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Converting fractions to decimal examples. Converting a decimal fraction to a prime fraction and vice versa

Already in elementary school, students are exposed to 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 become acquainted with the first ones in elementary 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 as a decimal. 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 is better to start in chronological order, as they are 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 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 a comma is placed between the fractional and integer parts.

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 this 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 a 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 fraction by a 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 may end up with a very large common fraction and decimal notation will make the task faster and easier to calculate. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.

All fractions are divided into two types: ordinary and decimal. Fractions of this type are called ordinary: 9/8.3/4.1/2.1 3/4. They have a top number (numerator) and a bottom number (denominator). When the numerator is less than the denominator, the fraction is called proper; otherwise, the fraction is called improper. Fractions such as 1 7/8 consist of an integer part (1) and a fractional part (7/8) and are called mixed.

So, fractions are:

  1. Ordinary
    1. Correct
    2. Wrong
    3. Mixed
  2. Decimal

How to make a decimal from a fraction

A basic school mathematics course teaches how to convert a fraction to a decimal. Everything is extremely simple: you need to divide the numerator by the denominator “manually” or, if you’re really lazy, then using a microcalculator. Here's an example: 2/5=0.4;3/4=0.75; 1/2=0.5. It's not much harder to convert an improper fraction to a decimal. Example: 1 3/4= 7/4= 1.75. The last result can be obtained without division, if we take into account that 3/4 = 0.75 and add one: 1 + 0.75 = 1.75.

However, not all ordinary fractions are so simple. For example, let's try to convert 1/3 from ordinary fractions to decimals. Even someone who had a C in mathematics (using a five-point system) will notice that no matter how long the division continues, after zero and a comma there will be an infinite number of triples 1/3 = 0.3333…. . It is customary to read this way: zero point, three in period. It is written accordingly as follows: 1/3=0,(3). A similar situation will occur if you try to convert 5/6 into a decimal fraction: 5/6=0.8(3). Such fractions are called infinite periodic. Here is an example for the fraction 3/7: 3/7= 0.42857142857142857142857142857143…, that is, 3/7=0.(428571).

So, as a result of converting a common fraction into a decimal, you can get:

  1. non-periodic decimal fraction;
  2. periodic decimal fraction.

It should be noted that there are also infinite non-periodic fractions that are obtained by performing the following actions: taking the nth root, logarithm, potentiation. For example, √3= 1.732050807568877… . The famous number π≈ 3.1415926535897932384626433832795…. .

Let's now multiply 3 by 0,(3): 3×0,(3)=0,(9)=1. It turns out that 0,(9) is another form of writing unit. Likewise, 9=9/9.16=16.0, etc.

The question opposite to that given in the title of this article is also legitimate: “how to convert a decimal fraction into a regular one.” The answer to this question is given by an example: 0.5= 5/10=1/2. In the last example, we reduced the numerator and denominator of the fraction 5/10 by 5. That is, to turn a decimal into a common fraction, you need to represent it as a fraction with a denominator of 10.

It will be interesting to watch this video about what fractions are:

To learn how to convert a decimal fraction to a common fraction, see here:

In dry mathematical language, a fraction is a number that is represented as a part of one. Fractions are widely used in human life: we use fractions to indicate proportions in culinary recipes, give decimal scores in competitions, or use them to calculate discounts in stores.

Representation of fractions

There are at least two forms of writing one fractional number: in decimal form or in the form of an ordinary fraction. In decimal form, the numbers look like 0.5; 0.25 or 1.375. We can represent any of these values ​​as an ordinary fraction:

  • 0,5 = 1/2;
  • 0,25 = 1/4;
  • 1,375 = 11/8.

And if we easily convert 0.5 and 0.25 from an ordinary fraction to a decimal and back, then in the case of the number 1.375 everything is not obvious. How to quickly convert any decimal number to a fraction? There are three simple ways.

Getting rid of the comma

The simplest algorithm involves multiplying a number by 10 until the comma disappears from the numerator. This transformation is carried out in three steps:

Step 1: To begin with, we write the decimal number as a fraction “number/1”, that is, we get 0.5/1; 0.25/1 and 1.375/1.

Step 2: After this, multiply the numerator and denominator of the new fractions until the comma disappears from the numerators:

  • 0,5/1 = 5/10;
  • 0,25/1 = 2,5/10 = 25/100;
  • 1,375/1 = 13,75/10 = 137,5/100 = 1375/1000.

Step 3: We reduce the resulting fractions to a digestible form:

  • 5/10 = 1 × 5 / 2 × 5 = 1/2;
  • 25/100 = 1 × 25 / 4 × 25 = 1/4;
  • 1375/1000 = 11 × 125 / 8 × 125 = 11/8.

The number 1.375 had to be multiplied by 10 three times, which is no longer very convenient, but what do we have to do if we need to convert the number 0.000625? In this situation, we use the following method of converting fractions.

Getting rid of commas even easier

The first method describes in detail the algorithm for “removing” a comma from a decimal, but we can simplify this process. Again, we follow three steps.

Step 1: We count how many digits are after the decimal point. For example, the number 1.375 has three such digits, and 0.000625 has six. We will denote this quantity by the letter n.

Step 2: Now we just need to represent the fraction in the form C/10 n, where C are the significant digits of the fraction (without zeros, if any), and n is the number of digits after the decimal point. Eg:

  • for the number 1.375 C = 1375, n = 3, the final fraction according to the formula 1375/10 3 = 1375/1000;
  • for the number 0.000625 C = 625, n = 6, the final fraction according to the formula 625/10 6 = 625/1000000.

Essentially, 10n is a 1 with n zeros, so you don't have to bother raising the ten to the power - just 1 with n zeros. After this, it is advisable to reduce a fraction so rich in zeros.

Step 3: We reduce the zeros and get the final result:

  • 1375/1000 = 11 × 125 / 8 × 125 = 11/8;
  • 625/1000000 = 1 × 625/ 1600 × 625 = 1/1600.

The fraction 11/8 is an improper fraction because its numerator is greater than its denominator, which means we can isolate the whole part. In this situation, we subtract the whole part of 8/8 from 11/8 and get the remainder 3/8, therefore the fraction looks like 1 and 3/8.

Conversion by ear

For those who can read decimals correctly, the easiest way to convert them is by hearing. If you read 0.025 not as “zero, zero, twenty-five” but as “25 thousandths,” then you will have no problem converting decimals to fractions.

0,025 = 25/1000 = 1/40

Thus, reading a decimal number correctly allows you to immediately write it down as a fraction and reduce it if necessary.

Examples of using fractions in everyday life

At first glance, ordinary fractions are practically not used in everyday life or at work, and it is difficult to imagine a situation when you need to convert a decimal fraction into a regular fraction outside of school tasks. Let's look at a couple of examples.

Job

So, you work in a candy store and sell halva by weight. To make the product easier to sell, you divide the halva into kilogram briquettes, but few buyers are willing to purchase a whole kilogram. Therefore, you have to divide the treat into pieces each time. And if the next buyer asks you for 0.4 kg of halva, you will sell him the required portion without any problems.

0,4 = 4/10 = 2/5

Life

For example, you need to make a 12% solution to paint the model in the shade you want. To do this, you need to mix paint and solvent, but how to do it correctly? 12% is a decimal fraction of 0.12. Convert the number to a common fraction and get:

0,12 = 12/100 = 3/25

Knowing the fractions will help you mix the ingredients correctly and get the color you want.

Conclusion

Fractions are commonly used in everyday life, so if you frequently need to convert decimals to fractions, you'll want to use an online calculator that can instantly get the result as a reduced fraction.

Then press the buttons and the task is completed. The result will be either a whole number or a decimal fraction. A decimal fraction may have a long remainder after . In this case, the fraction must be rounded to the specific digit you need, using rounding (numbers up to 5 are rounded down, from 5 inclusive and more - up).

If you don't have a calculator at hand, you will have to. Write the numerator of the fraction with the denominator, with a corner between them indicating . For example, convert the fraction 10/6 to a number. First, divide 10 by 6. You get 1. Write the result in a corner. Multiply 1 by 6, you get 6. Subtract 6 from 10. You get a remainder of 4. The remainder must be divided by 6 again. Add the number 0 to 4, and divide 40 by 6. You get 6. Write 6 in the result, after the decimal point. Multiply 6 by 6. You get 36. Subtract 36 from 40. The remainder is again 4. You don’t need to continue further, since it becomes obvious that the result will be the number 1.66(6). Round this fraction to the digit you need. For example, 1.67. This is the final result.

Related article

Sources:

  • converting fractions with whole numbers

Fractions are used to represent numbers that consist of one or more parts of a unit. The term "fraction" comes from the Latin fractura, which means "to crush, break." There are differences between ordinary and decimal fractions. Moreover, in ordinary fractions, a unit can be divided into any number of parts, and in a decimal, this quantity must be a multiple of 10. Any fraction can be either ordinary or decimal.

You will need

  • To calculate the result you will need a calculator or a piece of paper and a pen.

Instructions

So, first, take a common fraction and divide it into parts. For example, 2 1\8, in which 2 is an integer part, and 1\8 is a fraction. From it you can see that the number was divided by 8, but only one was taken. The part taken is the numerator, and the number of parts divided by is the denominator.

note

There are often fractions that cannot be completely converted to decimals. In this case, rounding comes to the rescue. If you want to round to the nearest thousand, look at the fourth decimal place. If it is less than 5, then write down the answer, the first three digits after the decimal point without changing, otherwise you must add one to the last digit of the three. For example, 0.89643123 can be written as 0.896, but 0.89663123 is 0.897.

Helpful advice

If you are calculating the result manually, then before dividing the fraction it is better to reduce it as much as possible, and also separate whole parts from it.

Sources:

  • how to convert fractions

Fraction is one of the elements of formulas for entering in the Word word processor there is a Microsoft Equation tool. Using it, you can enter any complex mathematical or physical formulas, equations and other elements that include special characters.

Instructions

To launch the Microsoft Equation tool, you need to go to: “Insert” -> “Object”, in the dialog box that opens, on the first tab from the list you need to select Microsoft Equation and click “Ok” or double-click on the selected item. After launching the editor, a toolbar will open in front of you and an input field will be displayed: a dotted rectangle. The toolbar is divided into sections, each of which contains a set of action symbols or expressions. When you click on one of the sections, a list of tools located in it will expand. From the list that opens, select the desired symbol and click on it. Once selected, the specified symbol will appear in the selected rectangle in the document.

The section containing elements for writing fractions is located in the second line of the toolbar. When you hover your mouse over it, you will see the tooltip “Patterns of Fractions and Radicals”. Click the section once and expand the list. The drop-down menu contains templates for horizontal and oblique fractions. From the options that appear, you can choose the one that suits your task. Click on the desired option. After clicking, a fraction symbol and places for entering the numerator and denominator, framed by a dotted line, will appear in the input field that opens in the document. The default cursor is automatically placed in the numerator input field. Enter the numerator. In addition to numbers, you can also enter symbols, letters or action signs. They can be entered either from the keyboard or from the corresponding sections of the Microsoft Equation toolbar. After the numerator, press the TAB key to move to the denominator. You can also go by clicking in the field to enter the denominator. Once written, click the mouse pointer anywhere in the document, the toolbar will close, and entering the fraction will be completed. To edit, double-click on it with the left mouse button.

If, when you open the “Insert” -> “Object” menu, you do not find the Microsoft Equation tool in the list, you need to install it. Launch the installation disk, disk image, or Word distribution file. In the installer window that appears, select “Add or remove components. Add or remove individual components" and click "Next". In the next window, check the “Advanced application settings” option. Click Next. In the next window, find the “Office Tools” list item and click on the plus sign on the left. In the expanded list, we are interested in the “Formula Editor” item. Click on the icon next to “Formula Editor” and, in the menu that opens, click “Run from Computer”. After that, click “Update” and wait until the required component is installed.

Materials on fractions and study sequentially. Below you will find detailed information with examples and explanations.

1. Mixed number into a common fraction.Let's write the number in general form:

We remember a simple rule - we multiply the whole part by the denominator and add the numerator, that is:

Examples:


2. On the contrary, an ordinary fraction into a mixed number. *Of course, this can only be done with an improper fraction (when the numerator is greater than the denominator).

With “small” numbers, in general, no actions need to be taken; the result is “visible” immediately, for example, fractions:

*More details:

15:13 = 1 remainder 2

4:3 = 1 remainder 1

9:5 = 1 remainder 4

But if the numbers are more, then you can’t do without calculations. Everything is simple here - divide the numerator by the denominator with a corner until the remainder is less than the divisor. Division scheme:


For example:

*Our numerator is the dividend, the denominator is the divisor.


We get the whole part (incomplete quotient) and the remainder. We write down an integer, then a fraction (the numerator contains the remainder, but the denominator remains the same):

3. Convert decimal to ordinary.

Partially in the first paragraph, where we talked about decimal fractions, we already touched on this. We write it down as we hear it. For example - 0.3; 0.45; 0.008; 4.38; 10.00015

We have the first three fractions without an integer part. And the fourth and fifth ones have it, let’s convert them into ordinary ones, we already know how to do this:

*We see that fractions can also be reduced, for example 45/100 = 9/20, 38/100 = 19/50 and others, but we will not do this here. Regarding reduction, you will find a separate paragraph below, where we will analyze everything in detail.

4. Convert ordinary to decimal.

It's not that simple. With some fractions it is immediately obvious and clear what to do with it so that it becomes a decimal, for example:

We use our wonderful basic property of a fraction - we multiply the numerator and denominator by 5, 25, 2, 5, 4, 2, respectively, and we get:


If there is an entire part, then it’s also not complicated:

We multiply the fractional part by 2, 25, 2 and 5, respectively, and get:

And there are those for which without experience it is impossible to determine that they can be converted into decimals, for example:

What numbers should we multiply the numerator and denominator by?

Here again a proven method comes to the rescue - division by a corner, a universal method, you can always use it to convert a common fraction to a decimal:


This way you can always determine whether a fraction is converted to a decimal. The fact is that not every ordinary fraction can be converted to a decimal, for example, such as 1/9, 3/7, 7/26 are not converted. What then is the fraction obtained when dividing 1 by 9, 3 by 7, 5 by 11? My answer is infinite decimal (we talked about them in paragraph 1). Let's divide:


That's all! Good luck to you!

Sincerely, Alexander Krutitskikh.