How to convert an integer to a fraction. Converting a decimal fraction to a common fraction and vice versa: rule, examples

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:

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 ordinary fraction. Here are some more examples:

Examples of transition from decimal notation of fractions to ordinary ones

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:

1. 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 $n$.
2. 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.
3. 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:

\[\begin(align)& 2.15\to 0.15=\frac(15)(100)=\frac(3)(20)\to 2\frac(3)(20); \\& 13.8\to 0.8=\frac(8)(10)=\frac(4)(5)\to 13\frac(4)(5). \\\end(align)\]

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”. One way or another, the key word is “thousands”, 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 = 10 3, and 10 = 2 ∙ 5, therefore

\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]

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 "

Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are three types of fractions.

1. Common fractions , For example:

Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need to write down the answers to tasks “B”.

3. Mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted into ordinary fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? And how! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where a typical mistake, a blunder, if you will, lurks.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the two on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then in a hurry you can cross out the “a” in the expression

and get it again

Which would be categorically untrue. Because here all the numerator on "a" is already not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's all.

However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction does not translate. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this is useful information for self-testing. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It is not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's all. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.

Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do it? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always available.

3. The choice of the type of fractions to work with a task depends on the task itself. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

Let's wrap this up. In this lesson we refreshed our memory on key points about fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

In dry mathematical language, a fraction is a number that is represented as a part of one. Fractions are widely used in human life: we use fractions to indicate proportions in culinary recipes, give decimal scores 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, the 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 back, 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 simple 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 write the decimal number as a fraction “number/1”, that is, we get 0.5/1; 0.25/1 and 1.375/1.

Step 2: After this, multiply the numerator and denominator of the 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 × 5 / 2 × 5 = 1/2;
• 25/100 = 1 × 25 / 4 × 25 = 1/4;
• 1375/1000 = 11 × 125 / 8 × 125 = 11/8.

The number 1.375 had to be multiplied by 10 three times, which is no longer very convenient, but what do we have to do if we need to convert the number 0.000625? In this situation, we use the following method of converting fractions.

Getting rid of commas even easier

The first method describes in detail the algorithm for “removing” a comma from a decimal, but we can simplify this process. Again, we follow three steps.

Step 1: We count 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 quantity by the letter n.

Step 2: Now we just need 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 = 1375, n = 3, the final fraction according to the formula 1375/10 3 = 1375/1000;
• for the number 0.000625 C = 625, n = 6, the final fraction according to the formula 625/10 6 = 625/1000000.

Essentially, 10n is a 1 with n zeros, so you don't have to bother raising the ten to the power - just 1 with n zeros. After this, it is advisable to reduce a fraction so rich in zeros.

Step 3: We reduce the zeros and get the final result:

• 1375/1000 = 11 × 125 / 8 × 125 = 11/8;
• 625/1000000 = 1 × 625/ 1600 × 625 = 1/1600.

The fraction 11/8 is an improper fraction because its numerator is greater than its denominator, which means we can isolate 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.

Conversion by ear

For those who can read decimals correctly, the easiest way to convert them is by hearing. If you read 0.025 not as “zero, zero, twenty-five” but as “25 thousandths,” then you will have no problem converting decimals to fractions.

0,025 = 25/1000 = 1/40

Thus, reading a decimal number correctly allows you to immediately write it down as a fraction and reduce it if necessary.

Examples of using fractions in everyday life

At first glance, ordinary fractions are practically not used in everyday life or at work, and it is difficult to imagine a situation when you need to convert a decimal fraction into a regular fraction outside of school tasks. Let's look at a couple of examples.

Job

So, you work in a candy store and sell halva by weight. To make the product easier to sell, you divide the halva into kilogram briquettes, but few buyers are willing to purchase a whole kilogram. Therefore, you have to divide the treat into pieces each time. And if the next buyer asks you for 0.4 kg of halva, you will sell him the required portion without any problems.

0,4 = 4/10 = 2/5

Life

For example, you need to make a 12% solution to paint the model in the shade you want. To do this, you need to mix paint and solvent, but how to do it correctly? 12% is a decimal fraction of 0.12. Convert the number to a common fraction and get:

0,12 = 12/100 = 3/25

Knowing the fractions will help you mix the ingredients correctly and get the color you want.

Conclusion

Fractions are commonly used in everyday life, so if you frequently need to convert decimals to fractions, you'll want to use an online calculator that can instantly get the result as a reduced fraction.

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.

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 point five hundredths, we get a fraction. Three point seventeen thousandths, we get the fraction. The numbers before the decimal point are the whole part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a single-digit number after the decimal point, the denominator will be 10, if there is a two-digit number - 100, a three-digit number - 1000, etc. Some resulting fractions can be reduced. In our examples

Converting a fraction to a decimal

This is the reverse of the previous transformation. What is the characteristic of a decimal fraction? Its denominator is always 10, or 100, or 1000, or 10000, and so on. If your common fraction has a denominator like this, there's no problem. For example, or

If the fraction is, for example . In this case, it is necessary to use the basic property of a 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 decimals!
For example,

Converting a mixed fraction to an improper fraction

A mixed fraction, for example, can be easily converted to an improper fraction. To do this, you need to multiply the whole part by the denominator (bottom) and add it with the numerator (top), leaving the denominator (bottom) unchanged. That is

When converting a mixed fraction to an improper fraction, you can remember that you can use fraction addition

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

An improper fraction can be converted to a mixed fraction by highlighting the whole part. Let's look at an example. We determine how many integer times “3” fits into “23”. Or divide 23 by 3 on a calculator, the whole number 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”. It’s as if we find the extra that remains from the numerator “23” if we remove the maximum amount of “3”. We leave the denominator unchanged. Everything is done, write down the result