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. All this - decimals. 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 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
- We break the fraction into whole and fractional parts.
- Let's see how many digits are after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
- We write the fraction in ordinary form.
- Reduce the numerator and denominator of the fraction.
- We write down the resulting fraction.
- We check and divide top part fractions to the bottom. 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.
Author on Youtube: Anastasia Ivanova
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Convert decimal to normal
Every decimal fraction can be represented as a regular fraction. Just write using the denominator to do this.
The basic rule for converting a decimal to a regular fraction is to read the decimal, but it is usually written. For example:
2,3 - two points out of three tens
Since the fraction is complete, it can be converted to a mixed number or irregular fraction:
Converting a correct fraction to a decimal
A non-traditional fraction can be converted to a decimal, just as for conventional decimal notation, the denominator must be followed by one or more zeros, such as 10, 100, 1000, and so on.
How to convert total fraction to decimal
If we expand such a denominator with the primary factors, we get the same number of doublings and five:
100 = 10 10 = 2 5 2.5
1000 = 10 10 10 = 2 5 2 5 2 5
There are no other prime factors, so these extensions do not contain, so:
A regular fraction can be represented as a decimal only if its denominator contains no factors other than 2 and 5.
Let's take part:
When the denominator is extended to the main factors, the result is a product of 2 2:
If you multiply it by two fours, equate the number five to two, you will get one of the required denominators - 100.
To get a passage equal to this, the counter must be multiplied by the product of two five:
Let's look at another faction:
When the denominator is extended to the main factors, the product is 2.7, containing the number 7:
A factor of 7 will be present in the denominator to multiply it or the integers, so that a product containing only two and five will never occur.
Therefore, this fraction cannot be reduced to any of the necessary denominators: 10, 100, 1000, etc. This means that it cannot be represented as a decimal number.
A regular incompatible fraction cannot be represented as a decimal if its denominator contains at least one major factor from one to two.
Note that the rule only talks about irreversible fractions, since some fractions can be represented as decimal abbreviations.
Let's look at two parts:
Now all that's left is to multiply as phrasal fractions by 5 to get 10 in the denominator, and you can convert the fraction to a decimal:
How to convert a decimal fraction to a common fraction
It would seem that converting a decimal fraction into a regular fraction is an elementary topic, but many students do not understand it!
Therefore, today we will take a detailed look at several algorithms at once, with the help of which you will understand any fractions in just a second.
Let me remind you that there are at least two forms of writing the same fraction: common and decimal.
Decimal fractions are all kinds of constructions of the form 0.75; 1.33; and even −7.41. Here are examples of ordinary fractions that express the same numbers:
Now let's figure it out: how to move from decimal notation to regular notation?
And most importantly: how to do this as quickly as possible?
Basic algorithm
In fact, there are at least two algorithms. And we'll look at both now. Let's start with the first one - the simplest and most understandable.
To convert a decimal to a fraction, you need to follow three steps:
- Rewrite the original fraction as a new fraction: the original decimal fraction will remain in the numerator, and you need to put one in the denominator. In this case, the sign of the original number is also placed in the numerator.
For example:
- Multiply the numerator and denominator of the resulting fraction by 10 until the decimal point disappears from the numerator. Let me remind you: for each multiplication by 10, the decimal point is shifted to the right by one place. Of course, since the denominator is also multiplied, instead of the number 1 there will appear 10, 100, etc.
- Finally, we reduce the resulting fraction by standard scheme: divide the numerator and denominator by the numbers to which they are multiples. For example, in the first example 0.75=75/100, and both 75 and 100 are divisible by 25.
Therefore, we get $0.75=\frac(75)(100)=\frac(3\cdot 25)(4\cdot 25)=\frac(3)(4)$ - that’s the whole answer. :)
An important note about negative numbers. If in the original example there is a minus sign in front of the decimal fraction, then in the output there should also be a minus sign in front of the common fraction.
Converting a fraction to a decimal
Here are some more examples:
I would like to pay special attention to the last example. As you can see, the fraction 0.0025 contains many zeros after the decimal point. Because of this, you have to multiply the numerator and denominator by 10 as many as four times. Is it possible to somehow simplify the algorithm in this case?
Of course you can. And now we will look at an alternative algorithm - it is a little more difficult to understand, but after a little practice it works much faster than the standard one.
Faster way
This algorithm also has 3 steps.
To get a fraction from a decimal, do the following:
- Count how many digits are after the decimal point. For example, the fraction 1.75 has two such digits, and 0.0025 has four. Let's denote this quantity by the letter $n$.
- Rewrite the original number as a fraction of the form $\frac(a)(((10)^(n)))$, where $a$ are all the digits of the original fraction (without the “starting” zeros on the left, if any), and $n$ is the same number of digits after the decimal point that we calculated in the first step.
In other words, you need to divide the digits of the original fraction by one followed by $n$ zeros.
- If possible, reduce the resulting fraction.
That's all! At first glance, this scheme is more complicated than the previous one. But in fact it is both simpler and faster. Judge for yourself:
As you can see, in the fraction 0.64 there are two digits after the decimal point - 6 and 4.
Therefore $n=2$. If we remove the comma and zeros on the left (in this case, just one zero), we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, Therefore, the denominator is exactly one hundred. Well, then all that remains is to reduce the numerator and denominator. :)
One more example:
Here everything is a little more complicated.
Firstly, there are already 3 numbers after the decimal point, i.e. $n=3$, so you have to divide by $((10)^(n))=((10)^(3))=1000$. Secondly, if we remove the comma from the decimal notation, we get this: 0.004 → 0004. Remember that the zeros on the left must be removed, so in fact we have the number 4. Then everything is simple: divide, reduce and get the answer.
Finally, the last example:
The peculiarity of this fraction is the presence of a whole part.
Therefore, the output we get is an improper fraction of 47/25. You can, of course, try to divide 47 by 25 with a remainder and thus again isolate the whole part.
But why complicate your life if this can be done at the stage of transformation? Well, let's figure it out.
What to do with the whole part
In fact, everything is very simple: if we want to get a proper fraction, then we need to remove the whole part from it during the transformation, and then, when we get the result, add it again to the right before the fraction line.
For example, consider the same number: 1.88. Let's score by one (the whole part) and look at the fraction 0.88.
It can be easily converted:
Then we remember about the “lost” unit and add it to the front:
\[\frac(22)(25)\to 1\frac(22)(25)\]
That's all! The answer turned out to be the same as after selecting the whole part last time. A couple more examples:
\[\begin(align)& 2.15\to 0.15=\frac(15)(100)=\frac(3)(20)\to 2\frac(3)(20); \\& 13.8\to 0.8=\frac(8)(10)=\frac(4)(5)\to 13\frac(4)(5).
This is the beauty of mathematics: no matter which way you go, if all the calculations are done correctly, the answer will always be the same. :)
In conclusion, I would like to consider one more technique that helps many.
Transformations “by ear”
Let's think about what a decimal even is.
More precisely, how we read it. For example, the number 0.64 - we read it as "zero point 64 hundredths", right? Well, or just “64 hundredths”. The key word here is “hundredths”, i.e. number 100.
What about 0.004? This is “zero point 4 thousandths” or simply “four thousandths”.
Anyway, keyword- “thousandths”, i.e. 1000.
So what's the big deal? And the fact is that it is these numbers that ultimately “pop up” in the denominators at the second stage of the algorithm. Those. 0.004 is "four thousandths" or "4 divided by 1000":
Try to practice yourself - it's very simple. The main thing is to read the original fraction correctly. For example, 2.5 is "2 whole, 5 tenths", so
And some 1.125 is “1 whole, 125 thousandths”, so
In the last example, of course, someone will object that it is not obvious to every student that 1000 is divisible by 125.
But here you need to remember that 1000 = 103, and 10 = 2 ∙ 5, so
\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]
Thus, any power of ten is decomposed only into factors 2 and 5 - it is these factors that need to be looked for in the numerator, so that in the end everything is reduced.
This concludes the lesson.
Let's move on to a more complex reverse operation - see "Transition from an ordinary fraction to a decimal."
If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).
The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or completely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.
The remainder is always less than the divisor.
Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.
Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.
The quotient of natural numbers can be written as a fraction.
The numerator of a fraction is the dividend, and the denominator is the divisor.
Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.
The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n) \), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n)\)
The following rules are true:
To get the fraction \(\frac(m)(n)\), you need to divide the unit into n equal parts (shares) and take m such parts.
To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.
To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.
To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.
If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)
If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.
The last two transformations are called reducing a fraction.
If fractions need to be represented as fractions with the same denominator, then this action is called bringing fractions to a common denominator.
Proper and improper fractions. Mixed numbers
You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense dictates that the part should always be less than the whole, but what about fractions such as \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.
As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.
If a number consists of an integer part and a fraction, then such fractions are called mixed.
For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.
If the numerator of the fraction \(\frac(a)(b)\) is divisible by natural number n, then to divide this fraction by n, you need to divide its numerator by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)
If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)
Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.
Actions with fractions. Adding fractions.
You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)
To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.
Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)
If you need to add fractions with different denominators, they must first be reduced to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)
For fractions, as for natural numbers, the commutative and associative properties of addition are valid.
Adding mixed fractions
Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”
When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)
Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.
Subtracting fractions (fractional numbers)
Subtraction of fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)
The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.
Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)
Multiplying fractions
To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.
Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)
Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, a mixed fraction - as an improper fraction.
The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.
For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.
Division of fractions
Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).
If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.
For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).
Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)
It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)
Using reciprocal fractions, you can reduce division of fractions to multiplication.
The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.
Using letters, the rule for dividing fractions can be written as follows:
\(\large \frac(a)(b) : \frac(c)(d) = \frac(a)(b) \cdot \frac(d)(c) \)
If the dividend or divisor is a natural number or a mixed fraction, then in order to use the rule for dividing fractions, it must first be represented as an improper fraction.
A large number of students, and not only, are wondering how to convert a fraction into a number. To do this, there are several fairly simple and understandable ways. The choice of a specific method depends on the preferences of the decider.
First of all, you need to know how fractions are written. And they are written as follows:
- Ordinary. It is written with the numerator and denominator using a slant or a column (1/2).
- Decimal. It is written separated by commas (1.0, 2.5, and so on).
Before you start solving, you need to know what an improper fraction is, because it occurs quite often. It has a numerator greater than the denominator, for example, 15/6. Improper fractions can also be solved in these ways, without any effort or time.
A mixed number is when the result is a whole number and a fractional part, for example 52/3.
Any natural number can be written as a fraction with completely different natural denominators, for example: 1= 2/2=3/3 = etc.
You can also translate using a calculator, but not all of them have this function. There is a special engineering calculator that has such a function, but it is not always possible to use it, especially at school. Therefore, it is better to understand this topic.
The first thing you should pay attention to is what fraction it is. If it can be easily multiplied up to 10 by the same values as the numerator, then you can use the first method. For example: you multiply an ordinary ½ in the numerator and denominator by 5 and get 5/10, which can be written as 0.5.
This rule is based on the fact that a decimal always has a round value in its denominator, such as 10,100,1000, and so on.
It follows from this that if you multiply the numerator and the denominator, then you need to achieve exactly the same value in the denominator as a result of the multiplication, regardless of what comes out in the numerator.
It is worth remembering that some fractions cannot be converted; to do this, you need to check it before starting the solution.
For example: 1.3333, where the number 3 is repeated ad infinitum, and the calculator will not get rid of it either. The only solution to this problem is to round it to a whole number, if possible. If this is not possible, then you should return to the beginning of the example and check the correctness of the solution to the problem; perhaps an error was made.
Figure 1-3. Converting fractions by multiplication.
To consolidate the information described, let us consider next example translation:
- For example, you need to convert 6/20 to a decimal. The first step is to check it, as shown in Figure 1.
- Only after you are convinced that it can be decomposed, as in this case into 2 and 5, should you begin the translation itself.
- Most simple option will multiply the denominator, resulting in a result of 100, which is 5, since 20x5=100.
- Following the example in Figure 2, the result will be 0.3.
You can consolidate the result and review everything again according to Figure 3. In order to fully understand the topic and no longer resort to studying this material. This knowledge will help not only the child, but also the adult.
Translation by division
The second option for converting fractions is a little more complicated, but more popular. This method is mainly used by teachers in schools to explain. Overall, it is much easier to explain and quicker to understand.
It is worth remembering that to correctly convert a simple fraction, you must divide its numerator by its denominator. After all, if you think about it, the solution is the process of division.
In order to understand this simple rule, you need to consider the following example solution:
- Let's take 78/200, which needs to be converted to decimal. To do this, divide 78 by 200, that is, the numerator by the denominator.
- But before you start, it's worth checking, as shown in Figure 4.
- Once you are convinced that it can be solved, you should begin the process. To do this, it is worth dividing the numerator by the denominator in a column or corner, as shown in Figure 5. In primary schools, such division is taught, and there should be no difficulties with this.
Figure 6 shows examples of the most common examples; you can simply remember them so that, if necessary, you do not waste time solving them. After all, at school, for every test or independent work Little time is given to solve, so you shouldn’t waste it on something that you can learn and simply remember.
Interest transfer
Convert interest to decimal number also quite easy. This begins to be taught in the 5th grade, and in some schools even earlier. But if your child did not understand this topic during a math lesson, you can clearly explain it to him again. First, you should learn the definition of what a percentage is.
A percentage is one hundredth of a number; in other words, it is completely arbitrary. For example, from 100 it will be 1 and so on.
Figure 7 shows clear example transfer of interest.
To convert a percentage, you just need to remove the % sign and then divide it by 100.
Another example is shown in Figure 8.
If you need to carry out a reverse “conversion”, you need to do everything exactly the opposite. In other words, the number must be multiplied by one hundred and then a percentage symbol must be added.
And in order to convert the usual into percentages, you can also use this example. Only initially should you convert the fraction into a number and only then into a percentage.
Based on the above, you can easily understand the principle of translation. Using these methods, you can explain a topic to a child if he did not understand it or was not present in the lesson at the time of its completion.
And there will never be a need to hire a tutor to explain to your child how to convert a fraction into a number or percentage.
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.
Yandex.RTB R-A-339285-1
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
- Write down 0 and put a comma after it.
- 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
- Write down the number that is in the numerator.
- 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
- We prepare the fractional part of the number, if necessary.
- We write down the whole part of the original number and put a comma after it.
- 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.
Fundamentally new way converting an ordinary fraction to a decimal is reduced to dividing 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 see 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:
- A common fraction can be reduced to a final decimal if its denominator can be factored into prime factors of 2 and 5.
- 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:
- We get a remainder of 0, and this is where the division ends.
- 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
- In the numerator we write the number from the original decimal fraction, discarding the comma and all zeros on the left, if any.
- In the denominator we write one followed by as many zeros as there are digits after the decimal point in the original decimal fraction.
- 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.
- We write the decimal fraction itself into the numerator, discarding the comma: 3025.
- 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.
- 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.
- In the numerator we write the fraction 0, 0017, discarding the comma and zeros on the left. It will turn out to be 17.
- 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 it?
Let's formulate one more rule.
Rule for converting decimals to mixed numbers.
- The number before the decimal point in the fraction is written as the integer part of the mixed number.
- In the numerator we write the number after the decimal point in the fraction, discarding the zeros on the left if there are any.
- 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.
- We write the number 155 as an integer part.
- In the numerator we write the numbers after the decimal point, discarding the zero.
- 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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