Converting a fraction to a decimal

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

						2∙2∙5∙5

We succeeded because in this case the factorization 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 does not contain anything 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 on the example of the same fraction 11/4. Let's divide 11 by 4 "corner":

In the response line, we got the integer part ( 2 ), and we also have the remainder ( 3 ). Previously, we ended the division on this, but now we know that a comma and several zeros can be attributed to the dividend ( 11 ) on the right, which we will mentally do now. After the decimal point comes the tenth place. Zero, which stands for the dividend in this category, we will attribute 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 integer part in the answer line:

Now we attribute to the remainder ( 2 ) zero, which stands for the dividend in the hundredths place and bring the division to the end:

As a result, we get, as before,

Now let's 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 seen such a remnant before. Therefore, we can immediately say that if we continue our division by the “corner”, then the next digit in the answer line will be 4, then the number 5 will go, then again 4 and again 5, and so on, ad infinitum:

27 / 11 = 2,454545454545...

We have received the so-called periodical a decimal fraction with a period of 45. For such fractions, a more compact notation is used, in which the period is written out only once, but at the same time it is enclosed in parentheses:

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

Generally speaking, if we divide one natural number by a “corner”, writing the answer as 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, which we have already met before (the set of possible residues is limited, since they are all obviously less than the divisor). In the first case, the result of division is a final decimal fraction, in the second case, a periodic one.

Converting a Periodic Decimal to a Common 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 comma and the opening parenthesis that indicates 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- "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 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.

	245 − 2
	10 ∙ 99

We deliberately do not bring the calculations to the end yet, so that it can be clearly seen how this result can be written out immediately, omitting intermediate arguments. Decreasing in the numerator ( 245 ) is the fractional part of the number

a = 0,2(45)

if you delete 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 one, 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 there is a period in the numerator, and as many nines in the denominator as there are digits in the period. After reducing by 9, the resulting fraction is equal to

In the same way,

A decimal has two parts separated by commas. The first part is an integer unit, the second part is tens (if the number after the decimal point is one), hundreds (two numbers after the decimal point, like two zeros in a hundred), thousandths, etc. Let's look at examples of decimals: 0, 2; 7, 54; 235.448; 5.1; 6.32; 0.5. These are all decimals. How do you convert a decimal to a common 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 integer units the fraction has. In our case, they are not. The second number shows tens. The fraction even reads zero point five tenths. Decimal number convert to fraction now it will not be difficult, we write 5/10. If you see that the numbers have a common divisor, you can reduce the fraction. We have this number 5, dividing both parts of the fraction by 5, we get - 1/2.

Example two

Let's take a more complex fraction - 2.25. It is read like this - two whole and twenty-five hundredths. Pay attention - hundredths, since there are two numbers after the decimal point. Now you can convert to a common fraction. We write down - 2 25/100. The integer part is 2, the fractional part is 25/100. As in the first example, this part can be shortened. The common divisor for 25 and 100 is 25. Note that we always choose the greatest common divisor. Dividing both parts 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 whole and one hundred and twelve thousandths. Why thousandths, I think, is clear. Now we write down 4 112/1000. According to 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 integer and fractional parts.
2. We look at how many digits after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
3. We write the fraction in the usual form.
4. We reduce the numerator and denominator of the fraction.
5. Write down the resulting fraction.
6. We perform a check, divide the upper part of the fraction by the lower one. If there is an integer part, add to the resulting decimal fraction. It turned out the original version - great, so you did everything right.

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

A fraction can be converted to an integer or a decimal. An improper fraction, the numerator of which is greater than the denominator and is divisible by it without a remainder, is converted into an integer, for example: 20/5. Divide 20 by 5 and get the number 4. If the fraction is correct, that is, the numerator is less than the denominator, then convert it to a number (decimal fraction). You can learn more 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 a given fraction into a decimal fraction. To do this, pay attention to the denominator (the number that is under the line or to the right of the oblique). If the denominator can be decomposed into factors (in our example - 2 and 5), which can be repeated, then this fraction can really be converted into a final decimal fraction. For example: 11/40 =11/(2∙2∙2∙5). This common fraction will be converted into a number (decimal fraction) with a finite number of decimal places. But the fraction 17/60 =17/(5∙2∙2∙3) will be translated 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 sign after the decimal point, since there are an infinite number of such signs. Therefore, to solve problems, you usually need to round the value to hundredths or thousandths. Further, it is necessary to multiply both the numerator and the denominator by such a number that the denominator will have 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 to a number is simpler: you need to divide the numerator by the denominator. To apply this method, we simply perform the division, and the resulting number will be the desired decimal fraction. For example, you need to convert the fraction 2/15 to a number. We divide 2 by 15. We get 0, 1333 ... - an infinite fraction. We write it down like this: 0.13(3). If the fraction is incorrect, that is, the numerator is greater than the denominator (for example, 345/100), then as a result of converting it to a number, an integer numerical value or a decimal fraction with an integer fractional part will be obtained. In our example, this will be 3.45. To convert a mixed fraction like 3 2 / 7 to a number, you must first convert it to an improper fraction: (3∙7+2)/7 =23/7. Next, we divide 23 by 7 and get the number 3.2857143, which we reduce to 3.29.

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

Decimal numbers such as 0.2; 1.05; 3.017 etc. as they are heard, so they are written. Zero point two, we get a fraction. One whole five hundredths, we get a fraction. Three whole seventeen thousandths, we get a fraction. The digits before the decimal point in a decimal number are the integer part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a one-digit number after the decimal point, the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some of the resulting fractions can be reduced. In our examples

Converting a fraction to a decimal number

This is the reverse of the previous transformation. What is a decimal fraction? Her denominator is always 10, or 100, or 1000, or 10,000, and so on. If your usual fraction has such a denominator, there is no problem. For example, or

If a fraction, for example . In this case, you need to use the basic property of the fraction and convert the denominator to 10 or 100, or 1000 ... In our example, if we multiply the numerator and denominator by 4, we get a fraction that can be written as a decimal number 0.12.

Some fractions are easier to divide than to convert the denominator. For example,

Some fractions cannot be converted to decimal numbers!
For example,

Converting a mixed fraction to an improper

A mixed fraction, such as , is easily converted to an improper fraction. To do this, you need to multiply the integer part by the denominator (bottom) and add it to the numerator (top), leaving the denominator (bottom) unchanged. That is

When converting a mixed fraction to an improper one, you can remember that you can use the addition of fractions

Converting an improper fraction to a mixed one (highlighting the whole part)

An improper fraction can be converted to a mixed fraction by highlighting the whole part. Consider an example, . Determine how many integer times "3" fit in "23". Or we divide 23 by 3 on the calculator, the whole number up to the decimal point is the desired one. This is "7". Next, we determine the numerator of the future fraction: we multiply the resulting "7" by the denominator "3" and subtract the result from the numerator "23". How would we find the excess that remains from the numerator "23", if we remove the maximum number of "3". The denominator is left unchanged. Everything is done, write down the result

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 distinguish the upper number (numerator) and the lower number (denominator). When the numerator is less than the denominator, the fraction is called proper, otherwise the fraction is improper. Fractions such as 1 7/8 consist of an integer part (1) and a fractional part (7/8) and are called mixed.

So the fractions are:

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

How to convert a common fraction to a decimal

How to convert an ordinary fraction to a decimal, teaches a basic school mathematics course. Everything is extremely simple: you need to divide the numerator by the denominator "manually" or, if you are completely lazy, then on a microcalculator. Here is an example: 2/5=0.4; 3/4=0.75; 1/2=0.5. It's not much harder to convert to a decimal improper fraction. 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 those who had a triple in mathematics (according to 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 as follows: zero integers, three in a 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…, i.e. 3/7=0,(428571).

So, as a result of the transformation of an ordinary fraction into a decimal, one can get:

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

It should be noted that there are also infinite non-periodic fractions, which are obtained by performing such actions: taking the root of the n-th degree, taking logarithms, potentiating. 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 a different form of writing unity. Similarly, 9=9/9.16=16.0, etc.

The question opposite to the one 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 gives 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 fraction into an ordinary one, you need to represent it as a fraction with a denominator of 10.

It will be interesting to watch a video about what fractions are in general:

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