Operations with decimal fractions. Division by decimal Counting in columns online

A columnar calculator for Android devices will become a wonderful assistant for modern schoolchildren. The program not only gives the correct answer to mathematical operation, but also clearly demonstrates its step-by-step solution. If you need more complex calculators, you can look at an advanced engineering calculator.

Peculiarities

The main feature of the program is the uniqueness of the calculation of mathematical operations. Displaying the calculation process in a column allows students to familiarize themselves with it in more detail, understand the solution algorithm, and not just get the finished result and copy it into a notebook. This feature has a huge advantage over other calculators because... Quite often at school, teachers require that intermediate calculations be written down in order to make sure that the student performs them in his head and really understands the algorithm for solving problems. By the way, we have another program of a similar kind -.

To start using the program, you need to download a column calculator for Android. You can do this on our website absolutely free of charge without additional registrations or SMS. After installation, the main page will open in the form of a notebook sheet in a cage, on which, in fact, the results of calculations and their detailed solution. At the bottom there is a panel with buttons:

1. Numbers.
2. Signs of arithmetic operations.
3. Deleting previously entered characters.

Input is carried out according to the same principle as on. The only difference is in the application interface - all mathematical calculations and their results are displayed in a virtual student notebook.

The application allows you to quickly and correctly perform standard mathematical calculations for a schoolchild:

• multiplication;
• division;
• addition;
• subtraction.

A nice addition to the app is the daily reminder function. homework mathematics. If you want, do your homework. To enable it, go to the settings (click the gear-shaped button) and check the reminder box.

Advantages and disadvantages

1. Helps the student not only quickly obtain the correct result of mathematical calculations, but also understand the principle of calculation itself.
2. A very simple, intuitive interface for every user.
3. You can install the application even on the most budget Android device with operating system 2.2 and later.
4. The calculator saves a history of mathematical calculations performed, which can be cleared at any time.

The calculator is limited in mathematical operations, so it won’t be possible to use it for complex calculations that an engineering calculator could handle. However, given the purpose of the application itself - to clearly demonstrate to primary school students the principle of columnar calculations, this should not be considered a disadvantage.

The application will also be an excellent assistant not only for schoolchildren, but also for parents who want to interest their child in mathematics and teach him to perform calculations correctly and consistently. If you have already used the Column Calculator application, leave your impressions below in the comments.

Already in primary school students encounter fractions. And then they appear in every topic. You cannot forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are not complicated, the main thing is to understand everything in order.

Why are fractions needed?

The world around us consists of entire 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 pieces. Consider a situation where his tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It can easily be divided into three. But it will not be possible to give five people a whole number of chocolate slices.

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 made up of parts of a unit. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written at the top (left) is called the numerator. What is at the bottom (right) is the denominator.

Essentially, the slash turns out to be a division sign. That is, the numerator can be called the dividend, and the denominator can be called the divisor.

What fractions are there?

In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren first meet in primary school, calling them simply "fractions". The latter will be learned in 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a line. For example, 4/7. A decimal is a number in which the fractional part has a positional notation and is separated from the whole number by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.

Every simple fraction can be written in decimal form. This statement is almost always true in reverse. There are rules that allow you to write a decimal fraction as a common fraction.

What subtypes do these types of fractions have?

It's better to start in 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 its denominator.

Wrong. Its numerator is greater than or equal to its denominator.

Reducible/irreducible. It may turn out to be either right or wrong. Another important thing is whether the numerator and denominator have common factors. If there are, then it is necessary to divide both parts of the fraction by them, that is, reduce it.

Mixed. An integer number is assigned to its usual regular (irregular) fractional part. Moreover, it is always on the left.

Composite. It is formed from two fractions divided by each other. That is, it contains three fractional lines at once.

Decimal fractions have only two subtypes:

finite, that is, one whose 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 a decimal fraction to a common fraction?

If this is a finite number, then an association is applied based on the rule - 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 bar.

As a hint about the required denominator, you need to remember that it is always one and several zeros. You need to write as many of the latter as there are digits in the fractional part of the number in question.

How to convert decimal fractions into ordinary fractions if their integer 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. All that remains is to write down the fractional parts. The first number will have a denominator of 10, the second will have a denominator of 100. That is, the given examples will have the following numbers as answers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it needs to be written as 1/20.

How can you convert a decimal fraction into an ordinary fraction if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer gives the following mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal fraction to an ordinary fraction?

If it is non-periodic, then such an operation will not be possible. This fact is due to the fact that each decimal fraction is always converted to either a finite or a periodic fraction.

The only thing you can do with such a fraction is 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 converted into ordinary fractions. This needs to be remembered.

How to write an infinite periodic fraction as an ordinary fraction?

In these numbers, there are always one or more digits after the decimal point that are repeated. They are called a period. For example, 0.3(3). Here "3" is in the period. They are classified as rational because 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 some numbers, and then the repetition begins.

The rule by which you need to write an infinite decimal as a common fraction will be different for the two types of numbers indicated. It is quite easy to write pure periodic fractions as ordinary fractions. As with finite ones, they need to be converted: write down the period in the numerator, and the denominator will be the number 9, repeated as many times as the number of digits the period contains.

For example, 0,(5). The number does not have an integer part, so you need to immediately start with the fractional part. Write 5 as the numerator and 9 as the denominator. That is, the answer will be the fraction 5/9.

The rule on how to write an ordinary decimal periodic fraction that is mixed.

Look at the length of the period. That's how many 9s the denominator will have.

Write down the denominator: first nines, then zeros.

To determine the numerator, you need to write down the difference of two numbers. All numbers after the decimal point will be minified, along with the period. Deductible - 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 contains one digit. So there will be one zero. There is also only one number 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, you need to subtract 5 from 58. It turns out 53. For example, you would have to write the answer as 53/90.

How are fractions converted to decimals?

The simplest option is a number whose denominator is the number 10, 100, etc. Then the denominator is simply discarded, and between the fractional and whole in parts a comma is added.

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. You just need to multiply not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two possible answers: a finite or a periodic decimal fraction.

Operations with ordinary fractions

Addition and subtraction

Students become acquainted with them earlier than others. Moreover, at first the fractions have the same denominators, and then they have different ones. General rules can be reduced to such a plan.

Find the least common multiple of the denominators.

Write additional factors for all ordinary fractions.

Multiply the numerators and denominators by the factors specified for them.

Add (subtract) the numerators of the fractions and leave the common denominator unchanged.

If the numerator of the minuend is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.

In the first case, you need to borrow one from the whole part. Add the denominator to the numerator of the fraction. And then do the subtraction.

In the second, it is necessary to apply the rule of subtracting a larger number from a smaller number. That is, from the module of the subtrahend, subtract the module of the minuend, and in response put a “-” sign.

Look carefully at the result of addition (subtraction). If you get an improper fraction, then you need to select the whole part. That is, divide the numerator by the denominator.

Multiplication and division

To perform them, fractions do not need to be reduced to common denominator. This makes it easier to perform actions. But they still require you to follow the rules.

When multiplying fractions, you need to look at the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.

Multiply the numerators.

Multiply the denominators.

If the result is a reducible fraction, then it must be simplified again.

When dividing, you must first replace division with multiplication, and the divisor (second fraction) with the reciprocal fraction (swap the numerator and denominator).

Then proceed as with multiplication (starting from point 1).

In tasks where you need to multiply (divide) by a whole number, the latter should be written as an improper fraction. That is, with a denominator of 1. Then act as described above.



Operations with decimals

Addition and subtraction

Of course, you can always convert a decimal into a fraction. And act according to the plan already described. 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. Add the missing number of zeros to it.

Write the fractions so that the comma is below the comma.

Add (subtract) like natural numbers.

Remove the comma.

Multiplication and division

It is important that you do not need to add zeros here. Fractions should be left as they are given in the example. And then go according to plan.

To multiply, you need to write the fractions one below the other, ignoring the commas.

Multiply like natural numbers.

Place 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 transform 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 a decimal by natural number.

Place a comma in your answer at the moment when the division of the whole part ends.



What if one example contains both types of fractions?

Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. In such tasks there are two possible solutions. You need to objectively weigh the numbers and choose the optimal one.

First way: represent ordinary decimals

It is suitable if division or translation results in finite fractions. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you don’t like working with ordinary fractions, you will have to count them.

Second way: write decimal fractions as ordinary

This technique turns out to be convenient if the part after the decimal point contains 1-2 digits. If there are more of them, you can get a very large ordinary fraction and decimal notations will allow you to calculate the task faster and easier. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.

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 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

Instructions

Learn to convert decimal fractions to ordinary fractions. Count how many characters are separated by a comma. One digit to the right of the decimal point means that the denominator is 10, two means 100, three means 1000, and so on. For example, the decimal fraction 6.8 is like "six point eight". When converting it, first write the number of whole units - 6. Write 10 in the denominator. The number 8 will appear in the numerator. It turns out that 6.8 = 6 8/10. Remember the rules of abbreviation. If the numerator and denominator are divisible by the same number, then the fraction can be reduced by common divisor. In this case, the number is 2. 6 8/10 = 6 2/5.

Try adding decimals. If you do this in a column, then be careful. The digits of all numbers must be strictly below each other - under the comma. The addition rules are exactly the same as when operating with . Add another decimal fraction to the same number 6.8 - for example, 7.3. Write a three under an eight, a comma under a comma, and a seven under a six. Start adding from the last digit. 3+8=11, that is, write down 1, remember 1. Next, add 6+7, you get 13. Add what was left in your mind and write down the result - 14.1.

Subtraction follows the same principle. Write the digits under each other, and the comma under the comma. Always use it as a guide, especially if the number of digits after it in the minuend is less than in the subtrahend. Subtract from the given number, for example, 2.139. Write the two under the six, the one under the eight, and the remaining two digits under the next digits, which can be designated zeros. It turns out that the minuend is not 6.8, but 6.800. By performing this action, you will receive a total of 4.661.

Operations with negative decimals are performed in the same way as with whole numbers. When adding, the minus is placed outside the brackets, and the given numbers are written in the brackets, and a plus is placed between them. The result is a negative number. That is, when you add -6.8 and -7.3 you will get the same result of 14.1, but with a “-” sign in front of it. If the subtrahend is greater than the minuend, then the minus is also taken out of the bracket, from more the lesser is deducted. Subtract -7.3 from 6.8. Transform the expression as follows. 6.8 - 7.3= -(7.3 - 6.8) = -0.5.

To multiply decimals, forget about the decimal point for a moment. Multiply them as if you were looking at whole numbers. After this, count the number of digits to the right after the decimal point in both factors. Separate the same number of characters in the work. Multiplying 6.8 and 7.3 gives you a total of 49.64. That is, to the right of the decimal point you will have 2 signs, while in the multiplicand and the multiplier there were one each.

Divide the given fraction by some integer. This action is performed in exactly the same way as with integers. The main thing is not to forget about the comma and put 0 at the beginning if the number of whole units is not divisible by the divisor. For example, try dividing the same 6.8 by 26. Put 0 at the beginning, since 6 is less than 26. Separate it with a comma, then tenths and hundredths will follow. The result will be approximately 0.26. In fact, in this case, an infinite non-periodic fraction is obtained, which can be rounded to the desired degree of accuracy.

When dividing two decimal fractions, use the property that when the dividend and divisor are multiplied by the same number, the quotient does not change. That is, convert both fractions to whole numbers, depending on how many decimal places there are. If you want to divide 6.8 by 7.3, just multiply both numbers by 10. It turns out that you need to divide 68 by 73. If one of the numbers has more decimal places, convert it to an integer first, and then second number. Multiply it by the same number. That is, when dividing 6.8 by 4.136, increase the dividend and divisor not by 10, but by 1000 times. Divide 6800 by 1436 to get 4.735.

The online fraction calculator allows you to perform simple arithmetic operations with fractions: adding fractions, subtracting fractions, multiplying fractions, dividing fractions. To make calculations, fill in the fields corresponding to the numerators and denominators of the two fractions.

Fractions in mathematics is a number representing a part of a unit or several parts of it.

A common fraction is written as two numbers, usually separated by a horizontal line indicating the division sign. The number above the line is called the numerator. The number below the line is called the denominator. The denominator of a fraction shows the number of equal parts into which the whole is divided, and the numerator of the fraction shows the number of these parts of the whole taken.

Fractions can be regular or improper.

• A fraction whose numerator is less than its denominator is called a proper fraction.
• An improper fraction is when the numerator of a fraction is greater than its denominator.

A mixed fraction is a fraction written as an integer and a proper fraction, and is understood as the sum of this number and the fractional part. Accordingly, a fraction that does not have an integer part is called a simple fraction. Any mixed fraction can be converted to an improper fraction.

In order to convert a mixed fraction into a common fraction, you need to add the product of the whole part and the denominator to the numerator of the fraction:

How to convert a common fraction to a mixed fraction

In order to translate common fraction mixed, you need:

1. Divide the numerator of a fraction by its denominator
2. The result of division will be the whole part
3. The balance of the department will be the numerator

How to convert a fraction to a decimal

In order to convert a fraction to a decimal, you need to divide its numerator by its denominator.

In order to convert a decimal fraction to an ordinary fraction, you must:

How to convert a fraction to a percentage

In order to convert a common or mixed fraction to a percentage, you need to convert it to a decimal fraction and multiply by 100.

How to convert percentages to fractions

In order to convert percentages into fractions, you need to obtain a decimal fraction from the percentage (dividing by 100), then convert the resulting decimal fraction into an ordinary fraction.

Adding Fractions

The algorithm for adding two fractions is as follows:

1. Perform addition of fractions by adding their numerators.

Subtracting Fractions

Algorithm for subtracting two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. Reduce fractions to a common denominator. To do this, you need to multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction.
3. Subtract one fraction from another by subtracting the numerator of the second fraction from the numerator of the first.
4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Multiplying fractions

Algorithm for multiplying two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
3. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Division of fractions

Algorithm for dividing two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. To divide fractions, you need to transform the second fraction by swapping its numerator and denominator, and then multiply the fractions.
3. Multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second.
4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Online calculators and converters: