Of the many fractions found in arithmetic, those with 10, 100, 1000 in the denominator deserve special attention - in general, any power of ten. These fractions have a special name and notation.
A decimal is any number whose denominator is a power of ten.
Decimal examples:
Why was it necessary to isolate such fractions at all? Why do they need their own entry form? There are at least three reasons for this:
- Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, bring the fractions to a common denominator. In decimal fractions, none of this is required;
- Reduction of calculations. Decimals add and multiply according to their own rules, and with a little practice you will be able to work with them much faster than with ordinary ones;
- Ease of recording. Unlike ordinary fractions, decimals are written in one line without loss of clarity.
Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you demand change in the amount of 2/3 rubles in a store :)
Rules for writing decimal fractions
The main advantage of decimal fractions is a convenient and visual notation. Namely:
Decimal notation is a form of decimal notation where the integer part is separated from the fractional part using a regular dot or comma. In this case, the separator itself (dot or comma) is called the decimal point.
For example, 0.3 (read: “zero integer, 3 tenths”); 7.25 (7 integers, 25 hundredths); 3.049 (3 integers, 49 thousandths). All examples are taken from the previous definition.
In writing, a comma is usually used as a decimal point. Here and below, the comma will also be used throughout the site.
To write an arbitrary decimal in this form, you need to follow three simple steps:
- Write out the numerator separately;
- Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
- If the decimal point has shifted, and after it there are zeros at the end of the record, they must be crossed out.
It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, any number of zeros can be assigned to the left of any number without harm to health. It's ugly, but sometimes useful.
At first glance, this algorithm may seem rather complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:
A task. For each fraction, indicate its decimal notation:
The numerator of the first fraction: 73. We shift the decimal point by one sign (because the denominator is 10) - we get 7.3.
The numerator of the second fraction: 9. We shift the decimal point by two digits (because the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange notation like “.09”.
The numerator of the third fraction: 10029. We shift the decimal point by three digits (because the denominator is 1000) - we get 10.029.
The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10.500. There are extra zeros at the end of the number. We cross them out - we get 10.5.
Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as is done in the last example. However, in no case should you do this with zeros that are inside the number (which are surrounded by other digits). That is why we got 10.029 and 10.5, and not 1.29 and 1.5.
So, we figured out the definition and form of recording decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.
Change from fractions to decimals
Consider a simple numerical fraction of the form a / b . You can use the basic property of a fraction and multiply the numerator and denominator by such a number that you get a power of ten below. But before doing so, please read the following:
There are denominators that are not reduced to the power of ten. Learn to recognize such fractions, because they cannot be worked with according to the algorithm described below.
That's it. Well, how to understand whether the denominator is reduced to the power of ten or not?
The answer is simple: factor the denominator into prime factors. If only factors 2 and 5 are present in the expansion, this number can be reduced to the power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the degree of ten.
A task. Check if the specified fractions can be represented as decimals:
We write out and factorize the denominators of these fractions:
20 \u003d 4 5 \u003d 2 2 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.
12 \u003d 4 3 \u003d 2 2 3 - there is a "forbidden" factor 3. The fraction cannot be represented as a decimal.
640 \u003d 8 8 10 \u003d 2 3 2 3 2 5 \u003d 2 7 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction is represented as a decimal.
48 \u003d 6 8 \u003d 2 3 2 3 \u003d 2 4 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.
So, we figured out the denominator - now we will consider the entire algorithm for switching to decimal fractions:
- Factorize the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
- Count how many twos and fives are present in the decomposition (there will be no other numbers there, remember?). Choose such an additional multiplier so that the number of twos and fives is equal.
- Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.
Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest such factor from all possible ones.
And one more thing: if there is an integer part in the original fraction, be sure to convert this fraction to an improper one - and only then apply the described algorithm.
A task. Convert these numbers to decimals:
Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, a fraction can be represented as a decimal. There are two twos and no fives in the expansion, so the additional factor is 5 2 = 25. The number of twos and fives will be equal to it. We have:
Now let's deal with the second fraction. To do this, note that 24 \u003d 3 8 \u003d 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.
The last two fractions have denominators 5 (a prime number) and 20 = 4 5 = 2 2 5 respectively - only twos and fives are present everywhere. At the same time, in the first case, “for complete happiness”, there is not enough multiplier 2, and in the second - 5. We get:
Change from decimals to ordinary
The reverse conversion - from decimal notation to normal - is much easier. There are no restrictions and special checks, so you can always convert a decimal fraction into a classic "two-story" one.
The translation algorithm is as follows:
- Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing - do not overdo it and do not cross out the internal zeros surrounded by other numbers;
- Calculate how many digits are in the original decimal fraction after the decimal point. Take the number 1 and add as many zeros to the right as you counted the characters. This will be the denominator;
- Actually, write down the fraction whose numerator and denominator we just found. Reduce if possible. If there was an integer part in the original fraction, now we will get an improper fraction, which is very convenient for further calculations.
A task. Convert decimals to ordinary: 0.008; 3.107; 2.25; 7.2008.
We cross out the zeros on the left and the commas - we get the following numbers (these will be numerators): 8; 3107; 225; 72008.
In the first and second fractions after the decimal point there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Once again, I note that any decimal fraction can be represented as an ordinary one. The reverse transformation is not always possible.
In this tutorial, we'll look at each of these operations one by one.
Lesson contentAdding decimals
As we know, a decimal fraction consists of an integer part and a fractional part. When adding decimals, the integer and fractional parts are added separately.
For example, let's add the decimals 3.2 and 5.3. It is more convenient to add decimal fractions in a column.
First, we write these two fractions in a column, while the integer parts must be under the integer parts, and the fractional ones under the fractional parts. In school, this requirement is called "comma under comma" .
Let's write the fractions in a column so that the comma is under the comma:
We add the fractional parts: 2 + 3 = 5. We write down the five in the fractional part of our answer:
Now we add up the integer parts: 3 + 5 = 8. We write the eight in the integer part of our answer:
Now we separate the integer part from the fractional part with a comma. To do this, we again follow the rule "comma under comma" :
Got the answer 8.5. So the expression 3.2 + 5.3 is equal to 8.5
3,2 + 5,3 = 8,5
In fact, not everything is as simple as it seems at first glance. Here, too, there are pitfalls, which we will now talk about.
Places in decimals
Decimals, like ordinary numbers, have their own digits. These are tenth places, hundredth places, thousandth places. In this case, the digits begin after the decimal point.
The first digit after the decimal point is responsible for the tenths place, the second digit after the decimal point for the hundredths place, the third digit after the decimal point for the thousandths place.
The digits in decimal fractions store some useful information. In particular, they report how many tenths, hundredths, and thousandths are in a decimal.
For example, consider the decimal 0.345
The position where the triple is located is called tenth place
The position where the four is located is called hundredths place
The position where the five is located is called thousandths
Let's look at this figure. We see that in the category of tenths there is a three. This suggests that there are three tenths in the decimal fraction 0.345.
If we add the fractions, and then we get the original decimal fraction 0.345
We first got the answer, but converted it to decimal and got 0.345.
Adding decimals follows the same rules as adding ordinary numbers. The addition of decimal fractions occurs by digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.
Therefore, when adding decimal fractions, it is required to follow the rule "comma under comma". 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 the fractional parts 5 + 4 = 9. We write the nine in the fractional part of our answer:
Now we add up the integer parts 1 + 3 = 4. We write down the four in the integer part of our answer:
Now we separate the integer part from the fractional part with a comma. To do this, we again observe the rule "comma under a comma":
Got the answer 4.9. So the value of the expression 1.5 + 3.4 is 4.9
Example 2 Find the value of the expression: 3.51 + 1.22
We write this expression in a column, observing the rule "comma under a comma"
First of all, we add the fractional part, namely the hundredths 1+2=3. We write the triple in the hundredth part of our answer:
Now add tenths of 5+2=7. We write down the seven in the tenth part of our answer:
Now add the whole parts 3+1=4. We write down the four in the integer part of our answer:
We separate the integer part from the fractional part with a comma, observing the “comma under the comma” rule:
Got the answer 4.73. So the value of the expression 3.51 + 1.22 is 4.73
3,51 + 1,22 = 4,73
As with ordinary numbers, when adding decimal fractions, . In this case, one digit is written in the answer, and the rest are transferred to the next digit.
Example 3 Find the value of the expression 2.65 + 3.27
We write this expression in a column:
Add hundredths of 5+7=12. The number 12 will not fit in the hundredth part of our answer. Therefore, in the hundredth part, we write the number 2, and transfer the unit to the next bit:
Now we add the tenths of 6+2=8 plus the unit that we got from the previous operation, we get 9. We write the number 9 in the tenth of our answer:
Now add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:
Got the answer 5.92. So the value of the expression 2.65 + 3.27 is 5.92
2,65 + 3,27 = 5,92
Example 4 Find the value of the expression 9.5 + 2.8
Write this expression in a column
We add the fractional parts 5 + 8 = 13. The number 13 will not fit in the fractional part of our answer, so we first write down the number 3, and transfer the unit to the next digit, or rather transfer it to the integer part:
Now we add the integer parts 9+2=11 plus the unit that we got from the previous operation, we get 12. We write the number 12 in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 12.3. So the value of the expression 9.5 + 2.8 is 12.3
9,5 + 2,8 = 12,3
When adding decimal fractions, the number of digits after the decimal point in both fractions must be the same. If there are not enough digits, then these places in the fractional part are filled with zeros.
Example 5. Find the value of the expression: 12.725 + 1.7
Before writing this expression in a column, let's make the number of digits after the decimal point in both fractions the same. The decimal fraction 12.725 has three digits after the decimal point, while the fraction 1.7 has only one. So in the fraction 1.7 at the end you need to add two zeros. Then we get the fraction 1,700. Now you can write this expression in a column and start calculating:
Add thousandths of 5+0=5. We write the number 5 in the thousandth part of our answer:
Add hundredths of 2+0=2. We write the number 2 in the hundredth part of our answer:
Add tenths of 7+7=14. The number 14 will not fit in a tenth of our answer. Therefore, we first write down the number 4, and transfer the unit to the next bit:
Now we add the integer parts 12+1=13 plus the unit that we got from the previous operation, we get 14. We write the number 14 in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 14,425. So the value of the expression 12.725+1.700 is 14.425
12,725+ 1,700 = 14,425
Subtraction of decimals
When subtracting decimal fractions, you must follow the same rules as when adding: “a comma under a comma” and “an equal number of digits after a decimal point”.
Example 1 Find the value of the expression 2.5 − 2.2
We write this expression in a column, observing the “comma under comma” rule:
We calculate the fractional part 5−2=3. We write the number 3 in the tenth part of our answer:
Calculate the integer part 2−2=0. We write zero in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
We got the answer 0.3. So the value of the expression 2.5 − 2.2 is equal to 0.3
2,5 − 2,2 = 0,3
Example 2 Find the value of the expression 7.353 - 3.1
In this expression different amount digits after the decimal point. In the fraction 7.353 there are three digits after the decimal point, and in the fraction 3.1 there is only one. This means that in the fraction 3.1, two zeros must be added at the end to make the number of digits in both fractions the same. Then we get 3,100.
Now you can write this expression in a column and calculate it:
Got the answer 4,253. So the value of the expression 7.353 − 3.1 is 4.253
7,353 — 3,1 = 4,253
As with ordinary numbers, sometimes you will have to borrow one from the adjacent bit if subtraction becomes impossible.
Example 3 Find the value of the expression 3.46 − 2.39
Subtract hundredths of 6−9. From the number 6 do not subtract the number 9. Therefore, you need to take a unit from the adjacent digit. Having borrowed one from the neighboring digit, the number 6 turns into the number 16. Now we can calculate the hundredths of 16−9=7. We write down the seven in the hundredth part of our answer:
Now subtract tenths. Since we took one unit in the category of tenths, the figure that was located there decreased by one unit. In other words, the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3=0. We write zero in the tenth part of our answer:
Now subtract the integer parts 3−2=1. We write the unit in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 1.07. So the value of the expression 3.46−2.39 is equal to 1.07
3,46−2,39=1,07
Example 4. Find the value of the expression 3−1.2
This example subtracts a decimal from an integer. Let's write this expression in a column so that the integer part of the decimal fraction 1.23 is under the number 3
Now let's make the number of digits after the decimal point the same. To do this, after the number 3, put a comma and add one zero:
Now subtract tenths: 0−2. Do not subtract the number 2 from zero. Therefore, you need to take a unit from the adjacent digit. By borrowing one from the adjacent digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write down the eight in the tenth part of our answer:
Now subtract the whole parts. Previously, the number 3 was located in the integer, but we borrowed one unit from it. As a result, it turned into the number 2. Therefore, we subtract 1 from 2. 2−1=1. We write the unit in the integer part of our answer:
Separate the integer part from the fractional part with a comma:
Got the answer 1.8. So the value of the expression 3−1.2 is 1.8
Decimal multiplication
Multiplying decimals is easy and even fun. To multiply decimals, you need to multiply them like regular numbers, ignoring the commas.
Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in both fractions, then count the same number of digits on the right in the answer and put a comma.
Example 1 Find the value of the expression 2.5 × 1.5
We multiply these decimal fractions as ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:
We got 375. In this number, it is necessary to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in fractions of 2.5 and 1.5. In the first fraction there is one digit after the decimal point, in the second fraction there is also one. A total of two numbers.
We return to the number 375 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 3.75. So the value of the expression 2.5 × 1.5 is 3.75
2.5 x 1.5 = 3.75
Example 2 Find the value of the expression 12.85 × 2.7
Let's multiply these decimals, ignoring the commas:
We got 34695. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 12.85 and 2.7. In the fraction 12.85 there are two digits after the decimal point, in the fraction 2.7 there is one digit - a total of three digits.
We return to the number 34695 and begin to move from right to left. We need to count three digits on the right and put a comma:
Got the answer 34,695. So the value of the expression 12.85 × 2.7 is 34.695
12.85 x 2.7 = 34.695
Multiplying a decimal by a regular number
Sometimes there are situations when you need to multiply a decimal fraction by a regular number.
To multiply a decimal and an ordinary number, you need to multiply them, regardless of the comma in the decimal. Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then in the answer, count the same number of digits to the right and put a comma.
For example, multiply 2.54 by 2
We multiply the decimal fraction 2.54 by the usual number 2, ignoring the comma:
We got the number 508. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. The fraction 2.54 has two digits after the decimal point.
We return to the number 508 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 5.08. So the value of the expression 2.54 × 2 is 5.08
2.54 x 2 = 5.08
Multiply decimals by 10, 100, 1000
Multiplying decimals by 10, 100, or 1000 is done in the same way as multiplying decimals by regular numbers. It is necessary to perform the multiplication, ignoring the comma in the decimal fraction, then in the answer, separate the integer part from the fractional part, counting the same number of digits on the right as there were digits after the decimal point in the decimal fraction.
For example, multiply 2.88 by 10
Let's multiply the decimal fraction 2.88 by 10, ignoring the comma in the decimal fraction:
We got 2880. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.88. We see that in the fraction 2.88 there are two digits after the decimal point.
We return to the number 2880 and begin to move from right to left. We need to count two digits from the right and put a comma:
Got the answer 28.80. We discard the last zero - we get 28.8. So the value of the expression 2.88 × 10 is 28.8
2.88 x 10 = 28.8
There is a second way to multiply decimal fractions by 10, 100, 1000. This method is much simpler and more convenient. It consists in the fact that the comma in the decimal fraction moves to the right by as many digits as there are zeros in the multiplier.
For example, let's solve the previous example 2.88×10 in this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 2.88 we move the decimal point to the right by one digit, we get 28.8.
2.88 x 10 = 28.8
Let's try to multiply 2.88 by 100. We immediately look at the factor 100. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 2.88 we move the decimal point to the right by two digits, we get 288
2.88 x 100 = 288
Let's try to multiply 2.88 by 1000. We immediately look at the factor 1000. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 2.88 we move the decimal point to the right by three digits. The third digit is not there, so we add another zero. As a result, we get 2880.
2.88 x 1000 = 2880
Multiplying decimals by 0.1 0.01 and 0.001
Multiplying decimals by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal by a decimal. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits on the right as there are digits after the decimal point in both fractions.
For example, multiply 3.25 by 0.1
We multiply these fractions like ordinary numbers, ignoring the commas:
We got 325. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 3.25 and 0.1. In the fraction 3.25 there are two digits after the decimal point, in the fraction 0.1 there is one digit. A total of three numbers.
We return to the number 325 and begin to move from right to left. We need to count three digits on the right and put a comma. After counting three digits, we find that the numbers are over. In this case, you need to add one zero and put a comma:
We got the answer 0.325. So the value of the expression 3.25 × 0.1 is 0.325
3.25 x 0.1 = 0.325
There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction moves to the left by as many digits as there are zeros in the multiplier.
For example, let's solve the previous example 3.25 × 0.1 in this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 3.25 we move the decimal point to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits before the three. In this case, add one zero and put a comma. As a result, we get 0.325
3.25 x 0.1 = 0.325
Let's try multiplying 3.25 by 0.01. Immediately look at the multiplier of 0.01. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 3.25 we move the comma to the left by two digits, we get 0.0325
3.25 x 0.01 = 0.0325
Let's try multiplying 3.25 by 0.001. Immediately look at the multiplier of 0.001. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 3.25 we move the decimal point to the left by three digits, we get 0.00325
3.25 × 0.001 = 0.00325
Do not confuse multiplying decimals by 0.1, 0.001 and 0.001 with multiplying by 10, 100, 1000. Common Mistake most people.
When multiplying by 10, 100, 1000, the comma is moved to the right by as many digits as there are zeros in the multiplier.
And when multiplying by 0.1, 0.01 and 0.001, the comma is moved to the left by as many digits as there are zeros in the multiplier.
If at first it is difficult to remember, you can use the first method, in which the multiplication is performed as with ordinary numbers. In the answer, you will need to separate the integer part from the fractional part by counting as many digits on the right as there are digits after the decimal point in both fractions.
Dividing a smaller number by a larger one. Advanced level.
In one of the previous lessons, we said that when dividing a smaller number by a larger one, a fraction is obtained, in the numerator of which is the dividend, and in the denominator is the divisor.
For example, to divide one apple into two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. The result is a fraction. So each friend will get an apple. In other words, half an apple. A fraction is the answer to a problem how to split one apple between two
It turns out that you can solve this problem further if you divide 1 by 2. After all, a fractional bar in any fraction means division, which means that this division is also allowed in a fraction. But how? We are used to the fact that the dividend is always greater than the divisor. And here, on the contrary, the dividend is less than the divisor.
Everything will become clear if we remember that a fraction means crushing, dividing, dividing. This means that the unit can be split into as many parts as you like, and not just into two parts.
When dividing a smaller number by a larger one, a decimal fraction is obtained, in which the integer part will be 0 (zero). The fractional part can be anything.
So, let's divide 1 by 2. Let's solve this example with a corner:
One cannot be divided into two just like that. If you ask a question "how many twos are in one" , then the answer will be 0. Therefore, in private we write 0 and put a comma:
Now, as usual, we multiply the quotient by the divisor to pull out the remainder:
The moment has come when the unit can be divided into two parts. To do this, add another zero to the right of the received one:
We got 10. We divide 10 by 2, we get 5. We write down the five in the fractional part of our answer:
Now we take out the last remainder to complete the calculation. Multiply 5 by 2, we get 10
We got the answer 0.5. So the fraction is 0.5
Half an apple can also be written using the decimal fraction 0.5. If we add these two halves (0.5 and 0.5), we again get the original one whole apple: 
This point can also be understood if we imagine how 1 cm is divided into two parts. If you divide 1 centimeter into 2 parts, you get 0.5 cm
Example 2 Find the value of expression 4:5
How many fives are in four? Not at all. We write in private 0 and put a comma:
We multiply 0 by 5, we get 0. We write zero under the four. Immediately subtract this zero from the dividend:
Now let's start splitting (dividing) the four into 5 parts. To do this, to the right of 4, we add zero and divide 40 by 5, we get 8. We write the eight in private.
We complete the example by multiplying 8 by 5, and get 40:
We got the answer 0.8. So the value of the expression 4: 5 is 0.8
Example 3 Find the value of expression 5: 125
How many numbers 125 are in five? Not at all. We write 0 in private and put a comma:
We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract from the five 0
Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write zero:
Divide 50 by 125. How many numbers 125 are in 50? Not at all. So in the quotient we again write 0
We multiply 0 by 125, we get 0. We write this zero under 50. Immediately subtract 0 from 50
Now we divide the number 50 into 125 parts. To do this, to the right of 50, we write another zero:
Divide 500 by 125. How many numbers are 125 in the number 500. In the number 500 there are four numbers 125. We write the four in private:
We complete the example by multiplying 4 by 125, and get 500
We got the answer 0.04. So the value of the expression 5: 125 is 0.04
Division of numbers without a remainder
So, let's put a comma in the quotient after the unit, thereby indicating that the division of integer parts is over and we proceed to the fractional part:
Add zero to the remainder 4
Now we divide 40 by 5, we get 8. We write the eight in private:
40−40=0. Received 0 in the remainder. So the division is completely completed. Dividing 9 by 5 results in a decimal of 1.8:
9: 5 = 1,8
Example 2. Divide 84 by 5 without a remainder
First we divide 84 by 5 as usual with a remainder:
Received in private 16 and 4 more in the balance. Now we divide this remainder by 5. We put a comma in the private, and add 0 to the remainder 4
Now we divide 40 by 5, we get 8. We write the eight in the quotient after the decimal point:
and complete the example by checking if there is still a remainder:
Dividing a decimal by a regular number
A decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by a regular number, first of all you need:
- divide the integer part of the decimal fraction by this number;
- after the integer part is divided, you need to immediately put a comma in the private part and continue the calculation, as in ordinary division.
For example, let's divide 4.8 by 2
Let's write this example as a corner:
Now let's divide the whole part by 2. Four divided by two is two. We write the deuce in private and immediately put a comma:
Now we multiply the quotient by the divisor and see if there is a remainder from the division:
4−4=0. Remainder zero. We do not write zero yet, since the solution is not completed. Then we continue to calculate, as in ordinary division. Take down 8 and divide it by 2
8: 2 = 4. We write the four in the quotient and immediately multiply it by the divisor:
Got the answer 2.4. Expression value 4.8: 2 equals 2.4
Example 2 Find the value of the expression 8.43:3
We divide 8 by 3, we get 2. Immediately put a comma after the two:
Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:
We divide 24 by 3, we get 8. We write the eight in private. We immediately multiply it by the divisor to find the remainder of the division:
24−24=0. The remainder is zero. Zero is not recorded yet. Take the last three of the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:
Got the answer 2.81. So the value of the expression 8.43: 3 is equal to 2.81
Dividing a decimal by a decimal
To divide a decimal fraction into a decimal fraction, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by a regular number.
For example, divide 5.95 by 1.7
Let's write this expression as a corner
Now, in the dividend and in the divisor, we move the comma to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we must move the comma to the right by one digit in the dividend and in the divisor. Transferring:
After moving the decimal point to the right by one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7, after moving the decimal point to the right by one digit, turned into the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculation is not difficult:
The comma is moved to the right to facilitate division. This is allowed due to the fact that when multiplying or dividing the dividend and the divisor by the same number, the quotient does not change. What does it mean?
This is one of interesting features division. It is called the private property. Consider expression 9: 3 = 3. If in this expression the dividend and the divisor are multiplied or divided by the same number, then the quotient 3 will not change.
Let's multiply the dividend and divisor by 2 and see what happens:
(9 × 2) : (3 × 2) = 18: 6 = 3
As can be seen from the example, the quotient has not changed.
The same thing happens when we carry a comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma one digit to the right in the dividend and divisor. After moving the comma, the fraction 5.91 was converted to the fraction 59.1 and the fraction 1.7 was converted to the usual number 17.
In fact, inside this process, multiplication by 10 took place. Here's what it looked like:
5.91 × 10 = 59.1
Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and divisor will be multiplied by. In other words, the number of digits after the decimal point in the divisor will determine how many digits in the dividend and in the divisor the comma will be moved to the right.
Decimal division by 10, 100, 1000
Dividing a decimal by 10, 100, or 1000 is done in the same way as . For example, let's divide 2.1 by 10. Let's solve this example with a corner:
But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the left by as many digits as there are zeros in the divisor.
Let's solve the previous example in this way. 2.1: 10. We look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 2.1, you need to move the comma to the left by one digit. We move the comma to the left by one digit and see that there are no more digits left. In this case, we add one more zero before the number. As a result, we get 0.21
Let's try to divide 2.1 by 100. There are two zeros in the number 100. So in the divisible 2.1, you need to move the comma to the left by two digits:
2,1: 100 = 0,021
Let's try to divide 2.1 by 1000. There are three zeros in the number 1000. So in the divisible 2.1, you need to move the comma to the left by three digits:
2,1: 1000 = 0,0021
Decimal division by 0.1, 0.01 and 0.001
Dividing a decimal by 0.1, 0.01, and 0.001 is done in the same way as . In the dividend and in the divisor, you need to move the comma to the right by as many digits as there are after the decimal point in the divisor.
For example, let's divide 6.3 by 0.1. First of all, we move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we move the commas in the dividend and in the divisor to the right by one digit.
After moving the decimal point to the right by one digit, the decimal fraction 6.3 turns into the usual number 63, and the decimal fraction 0.1, after moving the decimal point to the right by one digit, turns into one. And dividing 63 by 1 is very simple:
So the value of the expression 6.3: 0.1 is equal to 63
But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is transferred to the right by as many digits as there are zeros in the divisor.
Let's solve the previous example in this way. 6.3:0.1. Let's look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 6.3, you need to move the comma to the right by one digit. We move the comma to the right by one digit and get 63
Let's try to divide 6.3 by 0.01. Divisor 0.01 has two zeros. So in the divisible 6.3, you need to move the comma to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, one more zero must be added at the end. As a result, we get 630
Let's try dividing 6.3 by 0.001. There are three zeros in the 0.001 divisor. So in the divisible 6.3, you need to move the comma to the right by three digits:
6,3: 0,001 = 6300
Tasks for independent solution
Did you like the lesson?
Join our new group Vkontakte and start receiving notifications about new lessons
Already in primary school students are dealing with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of whole objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.
For example, chocolate consists of several slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.
By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.
What is a "fraction"?
This is a number consisting of parts of one. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one on the bottom (right) is the denominator.
In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.
What are the fractions?
In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren are first introduced to primary school, calling them simply "fractions". The second learn in the 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear that the two examples given are completely different numbers.
Every simple fraction can be written as a decimal. This statement is almost always true in reverse as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.
What subspecies do these types of fractions have?
Better start at chronological order as they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.
Correct. Its numerator is always less than the denominator.
Wrong. Its numerator is greater than or equal to the denominator.
Reducible / irreducible. It can be either right or wrong. Another thing is important, whether the numerator and denominator have common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.
Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.
Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.
Decimals have only two subspecies:
final, that is, one in which the fractional part is limited (has an end);
infinite - a number whose digits after the decimal point do not end (they can be written endlessly).
How to convert decimal to ordinary?
If this is a finite number, then an association based on the rule is applied - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional line.
As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.
How to convert decimal fractions to ordinary ones if their whole part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.
How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second, 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The response is like this mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal to a common fraction?
If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.
The only thing that is allowed to be done with such a fraction is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal - will never give the initial value. That is, infinite non-periodic fractions are not translated into ordinary fractions. This must be remembered.
How to write an infinite periodic fraction in the form of an ordinary?
In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as they can be converted into ordinary fractions.
Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with any numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.
For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.
A rule on how to write a common decimal fraction that is a mixed fraction.
Look at the length of the period. So much 9 will have a denominator.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - it is without a period.
For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period is one digit. So zero will be one. There is also only one digit in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.
To determine the numerator from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.
How are common fractions converted to decimals?
The simplest option is a number whose denominator is the number 10, 100, and so on. Then the denominator is simply discarded, and between the fractional and whole parts a comma is placed.
There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.
Operations with common fractions
Addition and subtraction
Students get to know them earlier than others. And at first the fractions have the same denominators, and then different. General rules can be reduced to such a plan.
Find the least common multiple of the denominators.
Write additional factors to all ordinary fractions.
Multiply the numerators and denominators by the factors defined for them.
Add (subtract) the numerators of fractions, and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then you need to find out whether we have a mixed number or a proper fraction.
In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.
In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
For their implementation, fractions do not need to be reduced to a common denominator. This makes it easier to take action. But they still have to follow the rules.
When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
Multiply numerators.
Multiply the denominators.
If you get a reducible fraction, then it is supposed to be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with a reciprocal (swap the numerator and denominator).
Then proceed as in multiplication (starting from step 1).
In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written in the form improper fraction. That is, with a denominator of 1. Then proceed as described above.
Operations with decimals
Addition and subtraction
Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.
Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Assign the missing number of zeros in it.
Write fractions so that the comma is under the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to append zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.
For multiplication, you need to write fractions one under the other, not paying attention to commas.
Multiply like natural numbers.
Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.
To divide, you must first convert the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.
Multiply the dividend by the same number.
Divide decimal by natural number.
Put a comma in the answer at the moment when the division of the whole part ends.
What if there are both types of fractions in one example?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.
First way: represent ordinary decimals
It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.
The second way: write decimal fractions as ordinary
This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, it can turn out to be very large. common fraction and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.
Division by a decimal is the same as division by a natural number.
Rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, it is necessary both in the dividend and in the divisor to move the comma as many digits to the right as there are in the divisor after the decimal point. After that, divide by a natural number.
Examples.
Perform division by decimal:
To divide by a decimal fraction, you need to move the comma as many digits to the right in both the dividend and the divisor as there are after the decimal point in the divisor, that is, by one sign. We get: 35.1: 1.8 \u003d 351: 18. Now we perform division by a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To perform the division of decimal fractions, both in the dividend and in the divisor, move the comma to the right by one sign: 14.76: 3.6 \u003d 147.6: 36. Now we perform on a natural number. Result: 14.76: 3.6 = 4.1.
To perform division by a decimal fraction of a natural number, it is necessary both in the dividend and in the divisor to move as many characters to the right as there are in the divisor after the decimal point. Since the comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 \u003d 7000: 175. We divide the resulting natural numbers with a corner: 70: 1.75 \u003d 7000: 175 \u003d 40.
4) 0,1218: 0,058
To divide one decimal fraction into another, we move the comma to the right both in the dividend and in the divisor by as many digits as there are in the divisor after the decimal point, that is, by three digits. Thus, 0.1218: 0.058 \u003d 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8
Fraction calculator designed for quick calculation of operations with fractions, it will help you easily add, multiply, divide or subtract fractions.
Modern schoolchildren begin to study fractions already in the 5th grade, and every year the exercises with them become more complicated. Mathematical terms and quantities that we learn in school are rarely useful to us in adulthood. However, fractions, unlike logarithms and degrees, are quite common in everyday life (measuring distance, weighing goods, etc.). Our calculator is designed for quick operations with fractions.
First, let's define what fractions are and what they are. Fractions are the ratio of one number to another; this is a number consisting of a whole number of fractions of a unit.
Fraction types:
- Ordinary
- Decimals
- mixed
Example ordinary fractions:
The top value is the numerator, the bottom is the denominator. The dash shows us that the top number is divisible by the bottom number. Instead of a similar writing format, when the dash is horizontal, you can write differently. You can put a slanted line, for example:
1/2, 3/7, 19/5, 32/8, 10/100, 4/1
Decimals are the most popular type of fractions. They consist of an integer part and a fractional part, separated by a comma.
Decimal example:
0.2 or 6.71 or 0.125
It consists of an integer and a fractional part. To find out the value of this fraction, you need to add the whole number and the fraction.
Example of mixed fractions:
The fraction calculator on our website is able to quickly perform any mathematical operations with fractions online:
- Addition
- Subtraction
- Multiplication
- Division
To carry out the calculation, you need to enter the numbers in the fields and select the action. For fractions, you need to fill in the numerator and denominator, an integer may not be written (if the fraction is ordinary). Don't forget to click on the "equal" button.
It is convenient that the calculator immediately provides a process for solving an example with fractions, and not just a ready-made answer. It is thanks to the expanded solution that you can use this material when solving school tasks and for a better mastering of the material covered.
You need to calculate the example:
After entering the indicators in the form fields, we get:
To make an independent calculation, enter the data in the form.
