With fractions, students get acquainted in the 5th grade. Previously, people who knew how to perform acts with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, etc. For several centuries, examples were considered too complicated. Now they have developed detailed rules for the transformation of fractions, addition, multiplication and other actions. It is enough to figure out the material a little, and the solution will be easy.
The ordinary fraction, which is called a simple fraction, is written as dividing two numbers: M and N.
M is a divisible, that is, the fractional numerator, and the divider N is called the denominator.
Eliminate the correct fractions (M< n) а также неправильные (m > n).
The correct fraction is less than the unit (for example 5/6 - this means that 5 parts are taken from the unit; 2/8 - from the unit to be taken 2 parts). The wrong fraction is equal to or more than 1 (8/7 - the unit will be 7/7 and the plus is taken another part).
So, one, this is when the numerator and the denominator coincided (3/3, 12/12, 100/100 and others).
Actions with ordinary fractions Grade 6
With ordinary fractions, you can carry out the following actions:
- Expand fraction. If you multiply the upper and lower part of the fraction on any identical number (not only for zero), then the fraction value will not change (3/5 \u003d 6/10 (simply multiplied by 2).
- Reducing fractions is similar to expansion, but they are divided into any number.
- Compare. If two fractions of the numerals are the same, then the larger will turn out to be with a smaller denominator. If the same denominators, it will be more fraction with the largest numerator.
- Perform addition and subtraction. With the same denominators, it is easy to do this (we summarize the upper parts, and the bottom does not change). If you have to find a common denominator and additional multipliers.
- Multiply and divide the fractions.
Examples of action with fractions Consider below.
Abbreviated fractions Grade 6
Reduce - it means to divide the upper and lower part of the fraction on any identical number.
The figure shows simple examples of reduction. In the first embodiment, you can immediately guess that the numerator and denominator are divided into 2.
On a note! If the number is even, it is divided in any way to 2. Even numbers - this is 2, 4, 6 ... 32 8 (ends on even), etc.
In the second case, with division 6 to 18, it is immediately seen that the numbers are divided by 2. Dividing, we obtain 3/9. This fraction is divided by another 3. Then in response it turns out 1/3. If you multiply both divisors: 2 by 3, then it will be released 6. It turns out that the fraction was divided into six. Such a gradual division is called sequential reduction of fractions on common dividers.
Someone will immediately divide on 6, someone will need dividing parts. The main thing is that in the end there is a fraction, which is no longer cut.
Note that if the number consists of numbers, when the number is addition, the number is divided by 3, then the initial can also be reduced by 3. Example: number 341. We fold the numbers: 3 + 4 + 1 \u003d 8 (8 to 3 is not divided, So, the number 341 cannot be reduced by 3 without a residue). Another example: 264. We fold: 2 + 6 + 4 \u003d 12 (divided by 3). We get: 264: 3 \u003d 88. It will simplify the reduction of large numbers.
In addition to the method of consistent reduction in the fraction on common divisors there are other ways.
Node is the biggest divider for the number. Having found a node for the denominator and the numerator, you can immediately reduce the fraction to the desired number. The search is carried out by gradual division of each number. Further look at what dividers coincide if there are several of them (as in the picture below), then you need to multiply.
Mixed fractions Grade 6
All incorrect fractions can be turned into mixed, highlighting the whole part in them. An integer is written on the left.
Often comes from the wrong fraction to make a mixed number. The process of conversion on the example below: 22/4 \u003d 22 Delimis by 4, we obtain 5 of the whole (5 * 4 \u003d 20). 22 - 20 \u003d 2. We obtain 5 whole and 2/4 (the denominator does not change). Since the fraction can be reduced, then we divide the upper and lower part to 2.
The mixed number is easy to turn into an irregular fraction (this is necessary when dividing and multiplying fractions). To do this: an integer multiply on the bottom of the fraction and add a numerator to this. Ready. The denominator does not change.
Calculations with fractions Grade 6
Mixed numbers can be folded. If the denominants are the same, then it is easy to do it: we fold entire parts and numerals, the denominator remains in place.
When adding numbers with different denominator, the process is more difficult. First, we give a number to one small denominator (nose).
In the example below, for numbers 9 and 6, the denominator will be 18. After that, additional multipliers are needed. To find them, 18 divided by 9, so there is an additional number - 2. It is multiplied by the Nizer 4, it turned out to be 8/18). The same is done with the second fraction. Transformed fractions are already folding (integers and numerals separately, the denominator does not change). In the example, the answer had to be converted to the correct fraction (initially the numerator turned out to be greater than the denominator).
Please note that with the difference of fractions, the algorithm of actions is the same.
When multiplying fractions, it is important to put both on one line. If the number is mixed, then turn it into a simple fraction. Next, multiply the upper and lower parts and write the answer. If you can see that the fraction can be reduced, then reduce immediately.
In the specified example, nothing had to shorten, simply recorded the answer and allocated the whole part.
In this example, I had to reduce the numbers under one feature. Although it is possible to cut the ready answer.
When dividing the algorithm is almost the same. First, we turn the mixed fraction in the wrong, then write the numbers under one feature, replacing the division by multiplication. Do not forget the upper and lower part of the second fraction to change places (this is a division rule of fractions).
If necessary, reducing the number (in the example below, it was reduced to the top five and a two). Incorrect fraction converting, highlighting the whole part.
Basic tasks for fractions Grade 6
The video shows a few more tasks. For clarity, graphic images of solutions will help to clearly imagine fractions.
Examples of multiplication of fractions Grade 6 with explanations
Protecting fractions are written under the same line. After that, they are reduced by dividing on the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided into five).
Comparison of fractions Grade 6
To compare the fractions, you need to remember two simple rules.
Rule 1. If different denominators
Rule 2. When the denominants are the same
For example, we compare the fractions 7/12 and 2/3.
- We look at the denominators, they do not coincide. So you need to find a common one.
- For fractions, the general denominator will be 12.
- We divide 12 first on the bottom of the first fraction: 12: 12 \u003d 1 (this is an additional factor for the 1st fraction).
- Now 12 divide by 3, we get 4 - add. Multiplier of the 2nd fraction.
- Multiply the figures obtained on the numerals to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 \u003d 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. It turned out: 7/12.< 8/12.
To represent the fraction better, it is possible for clarity to use pictures where the subject is divided into parts (for example, cake). If you need to compare 4/7 and 2/3, in the first case, the cake is divided into 7 parts and choose 4 of them. In the second - divide on 3 parts and take 2. The naked eye will understand that 2/3 will be more than 4/7.
Examples with fractions Grade 6 for training
You can perform the following tasks as a workout.
- Compare fractions
- perform multiplication
Tip: If it is difficult to find the smallest common denominator in fractions (especially if the values \u200b\u200bare small), then you can multiply the denominator of the first and second fraction. Example: 2/8 and 5/9. Find their denominator simply: 8 multiply by 9, it turns out 72.
Solving equations with fractions grade 6
In solving the equations, you need to recall the steps with fractions: multiplication, division, subtraction and addition. If one of the multipliers is unknown, the work (result) is divided into a well-known multiplier, that is, the fractions are variable (the second turns over).
If unknown is divisible, the denominator is multiplied by the divider, and for the search for a divider, you need to divide into a private.
Imagine simple examples of solving equations:
It only needs to make a difference in fractions, not leading to a common denominator.
The answer turned out in the form of incorrect fraction. It can be converted to 1 whole and 3/5.
In the second method, the numerator and denominator were multiplied by 4 to shorten the lower part, and not turn the denominator.
When the student goes into senior school, mathematics is divided into 2 subjects: algebra and geometry. Concepts are becoming more and more tasks. In some, there are difficulties with the perception of fractions. They missed the first lesson on this topic, and voila. Fruit? The question that will torment throughout the whole school life.
The concept of algebraic fraci
Let's start with the definition. Under algebraic fractionit is understood as the expression P / Q, where p is a numerator, and the q - denominator. Under an alphabone record, a number, numerical expression, numerical expression may be hidden.
Before wondering how to solve algebraic fractions, first need to understand that such an expression is part of the whole.
As a rule, the whole is 1. The number in the denominator shows how many parts were divided by a unit. The numerator is necessary in order to find out how many elements are taken. The fractional feature corresponds to the division sign. It is allowed to record a fractional expression as a mathematical operation "Decision". In this case, the numerator is divisible, denominator - divider.
Major rule of ordinary fractions
When students take this topic at school, they are given examples to consolidate. In order to solve them correctly and find various paths from complex situations, it is necessary to apply the basic property of fractions.
It sounds like this: if you multiply the numerator, and the denominator on the same number or expression (different from zero), then the value of the ordinary fraction will not change. A special case of this rule is the separation of both parts of the expression on the same number or polynomial. Such transformations are called identical equalities.
Below will be considered how to solve the addition and subtraction of algebraic fractions, to produce multiplication, division and reduction of fractions.
Mathematical transactions with fractions
Consider how to solve, the main property of algebraic fraction, how to apply it in practice. If you need to multiply two fractions, fold them, divide one to another or deduct, you must always stick to the rules.
So, for the addition operation and subtraction, an additional factor should be found to bring expressions to the general denominator. If initially fractions are given with the same q expressions, then you need to lower this item. When is the common denominator found how to solve algebraic fractions? You need to fold or subtract numerals. But! It must be remembered that if there is a sign "-" before the fraction, all signs in the numener are changing to the opposite. Sometimes you should not make any substitutions and mathematical operations. Enough to change the sign before the fraction.
Often used such a thing as reducing fractions. This means the following: if the numerator and the denominator are divided into an expression other than the unit (the same for both parts), then a new fraction is obtained. The divider and divider is less than the former, but due to the basic rules of fractions remain equal in the original example.
The purpose of this operation is to obtain a new non-interpretable expression. You can solve this task if you cut the numerator and the denominator to the greatest common divisor. The operation algorithm consists of two points:
- Finding a node for both parts of the fraction.
- The division of the numerator and the denominator for the found expression and the receipt of an unstable fraction equal to the previous one.
Below is the table in which the formulas are painted. For convenience, it can be printed and carry with you in the notebook. However, in order to solve the control or exam in the future in the future, there was no difficulty solving how to solve algebraic fractions, these formulas need to be learned by heart.
Some examples with solutions
From the theoretical point of view, the question of how to solve algebraic fractions. Examples given in the article will help better learn the material.
1. Transform fractions and lead them to a common denominator.
2. Convert fractions and lead them to a common denominator.
After studying the theoretical part and the search for practical issues should not be more.
Fraction - form of representation of a number in mathematics. The fractional feature indicates the operation of the division. Numerator The fraci is divisible, and denominator - Divider. For example, the fraction of the numerator is the number 5, and the denominator is 7.
Right It is called a fraction that has a numerator module larger than the denominator module. If the fraction is correct, then its module is always less than 1. All other fractions are wrong.
Fraction is called mixedif it is recorded as an integer and fraction. This is the same as the amount of this number and fractions:
The main property of the fraci
If the numerator and denominator of the fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Bringing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- The numerator of the first fraction multiply to the denominator of the second
- Numerator of the second fraction multiplying the denominator
- Rannels of both fractions replace their work
Actions with fractions
Addition. To fold two fractions, you need
- Folded new numerals of both fractions, and the denominator is left unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Bring a fraction to a common denominator
- Subtract from the numerator of the first fraction Numerator second, and the denominator is left unchanged
Example:
Multiplication. To multiply one fraction to another, multiply their numerators and denominators:
Division. To divide one fraction to another, the numerator of the first fraction should be multiplied by the second denominator, and the denominator of the first fraction is to multiply the second numerator:
The fractions are ordinary numbers, they can also be folded and deducted. But due to the fact that they are present a denominator, more complex rules are required here than for integers.
Consider the easiest case when there are two fractions with the same denominators. Then:
To fold the fractions with the same denominators, it is necessary to fold their numerals, and the denominator should be left unchanged.
To subtract fractions with the same denominators, it is necessary to deduct the numerator of the first fraction, and the denominator is again left unchanged.
Inside each expression, denominators are equal. By definition of addition and subtract fractions, we get:
As you can see, nothing complicated: just fold or deduct the numerals - and that's it.
But even in such simple actions, people manage to make mistakes. Most often forget that the denominator does not change. For example, when adding them, they are also started to fold, and this is rooted incorrectly.
Get rid of the bad habit of folding the denominators simply. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) Will lose meaning.
Therefore, remember times and forever: when adding and subtracting, the denominator does not change!
Also, many make mistakes when adding several negative fractions. There is a confusion with signs: where to put minus, and where - plus.
This problem is also solved very simple. It is enough to remember that the minus before the fraci sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:
- Plus, minus gives minus;
- Two negatives make an affirmative.
We will analyze all this on specific examples:
A task. Find the value of the expression:
In the first case, everything is simple, and in the second we will make minuses in fractions numerators:
What to do if the denominators are different
Directly fold the fractions with different denominants. At least, this method is unknown to me. However, the initial fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are considered in the lesson "bringing fractions to a common denominator", so here we will not stop at them. Better look at the examples:
A task. Find the value of the expression:
In the first case, we give the fractions to the overall denominator by the "Cross-Length" method. In the second we will look for Nok. Note that 6 \u003d 2 · 3; 9 \u003d 3 · 3. Recent multipliers in these decompositions are equal, and the first are mutually simple. Consequently, the NOC (6; 9) \u003d 2 · 3 · 3 \u003d 18.
What to do if the fraci has a whole part
I can deliver you: different denominators in fractions are not the biggest evil. Much more errors occur when a whole part is highlighted in the smoke smokers.
Of course, for such fractions there are their own algorithms for addition and subtraction, but they are quite complex and require a long study. Better use a simple scheme below:
- Translate all the fractions containing the whole part to the wrong. We obtain the normal terms (even if even with different denominators), which are considered according to the rules discussed above;
- Actually, calculate the amount or difference of fractions obtained. As a result, we practically find the answer;
- If this is all that was required in the task, perform the reverse transformation, i.e. We get rid of incorrect fraction, highlighting the whole part in it.
The rules for the transition to incorrect fractions and allocations of the whole part are described in detail in the lesson "What is the numerical fraction". If you do not remember - be sure to repeat. Examples:
A task. Find the value of the expression:
Everything is simple here. Dannels within each expression are equal, so it remains to translate all the fractions into the wrong and count. We have:
To simplify the calculations, I missed some obvious steps in the latest examples.
A little remark to the two latest examples, where the fractions are subtracted with a part highlighted. The minus before the second fraction means that the entire fraction is deducted, and not only her whole part.
Re-read this offer again, take a look at the examples - and think about it. It is here that beginners allow a huge number of errors. Such tasks adore in the tests. You will also repeatedly meet with them in tests to this lesson that will be published soon.
Summary: General computing scheme
In conclusion, I will give a general algorithm that will help to find the amount or difference between two or more fractions:
- If a whole part is highlighted in one or several fractions, translate these fractions into incorrect;
- Give all the fractions to the general denominator in any way convenient for you (if, of course, this did not make compilers of tasks);
- Fold or deduct the numbers obtained according to the rules of addition and subtract fractions with the same denominators;
- If possible, reduce the result. If the fraction was incorrect, highlight the whole part.
Remember that allocating the whole part is better at the very end of the task, immediately before recording a response.
Multiplication and division of fractions.
Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")
This operation is much more nicer addition-subtraction! Because it's easier. I remind you: To multiply the fraction on the fraction, you need to multiply the numerators (it will be the resultant) and the denominators (this will be the denominator). I.e:
For example:
Everything is extremely simple. And please do not look for a common denominator! Do not need him here ...
To divide the fraction for the fraction, you need to flip over second(This is important!) Fraction and multiply them, i.e.:
For example:
If multiplication or division with integers and fractions was caught - nothing terrible. As with the addition, we make a fraction with a unit in the denominator - and forward! For example:
In high schools, it is often necessary to deal with three-story (or even four-storey!) Droks. For example:
How to bring this fraction to a decent mind? Yes, very simple! Use division in two points:
But do not forget about the order of division! Unlike multiplication, it is very important here! Of course, 4: 2, or 2: 4 We are not confused. But in the three-story fraction it is easy to make a mistake. Note, for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
And what is the order of division? Or brackets, or (as here) the length of horizontal lines. Develop the eye meter. And if there are no brackets, nor dash, like:
then divide-multiply in a few, left to right!
And a very simple and important technique. In actions with degrees, he oh, how can I come in handy! We divide the unit to any fraction, for example, by 13/15:
The fraction turned over! And it always happens. When dividing 1 to any fraction, as a result, we get the same fraction only inverted.
That's all the actions with fractions. The thing is quite simple, but the mistakes gives more than enough. Please note the practical advice, and their (errors) will be less!
Practical Tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a harsh need! All calculations on the exam make as a full task, focusing and clearly. It is better to write two extra lines in the draft, than to accumulate when calculating the mind.
2. In the examples with different types of fractions - we turn to ordinary fractions.
3. All fractions cut until it stops.
4. Multi-storey fractional expressions are reduced to ordinary, using division in two points (follow the order of division!).
5. Unit of fraction divide in mind, just turning the fraction.
Here are the tasks you need to break. Answers are given after all tasks. Use the materials of this topic and practical advice. Count how many examples you could solve correctly. The first time! Without a calculator! And make faithful conclusions ...
Remember - the correct answer, the resulting from the second (even more - the third) times - not considered! Such is a harsh life.
So, we decide in the exam mode ! This is already prepared for the exam, by the way. We solve the example, check, solve the following. They decided everything - they checked again from the first to last. Only later We look at the answers.
Calculate:
Did you cut?
We are looking for answers that coincide with yours. I specifically recorded them in disarray, away from the temptation, so to speak ... So they are answered, the point with the comma is recorded.
0; 17/22; 3/4; 2/5; 1; 25.
And now we make conclusions. If everything happened - I am glad for you! Elementary calculations with fractions - not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) Inattention. But this resolved Problems.
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By the way, I have another couple of interesting sites for you.)
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