Students are introduced to fractions in 5th grade. Before people who knew how to perform actions with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, and so on. For several centuries, the examples were considered too complex. Now developed detailed rules on the conversion of fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the solution will be given easily.
An ordinary fraction, which is called a simple fraction, is written as a division of two numbers: m and n.
M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.
Select proper fractions (m< n) а также неправильные (m >n).
A proper fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). An improper fraction is equal to or greater than 1 (8/7 - the unit will be 7/7 and one more part is taken as a plus).
So, a unit is when the numerator and denominator matched (3/3, 12/12, 100/100 and others).
Actions with ordinary fractions Grade 6
With simple fractions, you can do the following:
- Expand fraction. If we multiply the top and lower part fractions by any identical number (but not by zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
- Reducing fractions is similar to expanding, but here they are divided by a number.
- Compare. If two fractions have the same numerator, then the fraction with the smaller denominator will be larger. If the denominators are the same, then the fraction with the largest numerator will be larger.
- Perform addition and subtraction. With the same denominators, this is easy to do (we sum the upper parts, and the lower part does not change). With different will have to find common denominator and extra multipliers.
- Multiply and divide fractions.
Examples of operations with fractions are considered below.
Reduced fractions Grade 6
To reduce means to divide the top and bottom of a fraction by some equal number.
The figure shows simple examples of reduction. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.
On a note! If the number is even, then it is divisible by 2 in any way. Even numbers are 2, 4, 6 ... 32 8 (ends in even), etc.
In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is also divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then 6 will come out. It turns out that the fraction was divided by six. This gradual division is called successive reduction of a fraction by common divisors.
Someone will immediately divide by 6, someone will need division by parts. The main thing is that at the end there is a fraction that cannot be reduced in any way.
Note that if the number consists of digits, the addition of which will result in a number divisible by 3, then the original can also be reduced by 3. Example: the number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, so the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divided by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.
In addition to the method of successive reduction of a fraction by common divisors, there are other ways.
GCD is the largest divisor for a number. Having found the GCD for the denominator and numerator, you can immediately reduce the fraction by the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors match, if there are several of them (as in the picture below), then you need to multiply.
Mixed fractions grade 6
All improper fractions can be converted into mixed fractions by isolating the whole part in them. The integer is written on the left.
Often you have to make a mixed number from an improper fraction. The conversion process in the example below: 22/4 = 22 divided by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be reduced, we divide the upper and lower parts by 2.
It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the whole number by the lower part of the fraction and add the numerator to this. Ready. The denominator does not change.
Calculations with fractions Grade 6
Mixed numbers can be added. If the denominators are the same, then this is easy to do: add up the integer parts and numerators, the denominator remains in place.
When adding numbers with different denominators the process is more difficult. First, we bring the numbers to one itself small denominator(NOZ).
In the example below, for the numbers 9 and 6, the denominator will be 18. After that, additional factors are needed. To find them, you should divide 18 by 9, so an additional number is found - 2. We multiply it by the numerator 4, we get the fraction 8/18). The same is done with the second fraction. We already add the converted fractions (whole numbers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper 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 place both under the same line. If the number is mixed, then we turn it into a simple fraction. Next, multiply the top and bottom parts and write down the answer. If it is clear that fractions can be reduced, then we reduce immediately.
In this example, we didn’t have to cut anything, we just wrote down the answer and highlighted the whole part.
In this example, I had to reduce the numbers under one line. Though it is possible to reduce also the ready answer.
When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper one, then we write the numbers under one line, replacing the division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this is the rule for dividing fractions).
If necessary, we reduce the numbers (in the example below, they reduced it by five and two). We transform the improper fraction by highlighting the integer part.
Basic tasks for fractions Grade 6
The video shows a few more tasks. For clarity, graphic images of solutions are used to help visualize fractions.
Examples of fraction multiplication Grade 6 with explanations
Multiplying fractions are written under one line. After that, they are reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).
Comparison of fractions Grade 6
To compare fractions, you need to remember two simple rules.
Rule 1. If the denominators are different
Rule 2. When the denominators are the same
For example, let's compare the fractions 7/12 and 2/3.
- We look at the denominators, they do not match. So you need to find a common one.
- For fractions, the common denominator is 12.
- We divide 12 first by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
- Now we divide 12 by 3, we get 4 - add. multiplier of the 2nd fraction.
- We multiply the resulting numbers by numerators to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. Turned out: 7/12< 8/12.
To represent fractions better, you can use drawings for clarity, where an object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case, the cake is divided into 7 parts and 4 of them are chosen. In the second, they divide into 3 parts and take 2. With the naked eye, it will be clear that 2/3 will be more than 4/7.
Examples with fractions grade 6 for training
As an exercise, you can perform the following tasks.
- Compare fractions
- do the multiplication
Tip: if it is difficult to find the lowest common denominator of fractions (especially if their values are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, you get 72.
Solving equations with fractions Grade 6
In solving equations, you need to remember the actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the known factor, that is, the fractions are multiplied (the second is turned over).
If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor, you need to divide the dividend by the quotient.
Imagine simple examples solving equations:
Here it is only required to produce the difference of fractions, without leading to a common denominator.
The answer is an improper 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 bottom rather than flip the denominator.
When a student moves to high school, mathematics is divided into 2 subjects: algebra and geometry. There are more and more concepts, tasks are becoming more difficult. Some people have difficulty understanding fractions. Missed the first lesson on this topic, and voila. fractions? A question that will torment throughout the school life.
The concept of algebraic fraction
Let's start with a definition. Under algebraic fraction P/Q expressions are understood, where P is the numerator and Q is the denominator. A number can be hidden under a letter entry, numeric expression, a numeric-literal expression.
Before you wonder how to solve algebraic fractions, first you need to understand that such an expression is part of a whole.
As a rule, the whole is 1. The number in the denominator shows how many parts the unit was divided into. The numerator is needed in order to find out how many elements are taken. The fractional bar corresponds to the division sign. It is allowed to record a fractional expression as a mathematical operation "Division". In this case, the numerator is the dividend, the denominator is the divisor.
The basic rule for common fractions
When students pass this topic at school, they are given examples to reinforce. To solve them correctly and find different ways from difficult situations, you need to apply the basic property of fractions.
It sounds like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), then the value common fraction Will not change. A special case of this rule is the division of both parts of the expression into the same number or polynomial. Such transformations are called identical equalities.
Below we will consider how to solve addition and subtraction of algebraic fractions, to perform multiplication, division and reduction of fractions.
Mathematical operations with fractions
Consider how to solve the main property of an algebraic fraction, how to apply it in practice. If you need to multiply two fractions, add them, divide one by the other, or subtract, you must always follow the rules.
So, for the operation of addition and subtraction, an additional factor should be found in order to bring the expressions to a common denominator. If initially the fractions are given with the same expressions Q, then you need to omit this item. When a common denominator is found, how to solve algebraic fractions? Add or subtract numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not perform any substitutions and mathematical operations. It is enough to change the sign in front of the fraction.
The term is often used as fraction reduction. This means the following: if the numerator and denominator are divided by an expression other than unity (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions, they remain equal to the original example.
The purpose of this operation is to obtain a new irreducible expression. Decide this task possible, if we reduce the numerator and denominator by the largest common divisor. The operation algorithm consists of two points:
- Finding the GCD for both parts of a fraction.
- Dividing the numerator and denominator by the found expression and obtaining an irreducible fraction equal to the previous one.
The table below shows the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future, when solving a test or exam, there will be no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.
Some examples with solutions
From a theoretical point of view, the question of how to solve algebraic fractions is considered. The examples given in the article will help you better understand the material.
1. Convert fractions and bring them to a common denominator.
2. Convert fractions and bring them to a common denominator.
After studying the theoretical part and considering practical issues should not occur again.
Fraction- a form of representation of a number in mathematics. The slash indicates the division operation. numerator fractions 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 if the modulus of the numerator is greater than the modulus of the denominator. If the fraction is correct, then the modulus of its value is always less than 1. All other fractions are wrong.
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 a fraction:
Basic 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,
Bringing 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
Actions with fractions
Addition. To add two fractions, you need
- Add new numerators of both fractions, and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another,
- Bring fractions to a common denominator
- Subtract the numerator of the second fraction 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:
Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:
Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.
Consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:
As you can see, nothing complicated: just add or subtract the numerators - and that's it.
But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is easy enough. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Let's analyze all this with specific examples:
Task. Find the value of the expression:
In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:
What if the denominators are different
You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:
Task. Find the value of the expression:
In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.
What if the fraction has an integer part
I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the integer part is highlighted in the fractional terms.
Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use a simple circuit below:
- Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.
The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:
Task. Find the value of the expression:
Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:
To simplify the calculations, I skipped some obvious steps in the last examples.
A little note to two recent examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.
Reread this sentence again, look at the examples, and think about it. This is where beginners allow great amount errors. They love to give such tasks to control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.
Summary: General Scheme of Computing
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
- Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.
Multiplication and division of fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:
For instance:
Everything is extremely simple. And please don't look for a common denominator! Don't need it here...
To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:
For instance:
If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For instance:
In high school, you often have to deal with three-story (or even four-story!) fractions. For instance:
How to bring this fraction to a decent form? Yes, very easy! Use division through two points:
But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:
In the first case (expression on the left):
In the second (expression on the right):
Feel the difference? 4 and 1/9!
What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:
then divide-multiply in order, left to right!
And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:
The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.
That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Note practical advice, and they (errors) will be less!
Practical Tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! Is not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.
2. In the examples with different types fractions - go to ordinary fractions.
3. We reduce all fractions to the stop.
4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.
Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.
So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. Only Then look at the answers.
Calculate:
Did you decide?
Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.
0; 17/22; 3/4; 2/5; 1; 25.
And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are 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 solvable Problems.
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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. Learning - with interest!)
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