Convert the given number to a decimal. Ordinary and decimal fractions and operations on them
A sufficient number of people are wondering how to convert an ordinary fraction to a decimal fraction. There are several ways. The choice of a specific method depends on the type of fraction that needs to be converted to another form, or rather, on the number in its denominator. However, for reliability, it is necessary to indicate that an ordinary fraction is a fraction that is written with a numerator and a denominator, for example, 1/2. More often, the line between the numerator and denominator is drawn horizontally rather than obliquely. The decimal fraction is written as an ordinary number with a comma: for example, 1.25; 0.35 etc.
So, in order to convert an ordinary fraction to a decimal without a calculator, you need:
Pay attention to the denominator of an ordinary fraction. If the denominator can be easily multiplied up to 10 by the same number as the numerator, then this method should be used, as the simplest. For example, the ordinary fraction 1/2 is easily multiplied in the numerator and denominator by 5, resulting in the number 5/10, which can already be written as a decimal fraction: 0.5. This rule is based on the fact that a decimal fraction always has a round number in the denominator: 10, 100, 1000 and the like. Therefore, if you multiply the numerator and denominator of a fraction, then it is necessary to achieve exactly such a number in the denominator as a result of multiplication, regardless of what is obtained in the numerator.
There are ordinary fractions, the calculation of which after multiplication presents certain difficulties. For example, it is quite difficult to determine by how much the fraction 5/16 should be multiplied to get one of the numbers above in the denominator. In this case, you should use the usual division, which is performed by a column. The answer should be a decimal fraction, which will mark the end of the transfer operation. In the above example, the result is a number equal to 0.3125. If calculations in a column present difficulties, then you can’t do without the help of a calculator.
Finally, there are ordinary fractions that are not converted to decimals. For example, when translating the common fraction 4/3, the result is 1.33333, where the three is repeated ad infinitum. The calculator will also not get rid of the repeating three. There are several such fractions, you just need to know them. The way out of the above situation can be rounding, if the conditions of the example or problem being solved allow rounding. If the conditions do not allow this, and the answer must be written exactly in the form of a decimal fraction, then the example or problem was solved incorrectly, and you should go back several steps to find the error.
Thus, converting an ordinary fraction to a decimal is quite easy, it is not difficult to cope with this task without the help of a calculator. It looks even easier to translate decimal fractions into ordinary ones by performing the reverse steps described in method 1.
Video: 6th grade. Converting an ordinary fraction to a decimal fraction.
In dry mathematical terms, a fraction is a number that is represented as a fraction of a unit. Fractions are widely used in human life: with the help of fractional numbers, we indicate proportions in culinary recipes, set decimal marks 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, 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 vice versa, 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 easy 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 will write the decimal number as a fraction “number / 1”, that is, we will get 0.5 / 1; 0.25/1 and 1.375/1.
Step 2: After that, multiply the numerator and denominator of 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 x 5 / 2 x 5 = 1/2;
- 25/100 = 1 x 25 / 4 x 25 = 1/4;
- 1375/1000 = 11 x 125 / 8 x 125 = 11/8.
The number 1.375 had to be multiplied by 10 three times, which is no longer very convenient, but what will we have to do if we need to convert the number 0.000625? In this situation, we use the following method for converting fractions.
Getting rid of the comma is even easier
The first method describes in detail the algorithm for "removing" a comma from a decimal fraction, however, we can simplify this process. Again, we follow three steps.
Step 1: We consider 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 number by the letter n.
Step 2: Now it is enough for us 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 \u003d 1375, n \u003d 3, the final fraction according to the formula 1375/10 3 \u003d 1375/1000;
- for the number 0.000625 C \u003d 625, n \u003d 6, the final fraction according to the formula 625/10 6 \u003d 625/1000000.
Essentially, 10 n is 1 with n zeros, so you don't have to worry about raising the tens to a power - just specify 1 with n zeros. After that, it is desirable to reduce the fraction so rich in zeros.
Step 3: Reduce the zeros and get the final result:
- 1375/1000 = 11 x 125 / 8 x 125 = 11/8;
- 625/1000000 = 1 x 625/ 1600 x 625 = 1/1600.
The fraction 11/8 is an improper fraction, since its numerator is greater than the denominator, which means that we can select 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.
Transformation by ear
For those who know how to read decimals correctly, it is easiest to convert them by ear. If you read 0.025 not as "zero, zero, twenty-five", but as "25 thousandths", then you will have no problem converting decimal numbers to common fractions.
0,025 = 25/1000 = 1/40
Thus, the correct reading of the decimal number allows you to immediately write it as an ordinary fraction and reduce it if necessary.
Examples of using fractions in everyday life
At first glance, common fractions are practically not used in everyday life or at work, and it is difficult to imagine a situation where you need to convert a decimal fraction to a common one outside of school problems. Let's look at a couple of examples.
Job
So, you work in a candy store and sell halva by weight. For ease of sale of the product, you divide halva into kilogram briquettes, but few buyers are ready to purchase a whole kilogram. Therefore, you have to divide the treat into pieces every time. And if another buyer asks you for 0.4 kg of halva, you will sell him the right portion without any problems.
0,4 = 4/10 = 2/5
Life
For example, you need to make a 12% solution for painting the model in the shade you need. To do this, you need to mix paint and thinner, but how to do it right? 12% is a decimal fraction of 0.12. We convert the number to an ordinary fraction and get:
0,12 = 12/100 = 3/25
Knowing the fractions, you can mix the components correctly and get the right color.
Conclusion
Fractions are widely used in everyday life, so if you often need to convert decimals to fractions, you will need an online calculator that can instantly get the result in the form of an already abbreviated fraction.
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, you will get an integer value or a decimal fraction with an integer fractional part. 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.
Already in elementary school, students are faced with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of whole 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 slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.
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 consisting of parts of one. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one on the bottom (right) is the denominator.
In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.
What are the fractions?
In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren get acquainted with the first ones in the elementary grades, calling them simply “fractions”. The second learn in the 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear 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 as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.
What subspecies do these types of fractions have?
It is 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 the denominator.
Wrong. Its numerator is greater than or equal to the denominator.
Reducible / irreducible. It can be either right or wrong. Another thing is important, whether the numerator and denominator have common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.
Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.
Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.
Decimals have only two subspecies:
final, that is, one in which the 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 decimal to ordinary?
If this is a finite number, then an association based on the rule is applied - 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 line.
As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.
How to convert decimal fractions to ordinary ones if their whole 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. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.
How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second - 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The answer is mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal to a common fraction?
If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.
The only thing that is allowed to be done with such a fraction is to 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 to ordinary fractions. This must be remembered.
How to write an infinite periodic fraction in the form of an ordinary?
In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as 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 any numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.
For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.
A rule on how to write a common decimal fraction that is a mixed fraction.
Look at the length of the period. So much 9 will have a denominator.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - 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 is one digit. So zero will be one. There is also only one digit 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 from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.
How are common fractions converted to decimals?
The simplest option is a number whose denominator is the number 10, 100, and so on. 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. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.
Operations with common fractions
Addition and subtraction
Students get to know them earlier than others. And at first the fractions have the same denominators, and then different. General rules can be reduced to such a plan.
Find the least common multiple of the denominators.
Write additional factors to all ordinary fractions.
Multiply the numerators and denominators by the factors defined for them.
Add (subtract) the numerators of fractions, and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then you need to find out whether we have a mixed number or a proper fraction.
In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.
In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
For their implementation, fractions do not need to be reduced to a common denominator. This makes it easier to take action. But they still have to follow the rules.
When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
Multiply numerators.
Multiply the denominators.
If you get a reducible fraction, then it is supposed to be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with a reciprocal (swap the numerator and denominator).
Then proceed as in multiplication (starting from step 1).
In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written as an improper fraction. That is, with a denominator of 1. Then proceed as described above.
Operations with decimals
Addition and subtraction
Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. 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. Assign the missing number of zeros in it.
Write fractions so that the comma is under the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to append zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.
For multiplication, you need to write fractions one under the other, not paying attention to commas.
Multiply like natural numbers.
Put 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 convert 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 a natural number.
Put a comma in the answer at the moment when the division of the whole part ends.
What if there are both types of fractions in one example?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.
First way: represent ordinary decimals
It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.
The second way: write decimal fractions as ordinary
This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, a very large ordinary fraction can turn out and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.
They are used extremely widely, and in various fields of human activity, be it scientific and applied calculations, the development and operation of various equipment, economic calculation, and so on. Due to various reasons, it is often necessary to carry out decimal inversion, as well as the process inverse to it. It should be noted that such transformations are produced relatively easily, and in accordance with certain rules and methods that have existed in mathematics for many hundreds of years.
Converting a decimal to a simple fraction
Decimal conversion into fraction "ordinary" is made quite easily and simply. For this, the following technique is used: the number that is 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 degree 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 1Fifty point twenty five hundredths equals fifty point 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 to a simple. Its implementation also does not cause any difficulties and is, in fact, a fairly simple arithmetic operation. In order to convert simple fraction to decimal you need to divide the numerator by its denominator in accordance with certain rules.
EXAMPLE 1Need to implement fraction conversion five eighths 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, in contrast to such a process as decimal conversion, this procedure can often last indefinitely. In such cases, it is said 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 the values of those decimal parts that are of practical interest in each specific case have already been obtained in its course.
EXAMPLE 1It is required to cut a piece of butter weighing one kilogram into nine parts of the same mass. When performing this procedure, it turns out that the mass of each of them is 1/9 of a kilogram. If according to all the rules to carry out transformation this ordinary fraction V decimal fraction, it turns out that the mass of each of the resulting parts is equal to zero integers 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 2Convert common fraction one eighth to a decimal.
When dividing one by eight, you get zero point one hundred and twenty-five thousandths, or rounded up - zero point thirteen hundredths.
