When talking about mathematics, one cannot help but remember fractions. A lot of attention and time is devoted to their study. Remember how many examples you had to solve in order to learn certain rules for working with fractions, how you memorized and applied the basic property of a fraction. How much nerve was spent finding the common denominator, especially if the examples had more than two terms!
Let's remember what it is and a little refresher on the basic information and rules of working with fractions.
Definition of fractions
Let's start, perhaps, with the most important thing - the definition. A fraction is a number that is made up of one or more parts of a unit. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the top (or first) is called the numerator, and the bottom (second) is called the denominator.
It is worth noting that the denominator shows how many parts the unit is divided into, and the numerator shows the number of shares or parts taken. Often fractions, if proper, are less than one.
Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we examine such a concept as the “basic property rational fraction", let's talk about the types of fractions and their features.
What are fractions?
There are several types of such numbers. First of all, these are ordinary and decimal. The first represent the type of recording we have already indicated using a horizontal or slash. The second type of fractions is indicated using the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.
It is worth noting here that in mathematics both decimal and common fractions. The main property of the fraction is valid only for the second option. In addition, ordinary fractions are divided into regular and improper numbers. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than one. In an improper fraction, on the contrary, the numerator is greater than the denominator, and the fraction itself is greater than one. In this case, an integer can be extracted from it. In this article we will consider only ordinary fractions.
Properties of Fractions
Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers were no exception. They have one important feature, with the help of which certain operations can be carried out on them. What is the main property of a fraction? The rule states that if its numerator and denominator are multiplied or divided by the same rational number, we will get a new fraction, the value of which will be equal to the value of the original one. That is, by multiplying two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, and they will be equal.
Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.
Operations
Although fractions seem more complex, they can also be used to perform basic math operations, such as addition and subtraction, multiplication, and division. In addition, there is such a specific action as reducing fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes working with fractions easier, easier and more interesting. That is why next we will consider the basic rules and algorithm of actions when working with such numbers.
But before we talk about mathematical operations such as addition and subtraction, let’s look at an operation such as reduction to a common denominator. This is where knowledge of what basic property of a fraction exists comes in handy.
Common denominator
In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. That is, the smallest number that is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write down on a line for one denominator, then for the second, and find the matching number among them. If the LCM is not found, that is, these numbers do not have a common multiple, you should multiply them, and the resulting value is considered the LCM.
So, we have found the LCM, now we need to find an additional factor. To do this, you need to alternately divide the LCM into the denominators of the fractions and write the resulting number over each of them. Next, you should multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the basic property of a fraction.
Addition
Now let's move directly to mathematical operations on fractional numbers. Let's start with the simplest one. There are several options for adding fractions. In the first case, both numbers have the same denominator. In this case, all that remains is to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.
If the fractions different denominators, you should bring them to a common value and only then perform addition. We discussed how to do this a little higher. In this situation, the basic property of a fraction will come in handy. The rule will allow you to bring numbers to a common denominator. The value will not change in any way.
Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.
Multiplication
It does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of a fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the denominators will become the new denominator. As you can see, nothing complicated.
The only thing that is required of you is knowledge of the multiplication tables, as well as attentiveness. In addition, after receiving the result, you should definitely check whether it is possible to reduce given number or not. We'll talk about how to reduce fractions a little later.
Subtraction
When performing, you should be guided by the same rules as when adding. So, in numbers with same denominator It is enough to subtract the numerator of the subtrahend from the numerator of the minuend. If the fractions have different denominators, you should reduce them to a common denominator and then perform this operation. As in the similar case with addition, you will need to use the basic property of algebraic fractions, as well as skills in finding LCM and common divisors for fractions.
Division
And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who have little understanding of how to work with fractions, especially addition and subtraction. When dividing, the same rule applies as multiplying by a reciprocal fraction. The main property of a fraction, as in the case of multiplication, will not be used for this operation. Let's take a closer look.
When dividing numbers, the dividend remains unchanged. The divisor fraction turns into its reciprocal, that is, the numerator and denominator change places. After this, the numbers are multiplied with each other.
Reduction
So, we have already examined the definition and structure of fractions, their types, the rules of operations on these numbers, and found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - dividing the numerator and denominator by the same number. Thus, the fraction is reduced without changing its properties.
Usually, when performing a mathematical operation, you should carefully look at the resulting result and find out whether it is possible to reduce the resulting fraction or not. Remember that the final result always contains a fractional number that does not require reduction.
Other operations
Finally, we note that we have not listed all operations on fractional numbers, mentioning only the most well-known and necessary. Fractions can also be compared, converted to decimals and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those we presented above.
Conclusions
We talked about fractional numbers and operations with them. We also examined the main property. But let us note that all these issues were considered by us in passing. We have given only the most well-known and used rules and given the most important, in our opinion, advice.
This article is intended to refresh information about fractions that you have forgotten rather than to give new information and fill your head with endless rules and formulas that, most likely, will never be useful to you.
We hope that the material presented in the article, simply and concisely, was useful to you.
When studying ordinary fractions, we come across the concepts of the basic properties of a fraction. A simplified formulation is necessary to solve examples with ordinary fractions. This article involves the consideration of algebraic fractions and the application of a basic property to them, which will be formulated with examples of the scope of its application.
Formulation and rationale
The main property of a fraction has the form:
Definition 1
When the numerator and denominator are simultaneously multiplied or divided by the same number, the value of the fraction remains unchanged.
That is, we get that a · m b · m = a b and a: m b: m = a b are equivalent, where a b = a · m b · m and a b = a: m b: m are considered fair. The values a, b, m are some natural numbers.
Dividing the numerator and denominator by a number can be represented as a · m b · m = a b . This is similar to solving the example 8 12 = 8: 4 12: 4 = 2 3. When dividing, an equality of the form a: m b is used: m = a b, then 8 12 = 2 · 4 2 · 4 = 2 3. It can also be represented in the form a · m b · m = a b, that is, 8 12 = 2 · 4 3 · 4 = 2 3.
That is, the main property of the fraction a · m b · m = a b and a b = a · m b · m will be considered in detail in contrast to a: m b: m = a b and a b = a: m b: m.
If the numerator and denominator contain real numbers, then the property is applicable. First you need to prove the validity of the written inequality for all numbers. That is, prove the existence of a · m b · m = a b for all real a , b , m , where b and m are nonzero values to avoid division by zero.
Evidence 1
Let a fraction of the form a b be considered part of the record z, in other words, a b = z, then it is necessary to prove that a · m b · m corresponds to z, that is, prove a · m b · m = z. Then this will allow us to prove the existence of the equality a · m b · m = a b .
The fraction bar means the division sign. Applying the connection with multiplication and division, we find that from a b = z after transformation we obtain a = b · z. By properties numerical inequalities both sides of the inequality should be multiplied by a number other than zero. Then we multiply by the number m, we get that a · m = (b · z) · m. By property, we have the right to write the expression in the form a · m = (b · m) · z. This means that from the definition it follows that a b = z. That's all the proof of the expression a · m b · m = a b .
Equalities of the form a · m b · m = a b and a b = a · m b · m make sense when instead of a , b , m there are polynomials, and instead of b and m they are non-zero.
The main property of an algebraic fraction: when we simultaneously multiply the numerator and denominator by the same number, we obtain an expression identical to the original one.
The property is considered valid, since actions with polynomials correspond to actions with numbers.
Example 1
Let's look at the example of the fraction 3 · x x 2 - x y + 4 · y 3. It is possible to convert to the form 3 · x · (x 2 + 2 · x · y) (x 2 - x y + 4 · y 3) · (x 2 + 2 · x · y).
Multiplication by the polynomial x 2 + 2 · x · y was performed. In the same way, the main property helps to get rid of x 2, present in a given fraction of the form 5 x 2 (x + 1) x 2 (x 3 + 3) to the form 5 x + 5 x 3 + 3. This is called simplification.
The main property can be written as expressions a · m b · m = a b and a b = a · m b · m, when a, b, m are polynomials or ordinary variables, and b and m must be non-zero.
Areas of application of the basic property of an algebraic fraction
The application of the main property is relevant for reducing a fraction to a new denominator or reducing a fraction.
Definition 2
Reducing to a common denominator is multiplying the numerator and denominator by a similar polynomial to obtain a new one. The resulting fraction is equal to the original one.
That is, a fraction of the form x + y · x 2 + 1 (x + 1) · x 2 + 1 when multiplied by x 2 + 1 and reduced to a common denominator (x + 1) · (x 2 + 1) will receive the form x 3 + x + x 2 · y + y x 3 + x + x 2 + 1 .
After carrying out operations with polynomials, we find that the algebraic fraction is transformed into x 3 + x + x 2 · y + y x 3 + x + x 2 + 1.
Reduction to a common denominator is also performed when adding or subtracting fractions. If fractional coefficients are given, then a simplification must first be made, which will simplify the appearance and the very determination of the common denominator. For example, 2 5 x y - 2 x + 1 2 = 10 2 5 x y - 2 10 x + 1 2 = 4 x y - 20 10 x + 5.
The application of the property when reducing fractions is carried out in 2 stages: decomposition of the numerator and denominator into factors to find the common m, and then proceed to the type of fraction a b, based on an equality of the form a · m b · m = a b.
If a fraction of the form 4 x 3 - x y 16 x 4 - y 2 after expansion is transformed into x (4 x 2 - y) 4 x 2 - y 4 x 2 + y, it is obvious that the general the multiplier will be the polynomial 4 x 2 − y. Then it will be possible to reduce the fraction according to its main property. We get that
x (4 x 2 - y) 4 x 2 - y 4 x 2 + y = x 4 x 2 + y. The fraction is simplified, then when substituting values it will be necessary to perform much less action than when substituting into the original one.
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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 encounter degrees with rational indicators yes 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? Yes! 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 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 inverse conversion, ordinary to decimal, some people can’t do it 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 it.
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 not translated. 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 useful 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 this? 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's 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. 7 multiplied by 1 ( whole part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of the common fraction. That's it. 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. There, by the way, about improper fractions you'll find out.
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 this? 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, 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 possible
3. The choice of the type of fractions to work with a task depends on the task itself. Subject to availability 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 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.
Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.
Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.
The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:
The main property of a fraction
If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Reducing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- Multiply the numerator of the first fraction by the denominator of the second
- Multiply the numerator of the second fraction by the denominator of the first
- Replace the denominators of both fractions with their product
Operations with fractions
Addition. To add two fractions you need
- Add the new numerators of both fractions and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Reduce fractions to a common denominator
- Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged
Example:
Multiplication. To multiply one fraction by another, multiply their numerators and denominators.
This topic is quite important; all further mathematics and algebra are based on the basic properties of fractions. The properties of fractions considered, despite their importance, are very simple.
To understand basic properties of fractions Let's consider a circle.
On the circle you can see that 4 parts or are shaded out of the possible eight. Let's write the resulting fraction \(\frac(4)(8)\)
On the next circle you can see that one of the two possible parts is shaded. Let's write the resulting fraction \(\frac(1)(2)\)
If we look closely, we will see that in the first case, that in the second case we have half the circle shaded, so the resulting fractions are equal to \(\frac(4)(8) = \frac(1)(2)\), that is it's the same number.
How to prove this mathematically? It’s very simple, remember the multiplication table and write the first fraction into factors.
\(\frac(4)(8) = \frac(1 \cdot \color(red) (4))(2 \cdot \color(red) (4)) = \frac(1)(2) \cdot \color(red) (\frac(4)(4)) =\frac(1)(2) \cdot \color(red)(1) = \frac(1)(2)\)
What have we done? We factored the numerator and denominator \(\frac(1 \cdot \color(red) (4))(2 \cdot \color(red) (4))\), and then divided the fractions \(\frac(1) (2) \cdot \color(red) (\frac(4)(4))\). Four divided by four is 1, and one multiplied by any number is the number itself. What we did in the above example is called reducing fractions.
Let's look at another example and reduce the fraction.
\(\frac(6)(10) = \frac(3 \cdot \color(red) (2))(5 \cdot \color(red) (2)) = \frac(3)(5) \cdot \color(red) (\frac(2)(2)) =\frac(3)(5) \cdot \color(red)(1) = \frac(3)(5)\)
We again factored the numerator and denominator and reduced the same numbers into numerators and denominators. That is, two divided by two gives one, and one multiplied by any number gives the same number.
The main property of a fraction.
This implies the main property of a fraction:
If both the numerator and the denominator of a fraction are multiplied by the same number (except zero), then the value of the fraction will not change.
\(\bf \frac(a)(b) = \frac(a \cdot n)(b \cdot n)\)
You can also divide the numerator and denominator by the same number at the same time.
Let's look at an example:
\(\frac(6)(8) = \frac(6 \div \color(red) (2))(8 \div \color(red) (2)) = \frac(3)(4)\)
If both the numerator and denominator of a fraction are divided by the same number (except zero), then the value of the fraction will not change.
\(\bf \frac(a)(b) = \frac(a \div n)(b \div n)\)
Fractions that have common prime factors in both numerators and denominators are called reducible fractions.
Example of a reducible fraction: \(\frac(2)(4), \frac(6)(10), \frac(9)(15), \frac(10)(5), …\)
There is also irreducible fractions.
Irreducible fraction is a fraction that does not have common prime factors in its numerators and denominators.
Example of an irreducible fraction: \(\frac(1)(2), \frac(3)(5), \frac(5)(7), \frac(13)(5), …\)
Any number can be expressed as a fraction because any number is divisible by one. For example:
\(7 = \frac(7)(1)\)
Questions to the topic:
Do you think any fraction can be reduced or not?
Answer: no, there are reducible fractions and irreducible fractions.
Check whether the equality is true: \(\frac(7)(11) = \frac(14)(22)\)?
Answer: write the fraction \(\frac(14)(22) = \frac(7 \cdot 2)(11 \cdot 2) = \frac(7)(11)\), yes that's fair.
Example #1:
a) Find a fraction with a denominator of 15 equal to the fraction \(\frac(2)(3)\).
b) Find a fraction with numerator 8 equal to the fraction \(\frac(1)(5)\).
Solution:
a) We need the number 15 in the denominator. Now the denominator has the number 3. What number do we need to multiply the number 3 by to get 15? Let's remember the multiplication table 3⋅5. We need to use the basic property of fractions and multiply both the numerator and the denominator of the fraction \(\frac(2)(3)\) by 5.
\(\frac(2)(3) = \frac(2 \cdot 5)(3 \cdot 5) = \frac(10)(15)\)
b) We need the number 8 to be in the numerator. Now the number 1 is in the numerator. What number should we multiply the number 1 by to get 8? Of course, 1⋅8. We need to use the basic property of fractions and multiply both the numerator and the denominator of the fraction \(\frac(1)(5)\) by 8. We get:
\(\frac(1)(5) = \frac(1 \cdot 8)(5 \cdot 8) = \frac(8)(40)\)
Example #2:
Find an irreducible fraction equal to the fraction: a) \(\frac(16)(36)\), b) \(\frac(10)(25)\).
Solution:
A) \(\frac(16)(36) = \frac(4 \cdot 4)(9 \cdot 4) = \frac(4)(9)\)
b) \(\frac(10)(25) = \frac(2 \cdot 5)(5 \cdot 5) = \frac(2)(5)\)
Example #3:
Write the number as a fraction: a) 13 b)123
Solution:
A) \(13 = \frac(13) (1)\)
b) \(123 = \frac(123) (1)\)
