How to make a decimal from a regular fraction. Converting a fraction to a decimal

Converting a Fraction to a Decimal

Let's say we want to convert the fraction 11/4 to a decimal. The easiest way to do it is this:

We succeeded because in this case the decomposition of the denominator into prime factors consists only of twos. We supplemented this expansion with two more fives, took advantage of the fact that 10 = 2∙5, and got a decimal fraction. Such a procedure is obviously possible if and only if the decomposition of the denominator into prime factors contains nothing but twos and fives. If any other prime number is present in the expansion of the denominator, then such a fraction cannot be converted to a decimal. Nevertheless, we will try to do this, but only in a different way, which we will get acquainted with using the example of the same fraction 11/4. Let's divide 11 by 4 using the “corner”:

In the response line we received the whole part (2), and we also have the remainder (3). Previously, we ended the division here, but now we know that we can add a comma and several zeros to the right of the dividend (11), which we will now mentally do. After the decimal point comes the tenths place. We add the zero that appears to the dividend in this digit to the resulting remainder (3):

Now the division can continue as if nothing had happened. You just need to remember to put a comma after the whole part in the answer line:

Now we add a zero to the remainder (2), which is in the hundredths place of the dividend and complete the division:

As a result, we get, as before,

Let's now try to calculate in exactly the same way what the fraction 27/11 is equal to:

We received the number 2.45 in the answer line, and the number 5 in the remainder line. But we have already encountered such a remnant before. Therefore, we can immediately say that if we continue our division with a “corner”, then the next number in the answer line will be 4, then the number 5 will come, then again 4 and again 5, and so on, ad infinitum:

27 / 11 = 2,454545454545...

We got the so-called periodic a decimal fraction with a period of 45. For such fractions, a more compact notation is used, in which the period is written only once, but it is enclosed in parentheses:

2,454545454545... = 2,(45).

Generally speaking, if we divide one natural number by another with a “corner”, writing the answer in the form of a decimal fraction, then only two outcomes are possible: (1) either sooner or later we will get zero in the remainder line, (2) or there will be such a remainder there, which we have already encountered before (the set of possible remainders is limited, since all of them are obviously smaller than the divisor). In the first case, the result of division is a finite decimal fraction, in the second case - a periodic one.

Convert periodic decimal to fraction

Let us be given a positive periodic decimal fraction with a zero integer part, for example:

a = 0,2(45).

How can I convert this fraction back to a common fraction?

Let's multiply it by 10 k, Where k is the number of digits between the decimal point and the opening parenthesis indicating the beginning of the period. In this case k= 1 and 10 k = 10:

a∙ 10 k = 2,(45).

Multiply the result by 10 n, Where n- the “length” of the period, that is, the number of digits enclosed between parentheses. In this case n= 2 and 10 n = 100:

a∙ 10 k ∙ 10 n = 245,(45).

Now let's calculate the difference

a∙ 10 k ∙ 10 n − a∙ 10 k = 245,(45) − 2,(45).

Since the fractional parts of the minuend and the subtrahend are the same, then the fractional part of the difference is equal to zero, and we arrive at a simple equation for a:

a∙ 10 k ∙ (10 n − 1) = 245 − 2.

This equation is solved using the following transformations:

a∙ 10 ∙ (100 − 1) = 245 − 2.

a∙ 10 ∙ 99 = 245 − 2.

We deliberately do not complete the calculations yet, so that it is clearly visible how this result can be immediately written down, omitting intermediate arguments. The minuend in the numerator (245) is the fractional part of the number

a = 0,2(45)

if you erase the brackets in her entry. The subtrahend in the numerator (2) is the non-periodic part of the number A, located between the comma and the opening parenthesis. The first factor in the denominator (10) is a unit, to which as many zeros are assigned as there are digits in the non-periodic part ( k). The second factor in the denominator (99) is as many nines as there are digits in the period ( n).

Now our calculations can be completed:

Here the numerator contains the period, and the denominator contains as many nines as there are digits in the period. After reduction by 9, the resulting fraction is equal to

In the same way,

It happens that for the convenience of calculations you need to convert an ordinary fraction to a decimal and vice versa. We will talk about how to do this in this article. Let's look at the rules for converting ordinary fractions to decimals and vice versa, and also give examples.

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We will consider converting ordinary fractions to decimals, following a certain sequence. First, let's look at how ordinary fractions with a denominator that is a multiple of 10 are converted into decimals: 10, 100, 1000, etc. Fractions with such denominators are, in fact, a more cumbersome notation of decimal fractions.

Next, we will look at how to convert ordinary fractions with any denominator, not just multiples of 10, into decimal fractions. Note that when converting ordinary fractions to decimals, not only finite decimals are obtained, but also infinite periodic decimal fractions.

Let's get started!

Translation of ordinary fractions with denominators 10, 100, 1000, etc. to decimals

First of all, let's say that some fractions require some preparation before converting to decimal form. What is it? Before the number in the numerator, you need to add so many zeros so that the number of digits in the numerator becomes equal to the number of zeros in the denominator. For example, for the fraction 3100, the number 0 must be added once to the left of the 3 in the numerator. Fraction 610, according to the rule stated above, does not need modification.

Let's look at one more example, after which we will formulate a rule that is especially convenient to use at first, while there is not much experience in converting fractions. So, the fraction 1610000 after adding zeros in the numerator will look like 001510000.

How to convert a common fraction with a denominator of 10, 100, 1000, etc. to decimal?

Rule for converting ordinary proper fractions to decimals

1. Write down 0 and put a comma after it.
2. We write down the number from the numerator that was obtained after adding zeros.

Now let's move on to examples.

Example 1: Converting fractions to decimals

Let's convert the fraction 39,100 to a decimal.

First, we look at the fraction and see that there is no need to carry out any preparatory actions - the number of digits in the numerator coincides with the number of zeros in the denominator.

Following the rule, we write 0, put a decimal point after it and write the number from the numerator. We get the decimal fraction 0.39.

Let's look at the solution to another example on this topic.

Example 2. Converting fractions to decimals

Let's write the fraction 105 10000000 as a decimal.

The number of zeros in the denominator is 7, and the numerator has only three digits. Let's add 4 more zeros before the number in the numerator:

0000105 10000000

Now we write down 0, put a decimal point after it and write down the number from the numerator. We get the decimal fraction 0.0000105.

The fractions considered in all examples are ordinary proper fractions. But how do you convert an improper fraction to a decimal? Let’s say right away that there is no need for preparation with adding zeros for such fractions. Let's formulate a rule.

Rule for converting ordinary improper fractions to decimals

1. Write down the number that is in the numerator.
2. We use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.

Below is an example of how to use this rule.

Example 3. Converting fractions to decimals

Let's convert the fraction 56888038009 100000 from an ordinary irregular fraction to a decimal.

First, let's write down the number from the numerator:

Now, on the right, we separate five digits with a decimal point (the number of zeros in the denominator is five). We get:

The next question that naturally arises is: how to convert a mixed number into a decimal fraction if the denominator of its fractional part is the number 10, 100, 1000, etc. To convert such a number to a decimal fraction, you can use the following rule.

Rule for converting mixed numbers to decimals

1. We prepare the fractional part of the number, if necessary.
2. We write down the whole part of the original number and put a comma after it.
3. We write down the number from the numerator of the fractional part along with the added zeros.

Let's look at an example.

Example 4: Converting mixed numbers to decimals

Let's convert the mixed number 23 17 10000 to a decimal fraction.

In the fractional part we have the expression 17 10000. Let's prepare it and add two more zeros to the left of the numerator. We get: 0017 10000.

Now we write down the whole part of the number and put a comma after it: 23, . .

After the decimal point, write down the number from the numerator along with zeros. We get the result:

23 17 10000 = 23 , 0017

Converting ordinary fractions to finite and infinite periodic fractions

Of course, you can convert to decimals and ordinary fractions with a denominator not equal to 10, 100, 1000, etc.

Often a fraction can be easily reduced to a new denominator, and then use the rule set out in the first paragraph of this article. For example, it is enough to multiply the numerator and denominator of the fraction 25 by 2, and we get the fraction 410, which is easily converted to the decimal form 0.4.

However, this method of converting a fraction to a decimal cannot always be used. Below we will consider what to do if it is impossible to apply the considered method.

A fundamentally new way to convert a fraction to a decimal is to divide the numerator by the denominator with a column. This operation is very similar to dividing natural numbers with a column, but has its own characteristics.

When dividing, the numerator is represented as a decimal fraction - a comma is placed to the right of the last digit of the numerator and zeros are added. In the resulting quotient, a decimal point is placed when the division of the integer part of the numerator ends. How exactly this method works will become clear after looking at the examples.

Example 5. Converting fractions to decimals

Let's convert the common fraction 621 4 to decimal form.

Let's represent the number 621 from the numerator as a decimal fraction, adding a few zeros after the decimal point. 621 = 621.00

Now let's divide 621.00 by 4 using a column. The first three steps of division will be the same as when dividing natural numbers, and we will get.

When we reach the decimal point in the dividend, and the remainder is different from zero, we put a decimal point in the quotient and continue dividing, no longer paying attention to the comma in the dividend.

As a result, we get the decimal fraction 155, 25, which is the result of reversing the common fraction 621 4

621 4 = 155 , 25

Let's look at another example to reinforce the material.

Example 6. Converting fractions to decimals

Let's reverse the common fraction 21 800.

To do this, divide the fraction 21,000 into a column by 800. The division of the whole part will end at the first step, so immediately after it we put a decimal point in the quotient and continue the division, not paying attention to the comma in the dividend until we get a remainder equal to zero.

As a result, we got: 21,800 = 0.02625.

But what if, when dividing, we still do not get a remainder of 0. In such cases, the division can be continued indefinitely. However, starting from a certain step, the residues will be repeated periodically. Accordingly, the numbers in the quotient will be repeated. This means that an ordinary fraction is converted into a decimal infinite periodic fraction. Let us illustrate this with an example.

Example 7. Converting fractions to decimals

Let's convert the common fraction 19 44 to a decimal. To do this, we perform division by column.

We see that during division, residues 8 and 36 are repeated. In this case, the numbers 1 and 8 are repeated in the quotient. This is the period in decimal fraction. When recording, these numbers are placed in brackets.

Thus, the original ordinary fraction is converted into an infinite periodic decimal fraction.

19 44 = 0 , 43 (18) .

Let us have an irreducible ordinary fraction. What form will it take? Which ordinary fractions are converted to finite decimals, and which ones are converted to infinite periodic ones?

First, let's say that if a fraction can be reduced to one of the denominators 10, 100, 1000..., then it will have the form of a final decimal fraction. In order for a fraction to be reduced to one of these denominators, its denominator must be a divisor of at least one of the numbers 10, 100, 1000, etc. From the rules for factoring numbers into prime factors it follows that the divisor of numbers is 10, 100, 1000, etc. must, when factored into prime factors, contain only the numbers 2 and 5.

Let's summarize what has been said:

1. A common fraction can be reduced to a final decimal if its denominator can be factored into prime factors of 2 and 5.
2. If, in addition to the numbers 2 and 5, there are other prime numbers in the expansion of the denominator, the fraction is reduced to the form of an infinite periodic decimal fraction.

Let's give an example.

Example 8. Converting fractions to decimals

Which of these fractions 47 20, 7 12, 21 56, 31 17 is converted into a final decimal fraction, and which one - only into a periodic one. Let's answer this question without directly converting a fraction to a decimal.

The fraction 47 20, as is easy to see, by multiplying the numerator and denominator by 5 is reduced to a new denominator 100.

47 20 = 235 100. From this we conclude that this fraction is converted to a final decimal fraction.

Factoring the denominator of the fraction 7 12 gives 12 = 2 · 2 · 3. Since the prime factor 3 is different from 2 and 5, this fraction cannot be represented as a finite decimal fraction, but will have the form of an infinite periodic fraction.

The fraction 21 56, firstly, needs to be reduced. After reduction by 7, we obtain the irreducible fraction 3 8, the denominator of which is factorized to give 8 = 2 · 2 · 2. Therefore, it is a final decimal fraction.

In the case of the fraction 31 17, factoring the denominator is the prime number 17 itself. Accordingly, this fraction can be converted into an infinite periodic decimal fraction.

An ordinary fraction cannot be converted into an infinite and non-periodic decimal fraction

Above we talked only about finite and infinite periodic fractions. But can any ordinary fraction be converted into an infinite non-periodic fraction?

We answer: no!

Important!

When converting an infinite fraction to a decimal, the result is either a finite decimal or an infinite periodic decimal.

The remainder of a division is always less than the divisor. In other words, according to the divisibility theorem, if we divide some natural number by the number q, then the remainder of the division in any case cannot be greater than q-1. After the division is completed, one of the following situations is possible:

1. We get a remainder of 0, and this is where the division ends.
2. We get a remainder, which is repeated upon subsequent division, resulting in an infinite periodic fraction.

There cannot be any other options when converting a fraction to a decimal. Let's also say that the length of the period (number of digits) in an infinite periodic fraction is always less than the number of digits in the denominator of the corresponding ordinary fraction.

Converting decimals to fractions

Now it's time to look at the reverse process of converting a decimal fraction into a common fraction. Let us formulate a translation rule that includes three stages. How to convert a decimal fraction to a common fraction?

Rule for converting decimal fractions to ordinary fractions

1. In the numerator we write the number from the original decimal fraction, discarding the comma and all zeros on the left, if any.
2. In the denominator we write one followed by as many zeros as there are digits after the decimal point in the original decimal fraction.
3. If necessary, reduce the resulting ordinary fraction.

Let's look at the application of this rule using examples.

Example 8. Converting decimal fractions to ordinary fractions

Let's imagine the number 3.025 as an ordinary fraction.

1. We write the decimal fraction itself into the numerator, discarding the comma: 3025.
2. In the denominator we write one, and after it three zeros - this is exactly how many digits are contained in the original fraction after the decimal point: 3025 1000.
3. The resulting fraction 3025 1000 can be reduced by 25, resulting in: 3025 1000 = 121 40.

Example 9. Converting decimal fractions to ordinary fractions

Let's convert the fraction 0.0017 from decimal to ordinary.

1. In the numerator we write the fraction 0, 0017, discarding the comma and zeros on the left. It will turn out to be 17.
2. We write one in the denominator, and after it we write four zeros: 17 10000. This fraction is irreducible.

If a decimal fraction has an integer part, then such a fraction can be immediately converted to a mixed number. How to do this?

Let's formulate one more rule.

Rule for converting decimal fractions to mixed numbers.

1. The number before the decimal point in the fraction is written as the integer part of the mixed number.
2. In the numerator we write the number after the decimal point in the fraction, discarding the zeros on the left if there are any.
3. In the denominator of the fractional part we add one and as many zeros as there are digits after the decimal point in the fractional part.

Let's take an example

Example 10: Converting a decimal to a mixed number

Let's imagine the fraction 155, 06005 as a mixed number.

1. We write the number 155 as an integer part.
2. In the numerator we write the numbers after the decimal point, discarding the zero.
3. We write one and five zeros in the denominator

Let's learn a mixed number: 155 6005 100000

The fractional part can be reduced by 5. We shorten it and get the final result:

155 , 06005 = 155 1201 20000

Converting infinite periodic decimals to fractions

Let's look at examples of how to convert periodic decimal fractions into ordinary fractions. Before we begin, let's clarify: any periodic decimal fraction can be converted to an ordinary fraction.

The simplest case is when the period of the fraction is zero. A periodic fraction with a zero period is replaced by a final decimal fraction, and the process of reversing such a fraction is reduced to reversing the final decimal fraction.

Example 11. Converting a periodic decimal fraction to a common fraction

Let us invert the periodic fraction 3, 75 (0).

Eliminating the zeros on the right, we get the final decimal fraction 3.75.

Converting this fraction to an ordinary fraction using the algorithm discussed in the previous paragraphs, we obtain:

3 , 75 (0) = 3 , 75 = 375 100 = 15 4 .

What if the period of the fraction is different from zero? The periodic part should be considered as the sum of the terms of a geometric progression, which decreases. Let's explain this with an example:

0 , (74) = 0 , 74 + 0 , 0074 + 0 , 000074 + 0 , 00000074 + . .

There is a formula for the sum of terms of an infinite decreasing geometric progression. If the first term of the progression is b and the denominator q is such that 0< q < 1 , то сумма равна b 1 - q .

Let's look at a few examples using this formula.

Example 12. Converting a periodic decimal fraction to a common fraction

Let us have a periodic fraction 0, (8) and we need to convert it to an ordinary fraction.

0 , (8) = 0 , 8 + 0 , 08 + 0 , 008 + . .

Here we have an infinite decreasing geometric progression with the first term 0, 8 and the denominator 0, 1.

Let's apply the formula:

0 , (8) = 0 , 8 + 0 , 08 + 0 , 008 + . . = 0 , 8 1 - 0 , 1 = 0 , 8 0 , 9 = 8 9

This is the required ordinary fraction.

To consolidate the material, consider another example.

Example 13. Converting a periodic decimal fraction to a common fraction

Let's reverse the fraction 0, 43 (18).

First we write the fraction as an infinite sum:

0 , 43 (18) = 0 , 43 + (0 , 0018 + 0 , 000018 + 0 , 00000018 . .)

Let's look at the terms in brackets. This geometric progression can be represented as follows:

0 , 0018 + 0 , 000018 + 0 , 00000018 . . = 0 , 0018 1 - 0 , 01 = 0 , 0018 0 , 99 = 18 9900 .

We add the result to the final fraction 0, 43 = 43 100 and get the result:

0 , 43 (18) = 43 100 + 18 9900

After adding these fractions and reducing, we get the final answer:

0 , 43 (18) = 19 44

To conclude this article, we will say that non-periodic infinite decimal fractions cannot be converted into ordinary fractions.

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A decimal fraction consists of two parts, separated by commas. The first part is a whole unit, the second part is tens (if there is one number after the decimal point), hundreds (two numbers after the decimal point, like two zeros in a hundred), thousandths, etc. Let's look at examples of decimal fractions: 0, 2; 7, 54; 235.448; 5.1; 6.32; 0.5. These are all decimal fractions. How to convert a decimal fraction to an ordinary fraction?

Example one

We have a fraction, for example, 0.5. As mentioned above, it consists of two parts. The first number, 0, shows how many whole units the fraction has. In our case there are none. The second number shows tens. The fraction even reads zero point five. Decimal number convert to fraction Now it won’t be difficult, we’ll write 5/10. If you see that the numbers have a common factor, you can reduce the fraction. We have this number 5, dividing both sides of the fraction by 5, we get - 1/2.

Example two

Let's take a more complex fraction - 2.25. It reads like this: two point two and twenty-five hundredths. Please note - hundredths, since there are two numbers after the decimal point. Now you can convert it to a common fraction. We write down - 2 25/100. The whole part is 2, the fractional part is 25/100. As in the first example, this part can be shortened. The common factor for the numbers 25 and 100 is the number 25. Note that we always choose the greatest common factor. Dividing both sides of the fraction by GCD, we got 1/4. So 2.25 is 2 1/4.

Example three

And to consolidate the material, let’s take the decimal fraction 4.112 - four point one and one hundred and twelve thousandths. Why thousandths, I think, is clear. Now we write down 4 112/1000. Using the algorithm, we find the gcd of the numbers 112 and 1000. In our case, this is the number 6. We get 4 14/125.

Conclusion

1. We break the fraction into whole and fractional parts.
2. Let's see how many digits are after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
3. We write the fraction in ordinary form.
4. Reduce the numerator and denominator of the fraction.
5. We write down the resulting fraction.
6. We check by dividing the upper part of the fraction by the lower part. If there is an integer part, add it to the resulting decimal fraction. The original version turned out great, which means you did everything right.

Using examples, I showed how you can convert a decimal fraction to an ordinary fraction. As you can see, this is very easy and simple to do.

A fraction can be converted to a whole number or to a decimal. An improper fraction, the numerator of which is greater than the denominator and is divisible by it without a remainder, is converted to a whole number, for example: 20/5. Divide 20 by 5 and get the number 4. If the fraction is proper, that is, the numerator is less than the denominator, then convert it to a number (decimal fraction). You can get more information about fractions from our section -.

Ways to convert a fraction to a number

• The first way to convert a fraction to a number is suitable for a fraction that can be converted to a number that is a decimal fraction. First, let's find out whether it is possible to convert the given fraction to a decimal fraction. To do this, let's pay attention to the denominator (the number that is below the line or to the right of the sloping line). If the denominator can be factorized (in our example - 2 and 5), which can be repeated, then this fraction can actually be converted into a final decimal fraction. For example: 11/40 =11/(2∙2∙2∙5). This common fraction will be converted to a number (decimal) with a finite number of decimal places. But the fraction 17/60 =17/(5∙2∙2∙3) will be converted into a number with an infinite number of decimal places. That is, when accurately calculating a numerical value, it is quite difficult to determine the final decimal place, since there are an infinite number of such signs. Therefore, solving problems usually requires rounding the value to hundredths or thousandths. Next, you need to multiply both the numerator and the denominator by such a number so that the denominator produces the numbers 10, 100, 1000, etc. For example: 11/40 = (11∙25)/(40∙25) = 275/1000 = 0.275
• The second way to convert a fraction into a number is simpler: you need to divide the numerator by the denominator. To apply this method, we simply perform division, and the resulting number will be the desired decimal fraction. For example, you need to convert the fraction 2/15 into a number. Divide 2 by 15. We get 0.1333... - an infinite fraction. We write it like this: 0.13(3). If the fraction is an improper fraction, that is, the numerator is greater than the denominator (for example, 345/100), then converting it to a number will result in a whole number value or a decimal fraction with a whole fractional part. In our example it will be 3.45. To convert a mixed fraction like 3 2 / 7 into a number, you must first convert it to an improper fraction: (3∙7+2)/7 = 23/7. Next, divide 23 by 7 and get the number 3.2857143, which we reduce to 3.29.

The easiest way to convert a fraction into a number is to use a calculator or other computing device. First we indicate the numerator of the fraction, then press the button with the “divide” icon and enter the denominator. After pressing the "=" key, we get the desired number.