First level
Converting Expressions. Detailed theory (2019)
Converting Expressions
We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:
“It’s much simpler,” we say, but such an answer usually doesn’t work.
Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).
But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.
Have you read it? If yes, then you are now ready.
Basic simplification operations
Now let's look at the basic techniques that are used to simplify expressions.
The simplest one is
1. Bringing similar
What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in the sum, similar terms are and.
Do you remember?
To bring similar means to add several similar terms to each other and get one term.
How can we put the letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .
Now try this expression: .
To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.
So, the rule for bringing similar ones is:
Examples:
Give similar ones:
Answers:
2. (and similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most an important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.
You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):
Solutions:
3. Reducing a fraction.
Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?
That's the beauty of downsizing.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).
To reduce a fraction you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be crossed out.
The principle, I think, is clear?
I would like to draw your attention to one typical mistake when abbreviating. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.
No abbreviations if the numerator or denominator is a sum.
For example: we need to simplify.
Some people do this: which is absolutely wrong.
Another example: reduce.
The “smartest” will do this: .
Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.
Here's another example: .
This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:
You can immediately divide it into:
To avoid such mistakes, remember easy way how to determine whether an expression is factorized:
The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).
To consolidate, solve a few yourself examples:
Answers:
1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be factorization:
4. Adding and subtracting fractions. Reducing fractions to a common denominator.
Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:
Answers:
1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:
It's a completely different matter if the fractions contain letters, for example:
Let's start with something simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:
Now in the numerator you can give similar ones, if any, and factor them:
Try it yourself:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
· first of all, we determine the common factors;
· then we write out all the common factors one at a time;
· and multiply them by all other non-common factors.
To determine the common factors of the denominators, we first factor them into prime factors:
Let us emphasize the common factors:
Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
· factor the denominators;
· determine common (identical) factors;
· write out all common factors once;
· multiply them by all other non-common factors.
So, in order:
1) factor the denominators:
2) determine common (identical) factors:
3) write out all the common factors once and multiply them by all other (non-underlined) factors:
So there's a common denominator here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to a degree
to a degree
to a degree
to a degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?
So, another unshakable rule:
When you reduce fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply by to get?
So multiply by. And multiply by:
We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.
What about the expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic “”).
So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.
We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).
The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:
Great! Then:
Another example:
Solution:
As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:
So let's write:
That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now let's bring it to a common denominator:
Got it? Let's check it now.
Tasks for independent solution:
Answers:
Here we need to remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. Incomplete square the sum is one of the factors in the expansion of the difference of cubes:
What to do if there are already three fractions?
Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:
Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.
We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:
Hmm... It’s clear what to do with fractions. But what about the two?
It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Exactly what is needed!
5. Multiplication and division of fractions.
Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:
Did you count?
It should work.
So, let me remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the expression in brackets is evaluated out of turn!
If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.
What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But this is not the same as an expression with letters?
No, it's the same! Only instead of arithmetic operations you need to do algebraic ones, that is, the actions described in previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.
Usually our goal is to represent the expression as a product or quotient.
For example:
Let's simplify the expression.
1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).
2) We get:
Multiplying fractions: what could be simpler.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:
Now I’ll show you the process, tinting the current action in red:
Finally, I will give you two useful tips:
1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.
2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And what was promised at the very beginning:
Solutions (brief):
If you have coped with at least the first three examples, then you have mastered the topic.
Now on to learning!
CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS
Basic simplification operations:
- Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
- Factorization: putting the common factor out of brackets, applying it, etc.
- Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Adding and subtracting fractions:
; - Multiplying and dividing fractions:
;
To understand how to reduce fractions, let's first look at an example.
To reduce a fraction means to divide the numerator and denominator by the same thing. Both 360 and 420 end in a digit, so we can reduce this fraction by 2. In the new fraction, both 180 and 210 are also divisible by 2, so we reduce this fraction by 2. In the numbers 90 and 105, the sum of the digits is divisible by 3, so both these numbers are divisible by 3, we reduce the fraction by 3. In the new fraction, 30 and 35 end in 0 and 5, which means both numbers are divisible by 5, so we reduce the fraction by 5. The resulting fraction of six-sevenths is irreducible. This is the final answer.
We can arrive at the same answer in a different way.
Both 360 and 420 end in zero, which means they are divisible by 10. We reduce the fraction by 10. In the new fraction, both the numerator 36 and the denominator 42 are divided by 2. We reduce the fraction by 2. In the next fraction, both the numerator 18 and the denominator 21 are divided by 3, which means we reduce the fraction by 3. We came to the result - six sevenths.
And one more solution.
Next time we'll look at examples of reducing fractions.
Let's understand what reducing fractions is, why and how to reduce fractions, and give the rule for reducing fractions and examples of its use.
Yandex.RTB R-A-339285-1
What is "reducing fractions"
Reduce fractionTo reduce a fraction is to divide its numerator and denominator by a common factor that is positive and different from one.
As a result of this action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.
For example, let's take the common fraction 6 24 and reduce it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12. In this example, we reduced the original fraction by 2.
Reducing fractions to irreducible form
In the previous example, we reduced the fraction 6 24 by 2, resulting in the fraction 3 12. It is easy to see that this fraction can be further reduced. Typically, the goal of reducing fractions is to end up with an irreducible fraction. How to reduce a fraction to its irreducible form?
This can be done by reducing the numerator and denominator by their greatest common factor (GCD). Then, by the property of the greatest common divisor, the numerator and denominator will have mutually prime numbers, and the fraction will be irreducible.
a b = a ÷ N O D (a , b) b ÷ N O D (a , b)
Reducing a fraction to an irreducible form
To reduce a fraction to its irreducible form, you need to divide its numerator and denominator by their gcd.
Let's return to the fraction 6 24 from the first example and bring it to its irreducible form. The greatest common divisor of the numbers 6 and 24 is 6. Let's reduce the fraction:
6 24 = 6 ÷ 6 24 ÷ 6 = 1 4
Reducing fractions is convenient to use so as not to work with in big numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. Reducing a fraction most often means reducing it to an irreducible form, and not simply reducing it by the common divisor of the numerator and denominator.
Rule for reducing fractions
To reduce fractions, just remember the rule, which consists of two steps.
Rule for reducing fractions
To reduce a fraction you need:
- Find the gcd of the numerator and denominator.
- Divide the numerator and denominator by their gcd.
Let's look at practical examples.
Example 1. Let's reduce the fraction.
Given the fraction 182 195. Let's shorten it.
Let's find the gcd of the numerator and denominator. To do this in in this case It is most convenient to use the Euclidean algorithm.
195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13
Divide the numerator and denominator by 13. We get:
182 195 = 182 ÷ 13 195 ÷ 13 = 14 15
Ready. We have obtained an irreducible fraction that is equal to the original fraction.
How else can you reduce fractions? In some cases, it is convenient to factor the numerator and denominator into prime factors, and then remove all common factors from the upper and lower parts of the fraction.
Example 2. Reduce the fraction
Given the fraction 360 2940. Let's shorten it.
To do this, imagine the original fraction in the form:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7
Let's get rid of the common factors in the numerator and denominator, resulting in:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49
Finally, let's look at another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, in each of which the fraction is reduced by some obvious common factor.
Example 3. Reduce the fraction
Let's reduce the fraction 2000 4400.
It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:
2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44
20 44 = 20 ÷ 2 44 ÷ 2 = 10 22
We reduce the resulting result again by 2 and obtain an irreducible fraction:
10 22 = 10 ÷ 2 22 ÷ 2 = 5 11
If you notice an error in the text, please highlight it and press Ctrl+Enter
Online calculator performs reduction of algebraic fractions in accordance with the rule of reducing fractions: replacing the original fraction with an equal fraction, but with a smaller numerator and denominator, i.e. Simultaneously dividing the numerator and denominator of a fraction by their common greatest common factor (GCD). The calculator also displays a detailed solution that will help you understand the sequence of the reduction.
Given:
Solution:
Performing fraction reduction
checking the possibility of performing algebraic fraction reduction
1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction
determining the greatest common divisor (GCD) of the numerator and denominator of an algebraic fraction
2) Reducing the numerator and denominator of a fraction
reducing the numerator and denominator of an algebraic fraction
3) Selecting the whole part of a fraction
separating the whole part of an algebraic fraction
4) Converting an algebraic fraction to a decimal fraction
converting an algebraic fraction to a decimal
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I. Procedure for reducing an algebraic fraction using an online calculator:
- To reduce an algebraic fraction, enter the values of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the whole part of the fraction. If the fraction is simple, then leave the whole part field blank.
- To specify a negative fraction, place a minus sign on the whole part of the fraction.
- Depending on the specified algebraic fraction, the following sequence of actions is automatically performed:
- determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
- reducing the numerator and denominator of a fraction by gcd;
- highlighting the whole part of a fraction, if the numerator of the final fraction is greater than the denominator.
- converting the final algebraic fraction to a decimal fraction rounded to the nearest hundredth.
II. For reference:
A fraction is a number consisting of one or more parts (fractions) of a unit. A common fraction (simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction) separated by a horizontal bar (the fraction bar) indicating the division sign. The numerator of a fraction is the number above the fraction line. The numerator shows how many shares were taken from the whole. The denominator of a fraction is the number below the fraction line. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have a whole part. A simple fraction can be proper or improper. A proper fraction is a fraction whose numerator is less than its denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. An improper fraction is a fraction in which the numerator is greater than or equal to the denominator, so an improper fraction is always greater than or equal to one. Example of improper fractions: 7/6, 8/7, 13/13. mixed fraction is a number that contains a whole number and a proper fraction, and denotes the sum of that whole number and the proper fraction. Any mixed fraction can be converted to an improper fraction. Example mixed fractions: 1¼, 2½, 4¾.
III. Note:
- Source data block highlighted yellow , intermediate calculation block allocated blue , the solution block is highlighted in green.
- To add, subtract, multiply and divide ordinary or mixed fractions, use the online fraction calculator with detailed solution.
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... empty space. 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 it lurks typical mistake, a blooper, if you will.
For example, you need to simplify the expression:
There’s nothing to think about here, cross out the letter “a” on top and the “2” 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 everything that has been said 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 helpful information for self-test. 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. Fraction- 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 entirely decimals, but um... some evil ones, go to ordinary ones, try them! 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 can 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. In the presence of different types fractions in one task, the most reliable thing is to move on 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 finish here. In this lesson we refreshed our memory key points by 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).
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