How to turn an improper fraction into a decimal. Operations with ordinary fractions

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:

Ordinary
1. Correct
2. Wrong
3. Mixed
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:

non-periodic decimal fraction;
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:

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:


						2∙2∙5∙5

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. The zero that appears at the dividend in this digit will be added 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.

	245 − 2
	10 ∙ 99

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,

A fraction is a number that is made up of one or more units. There are three types of fractions in mathematics: common, mixed and decimal.

Common fractions

An ordinary fraction is written as a ratio in which the numerator reflects how many parts are taken from the number, and the denominator shows how many parts the unit is divided into. If the numerator is less than the denominator, then we have a proper fraction. For example: ½, 3/5, 8/9.

If the numerator is equal to or greater than the denominator, then we are dealing with an improper fraction. For example: 5/5, 9/4, 5/2 Dividing the numerator can result in a finite number. For example, 40/8 = 5. Therefore, any whole number can be written as an ordinary improper fraction or a series of such fractions. Let's consider the entries of the same number in the form of a number of different ones.

Mixed fractions

In general, a mixed fraction can be represented by the formula:

Thus, a mixed fraction is written as an integer and an ordinary proper fraction, and such a notation is understood as the sum of the whole and its fractional part.

Decimals

A decimal is a special type of fraction in which the denominator can be represented as a power of 10. There are infinite and finite decimals. When writing this type of fraction, the whole part is first indicated, then the fractional part is recorded through a separator (period or comma).

The notation of a fractional part is always determined by its dimension. The decimal notation looks like this:

Rules for converting between different types of fractions

Converting a mixed fraction to a common fraction

A mixed fraction can only be converted to an improper fraction. To translate, it is necessary to bring the whole part to the same denominator as the fractional part. In general it will look like this:
Let's look at the use of this rule using specific examples:

Converting a common fraction to a mixed fraction

An improper fraction can be converted into a mixed fraction by simple division, resulting in the whole part and the remainder (fractional part).

For example, let's convert the fraction 439/31 to mixed:

Converting fractions

In some cases, converting a fraction to a decimal is quite simple. In this case, the basic property of a fraction is applied: the numerator and denominator are multiplied by the same number in order to bring the divisor to a power of 10.

For example:

In some cases, you may need to find the quotient by dividing by corners or using a calculator. And some fractions cannot be reduced to a final decimal. For example, the fraction 1/3 when divided will never give the final result.

They are used extremely widely, and in a wide variety of areas of human activity, be it scientific and applied computing, the development and operation of various equipment, economic calculations, and so on. Due to various reasons, it is often necessary to carry out decimal conversion, as well as the reverse process. It should be noted that similar transformation are produced relatively easily, and in accordance with certain rules and techniques that have existed in mathematics for many hundreds of years.

Converting a decimal fraction to a prime fraction

Decimal conversion into the “ordinary” fraction it is quite easy and simple. To do this, the following technique is used: the number located to the right of the decimal point of the original number is taken as the numerator of the new fraction; the number ten is used as the denominator, to a power equal to the number of digits of the numerator. As for the remaining whole part, it remains unchanged. If the integer part is equal to zero, then after the transformation it is simply omitted.

EXAMPLE 1

Fifty point twenty five equals fifty point one and twenty five divided by one hundred equals fifty point one fourth.

Converting a fraction to a decimal

Converting a Fraction to a Decimal, in fact, is the inverse converting a decimal fraction into a prime fraction. Its implementation also does not cause any difficulties and is, in fact, a fairly simple arithmetic operation. In order to convert a fraction to a decimal you need to divide the numerator by its denominator in accordance with certain rules.

EXAMPLE 1

Need to implement fraction conversion five eighths in decimal.

Dividing five by eight gives decimal zero point six hundred twenty-five thousandths.

0.625

Rounding the result of converting a fraction to a decimal

It should be noted that, unlike a process such as decimal conversion, this procedure can often last indefinitely. In such cases they say that the result of the procedure converting a fraction to a decimal may not be accurate. However, practice shows that in the vast majority of cases, obtaining a perfectly accurate result is not required. As a rule, the division process ends when it has already obtained the values of those decimal fractions that are of practical interest in each specific case.

EXAMPLE 1

You need to cut a piece of butter weighing one kilogram into nine pieces of equal weight. When performing this procedure, it turns out that the mass of each of them is 1/9 kilogram. If carried out according to all the rules transformation this common fraction V decimal fraction, then it turns out that the mass of each of the resulting parts is equal to zero whole and one in the period of a kilogram.

Rounding is carried out according to the standard rules provided for in arithmetic: if the first of the “discarded” digits has a value of 5 or more, then the last of the significant ones is increased by one. Otherwise it remains unchanged.

EXAMPLE 2

Convert fraction one eighth to a decimal fraction.

When one is divided by eight, the result is zero point one hundred twenty-five thousandths, or rounded - zero point thirteen hundredths.

An improper fraction is one of the formats for writing a common fraction. Like any ordinary fraction, it has a number above the line (numerator) and below it - the denominator. If the numerator is greater than the denominator, this is a hallmark of an incorrect fraction. A mixed fraction can be converted into this form. The decimal can also be represented in the irregular form of notation, but only if the separating point is preceded by a number other than zero.

Instructions

In a mixed fraction format, the numerator and denominator are separated from the whole part by a space. To convert such an entry to , first multiply its integer part (the number before the space) by the denominator of the fractional part. Add the resulting value to the numerator. The value calculated in this way will be the numerator of the improper fraction, and put the denominator of the mixed fraction into its denominator without any changes. For example, 5 7/11 in the ordinary irregular format can be written as follows: (5*11+7)/11 = 62/11.

To convert a decimal fraction into an incorrect ordinary notation, determine the number of digits after the decimal point separating the whole part from the fractional part - it is equal to the number of digits to the right of this decimal point. Use the resulting number as an indicator of the power to which you need to raise ten to calculate the denominator of the improper fraction. The numerator is obtained without any calculations - just remove the comma from the decimal fraction. For example, if the original decimal fraction is 12.585, the numerator of the corresponding irregular fraction should contain the number 10³ = 1000, and the denominator - 12585: 12.585 = 12585/1000.

Like any ordinary fractions, they can and should be reduced. To do this, after obtaining the result using the methods described in the previous two steps, try to select the greatest common divisor for the numerator and denominator. If you can do this, divide by what you found on both sides of the fraction line. For the example from the second step, this divisor will be the number 5, so the improper fraction can be reduced: 12.585 = 12585/1000 = 2517/200. But for the example from the first step there is no common divisor, so there is no need to reduce the resulting improper fraction.

Video on the topic

Decimal fractions are more convenient for automated calculations than natural fractions. Any natural fraction can be converted to natural numbers either without loss of precision or with precision to a specified number of decimal places, depending on the relationship between the numerator and denominator.

Instructions

If necessary, round the result to the required number of decimal places. The rounding rules are as follows: if the highest digit to be deleted contains a digit from 0 to 4, then the next highest digit (which is not deleted) does not change, and if the digit is from 5 to 9, it increases by one. If the last of these operations is subjected to the digit with the number 9, the unit is transferred to another, even more senior digit, like a column. Please note that rounding to the available number of familiar places does not always carry out this operation. Sometimes there are hidden bits in its memory that are not displayed on the indicator. Logarithmic, having low accuracy (up to two decimal places), often handles rounding in the right direction better.

If you find that a certain sequence of numbers is repeated after a decimal point, place that sequence in parentheses. They say about it that it is located "" because it repeats periodically. For example, number 53.7854785478547854... can be written as 53,(7854).

A proper fraction, the value of which is greater than one, consists of two parts: an integer and a fraction. First, divide the numerator of the fraction by its denominator. Then add the result of division to the whole part. After this, if necessary, round the result to the required number of decimal places or find the periodicity and highlight it in brackets.

Decimal fractions are easy to use. They are recognized by calculators and many computer programs. But sometimes it is necessary, for example, to draw up a proportion. To do this, you will have to convert the decimal fraction to a regular fraction. This will not be difficult if you take a short excursion into the school curriculum.

Instructions

Reduce the fractional part of the result. To do this, the numerator and denominator of the fraction must be divided by the same divisor. In this case it is the number "5". So "5/10" is converted to "1/2".

Choose a number so that the result of multiplying it by the denominator is 10. Reason backwards: is it possible to turn the number 4 into 10? Answer: no, because 10 is not divisible by 4. Then 100? Yes, 100 is divided by 4 without a remainder, the result is 25. Multiply the numerator and denominator by 25 and write the answer in decimal form:
¼ = 25/100 = 0.25.

It is not always possible to use the selection method; there are two more ways. Their principle is practically the same, only the recording differs. One of them is the gradual allocation of decimal places. Example: convert the fraction 1/8.