One of the most important science, the application of which can be seen in disciplines such as chemistry, physics and even biology, is a mathematician.The study of this science allows us to develop some mental qualities, improve abstract thinking and the ability to concentrate.One of the topics that deserve special attention in the course "Mathematics" - addition and subtraction of fractions.Many students study it causes difficulty.Perhaps our article will help you better understand this topic.
How subtract fractions with denominators equal
Fractions - it's the same number, with which you can do various things.They differ from the integers is in the presence of the denominator.That is why when performing operations with fractions need to explore some of the features and rules.The simplest case is the subtraction of fractions with denominators which are presented in the form of the same number.Perform this action will not be difficult if you know the simple rule:
- To subtract fractions from one second, you need to without decreasing the numerator of the fraction subtract the numerator of the fraction deductible.This number is written in the numerator a difference and leave the same denominator: k / m - b / m = (kb) / m.
Examples subtraction of fractions whose denominators are the same
Let's see how it looks on the example:
7/19 - 3/19 = (7 - 3) / 19 = 4/19.From
without decreasing the numerator of the fraction "7" subtract the numerator of the fraction deductible "3", get "4".This number we record the response in the numerator and the denominator of the set is the same number that was in the denominators of the first and second fractions - "19".
The picture below shows a few examples.
Consider a more complex example, which produced subtract fractions with the same denominator:
29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7) / 47= 9/47.From
without decreasing the numerator of the fraction "29" by subtracting the numerators turns all subsequent fractions - "3", "8", "2", "7".As a result, we get the result "9", which is written in the numerator of the answer and write in the denominator is the number that is in the denominators of these fractions - "47".
Addition of fractions with the same denominator
Addition and subtraction of fractions is carried out on the same principle.
- To fold fractions whose denominators are the same, you must add up the numerators.Received number - the sum of the numerator and the denominator will be the same: k / m + b / m = (k + b) / m.
Let's see how it looks on the example:
1/4 + 2/4 = 3/4.
to the numerator of the first term of the fraction - "1" - adding fraction the numerator of the second term - "2".The result - "3" - a record amount in the numerator and denominator of the reserve is the same as that present in the fractions - "4".
fractions with different denominators and subtraction
Action with fractions that have the same denominator, we have already considered.As you can see, knowing simple rules to solve similar examples quite easily.But what if you need to perform an action with fractions that have different denominators?Many high school students come to the difficulty to such examples.But here, if you know the principle of the solution, examples will no longer pose difficulty for you.Here, too, there is a rule, without which the solution of such fractions is simply impossible.
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To perform subtraction of fractions with different denominators, you must bring them to the same lowest common denominator.
learn how to do it, we'll talk more.
Property fraction
In order to bring some fractions to the same denominator, to be used in solving the main property of fractions: after the division or multiplication numerator and denominator by the same number will roll equal to this.
For example, the fraction may be 2/3 denominators such as "6", "9", "12" and t. D., I.e. it may take the form of any number that is a multiple of "3".After the numerator and denominator, we multiply by "2", you get the fraction 4/6.After the numerator and denominator of the original fraction, we multiply by "3", we get a 6/9, and if you produce a similar effect with the number "4", we get 8/12.One is equality can be written as:
2/3 = 4/6 = 6/9 = 8/12 ...
As a result several fractions to the same denominator
consider how to bring a few fractions to the samedenominator.For example, take the fraction shown in the image below.First we need to determine how many may be denominator for all of them.To help expand the available denominator factors.
denominator of the fraction 1/2 and 2/3 fractions can not be decomposed into factors.Denominator 7/9 multiplier has two 7/9 = 7 / (3 × 3), the denominator of the fraction 5/6 = 5 / (2 x 3).Now you need to determine what are the factors to be the lowest for all the four fractions.As in the first fraction in the denominator has the number "2", then it must be present in all the denominators in the fraction 7/9 has two triples, therefore, they are also both be present in the denominator.Given the above, we determine that the denominator is composed of three factors: 3, 2, and 3 is equal to 3 x 2 x 3 = 18.
Consider the first roll - 1/2.It has a denominator of '2', but no one digit "3", and should be two.For the denominator, we multiply by two triples, but, according to the property of the fraction, the numerator and we must multiply by two triples:
1/2 = (1 x 3 x 3) / (2 x 3 x 3) = 9/18.
produce similar action with the remaining fractions.
- 2/3 - denominator lacks one triple and one of two:
2/3 = (2 x 3 x 2) / (3 x 3 x 2) = 12/18. - 7/9 or 7 / (3 x 3) - in the denominator is missing twos:
7/9 = (7 x 2) / (9 x 2) = 14/18. - 5/6 or 5 / (2 x 3) - in the denominator is missing triples:
5/6 = (5 x 3) / (6 x 3) = 15/18.
All together it looks like this:
How to subtract and add up fractions with different denominators
As mentioned above, in order to perform the addition and subtraction of fractions with different denominators, they should lead to a common denominator, and then useRules subtract fractions with the same denominator, which has already been told.
look at an example: 4/18 - 3/15.
find the multiple of 18 and 15:
- Number 18 consists of a 3 x 2 x 3.
- Number 15 consists of 5 x 3.
- Total fold will consist of the following factors of 5 x 3 x 3 x 2 = 90.
When the denominator is found, it is necessary to calculate the multiplier, which will be different for each fraction, that is the number by which it will be necessary not only to multiply the denominator, but the numerator.To this number we found (common multiple), divided by the denominator of the fraction, which is necessary to identify the additional factors.
- 90 divided by 15. The resulting number "6" will be a factor for 3/15.
- 90 divided by 18. The resulting number "5" will be a factor for 4/18.
next stage of our solutions - each bringing to the denominator of the fraction "90".
How to do it, we said.Consider, as written in the Example:
(4 x 5) / (18 x 5) - (3 x 6) / (15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.
If fractions with small numbers, it is possible to identify a common denominator, as in the example shown in the image below.
Similarly produced and adding fractions with different denominators.
addition and subtraction of fractions with whole parts
subtraction of fractions and their addition, we have to thoroughly understand.But how to make a subtraction, if the fraction is the integer part?Again, use a few rules:
- all shot with the integer part, translated into wrong.In simple words, remove the integer part.To do this, multiply the number in the whole of the denominator of the fraction obtained by adding the product to the numerator.That number, which is obtained after these actions - the numerator improper fractions.The denominator remains unchanged.
- If fractions have different denominators, you should bring them to the same.
- Perform addition or subtraction with the same denominator.
- Upon receipt of improper fractions to allocate a part of the whole.
There is another way by which you can carry out addition and subtraction of fractions with integral parts.To do this, made a separate action with whole pieces, and separate operations with fractions, and the results are recorded together.
The example consists of fractions that have the same denominator.In the case where the denominators are different, they must be brought to the same, and then follow the steps as shown in the example.
subtraction of fractions of an integer
Another of the varieties of actions with fractions is the case when you need to take a fraction of a natural number.At first glance it seems like an example of difficult to resolve.However, it's pretty simple.To solve it is necessary to translate the integer fraction, and with the denominator of which is available at a fraction of the deductible.Next subtracts similar subtraction with the same denominator.For example, it looks like this:
7 - 4/9 = (7 x 9) / 9 - 4/9 = 53/9 - 4/9 = 49/9.
given in this article subtraction of fractions (Grade 6) is the basis for more complex examples, which are discussed in the following classes.Knowledge of this topic later used for solving functions, derivatives, and so on.It is therefore important to understand and understand the action with fractions, discussed above.
