotient. That is .
We divide into a column. After we remove the number 4 (the number of tenths is the first digit after the decimal point in the dividend 134.4), we put a comma in the quotient and continue the division:
Find the first digit of the quotient (the result of division). To do this, divide the first digit of the dividend by the divisor. Write the result under the divisor.
- In our example, the first digit of the dividend is 3. Divide 3 by 12. Since 3 is less than 12, the result of division will be 0. Write 0 under the divisor - this is the first digit of the quotient.
Multiply the result by the divisor. Write the result of the multiplication under the first digit of the dividend, since this is the digit you just divided by the divisor.
- In our example, 0 × 12 = 0, so write 0 under 3.
Subtract the result of the multiplication from the first digit of the dividend. Write your answer on a new line.
- In our example: 3 - 0 = 3. Write 3 directly below 0.
Move down the second digit of the dividend. To do this, write down the next digit of the dividend next to the result of the subtraction.
- In our example, the dividend is 30. The second digit of the dividend is 0. Move it down by writing a 0 next to the 3 (the result of the subtraction). You will receive the number 30.
Divide the result by the divisor. You will find the second digit of the quotient. To do this, divide the number located on the bottom line by the divisor.
- In our example, divide 30 by 12. 30 ÷ 12 = 2 plus some remainder (since 12 x 2 = 24). Write 2 after 0 under the divisor - this is the second digit of the quotient.
- If you can't find a suitable digit, go through the digits until the result of multiplying a digit by a divisor is smaller and closest to the number located last in the column. In our example, consider the number 3. Multiply it by the divisor: 12 x 3 = 36. Since 36 is greater than 30, the number 3 is not suitable. Now consider the number 2. 12 x 2 = 24. 24 is less than 30, so the number 2 is the correct solution.
Repeat the steps above to find the next number. The described algorithm is used in any long division problem.
- Multiply the second digit of the quotient by the divisor: 2 x 12 = 24.
- Write the result of the multiplication (24) under the last number in the column (30).
- Subtract the smaller number from the larger one. In our example: 30 - 24 = 6. Write the result (6) on a new line.
If there are digits left in the dividend that can be moved down, continue the calculation process. Otherwise, continue to the next step.
- In our example, you moved down the last digit of the dividend (0). So move on to the next step.
If necessary, use a decimal point to expand the dividend. If the dividend is divisible by the divisor, then on the last line you will get the number 0. This means that the problem has been solved, and the answer (in the form of an integer) is written under the divisor. But if at the very bottom of the column there is any figure other than 0, it is necessary to expand the dividend by adding a decimal point and adding 0. Let us remind you that this does not change the value of the dividend.
- In our example, the last line contains the number 6. Therefore, to the right of 30 (the dividend), write a decimal point, and then write 0. Also, place a decimal point after the found digits of the quotient, which you write under the divisor (don’t write anything after this comma yet!) .
Repeat the steps described above to find the next number. The main thing is not to forget to put a decimal point both after the dividend and after the found digits of the quotient. The rest of the process is similar to the process described above.
- In our example, move down the 0 (which you wrote after the decimal point). You will get the number 60. Now divide this number by the divisor: 60 ÷ 12 = 5. Write 5 after the 2 (and after the decimal point) under the divisor. This is the third digit of the quotient. So the final answer is 2.5 (the zero before the 2 can be ignored).
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 numbers included in significant part numbers are called significant figures. 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 I will publish a test on this topic in the near future. 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 by any in a 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 out 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 down 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 to the front so as not to leave a “bare” 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).
pay attention to last example: Since the decimal point moves in different directions, the total shift is found through the difference. This is very important point! Here's another example:
Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 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 classic 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 the decimal notation, reducible fractions appear. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction contains a prime number. There is 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.
Division by a decimal fraction is reduced to division by a natural number.
The rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are in the divisor after the decimal point. After this, divide by a natural number.
Examples.
Divide by decimal fraction:
To divide by a decimal, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are after the decimal point in the divisor, that is, by one digit. We get: 35.1: 1.8 = 351: 18. Now we perform the division with a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To divide decimal fractions, in both the dividend and the divisor we move the decimal point to the right one place: 14.76: 3.6 = 147.6: 36. Now we perform a natural number. Result: 14.76: 3.6 = 4.1.
To divide a natural number by a decimal fraction, you need to move both the dividend and the divisor to the right as many places as there are in the divisor after the decimal point. Since a comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 = 7000: 175. Divide the resulting natural numbers with a corner: 70: 1.75 = 7000: 175 = 40.
4) 0,1218: 0,058
To divide one decimal fraction by another, we move the decimal point to the right in both the dividend and the divisor by as many digits as there are in the divisor after the decimal point, that is, by three decimal places. Thus, 0.1218: 0.058 = 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8