One of the most difficult to understand the student are different actions with simple fractions.This is due to the fact that children are more difficult to think abstractly, and shot, in fact, it is for them and look.Therefore, presenting the material, teachers often resort to analogies and explain addition and subtraction of fractions are literally on the fingers.Although no rules and definitions can not do any lesson in school mathematics.
basic concepts
Before any action with fractions, it is advisable to learn a few basic definitions and rules.Initially, it is important to understand what fraction.Beneath it is understood a number representing one or more shares of the unit.For example, if a loaf cut into 8 pieces and 3 slices of them to put in a bowl, then 3/8 and it will be shot.And then writing it would be a simple fraction, where the number of the feature - is the numerator, and under it - the denominator.But if it is written as 0.375, it will be a decimal.
Besides simple fractions divided into regular, irregular and mixed.The former include all those, the numerator of which is less than the denominator.If on the contrary, the denominator is less than the numerator, it will be improper fraction.In the case before the right is an integer, talk about mixed numbers.Thus, the fraction 1/2 - right, and 7/2 - no.And if it is written in the form: 31/2, it will be mixed.
To make it easier to understand what is the addition of fractions, and can easily carry it out, it is important to remember the main property of fractions.Its essence is as follows.If the numerator and denominator multiplied by the same number, the roll will not change.This property allows you to perform simple actions with common and other fractions.In fact, this means that the 1/15 and 3/45, in fact, the same number.
Addition of fractions with the same denominator
Doing this usually does not cause much difficulty.Addition of fractions in this case is very much like a similar effect with integers.The denominator remains unchanged, and the numerators are simply added together.For example, if you need to add the fraction 2/7 and 3/7, the solution to the problem of school notebooks will be like this:
2/7 + 3/7 = (2 + 3) / 7 = 5/7.
Moreover, this addition of fractions can be explained with a simple example.Take the usual apple and cut, for example, into 8 pieces.Put first 3 parts separately and then add another 2. As a result, in the cup will be based on 5/8 of the whole apple.Samu arithmetic problem is recorded, as shown below:
3/8 + 2/8 = (3 + 2) / 8 = 5/8.
Addition of fractions with different denominators
But often there are problems more complicated, where you have to lay down together, for example, 5/9, and 3/5.Here and there are the first difficulties in the operations with fractions.After the addition of such numbers require additional knowledge.Now fully required to recall their basic properties.To add a fraction of example, for a start they need to bring to a common denominator.To do this, simply multiply 9 and 5 together, the numerator "5" times 5, and "3", respectively, 9. Thus, already forming such fraction: 25/45 and 27/45.Now only remains to add the numerators and get an answer 52/45.On a piece of paper would look like this example:
5/9 + 3/5 = (5 x 5) / (9 x 5) + (3 x 9) / (5 x 9) = 25/45 +27/45 = (25 + 27) / 45 = 52/45 = 17/45.
But the addition of fractions with denominators such does not always require a simple multiplication of the number below the line.Please look for the lowest common denominator.For example, as for the fractions 2/3 and 5/6.For them it will be the number 6. But it is not always the answer is obvious.In this case, it is worth remembering usually find the least common multiple (abbreviated as NOC) of two numbers.
It refers to the least common multiple of two integers.To find it, laid out the prime factors of each.Now discharged those that are provided at least once in each number.Multiplies them together and get the same denominator.In fact, it looks a little bit easier.
example, you want to lay down fractions 4/15 and 1/6.So, 15 is obtained by multiplying prime numbers 3 and 5, and six - two and three.So NOC for them to be 5 x 3 x 2 = 30. Now, divide 30 by the denominator of the first fraction, we get a multiplier for its numerator - 2. And the second shot is to be number 5. Thus, it remains to lay down common fractions 8/305/30 and 13/30 and get an answer.All very simple.The notebook also be the task be written as:
4/15 + 1/6 = (4 x 2) / (15 x 2) + (1 x 5) / (6 x 5) = 8/30 + 5/30= 13/30.
NOC (15, 6) = 30.
Addition of mixed numbers
now, knowing all the basic techniques in the addition of fractions, you can try your hand at a more complicated example.And it will be mixed numbers, which refers to the fraction of this kind: 22/3.Here, right in front of the entire shot was discharged.And many are confused when performing actions such numbers.In fact, it employs all of the same rule.
To fold between a mixed number, the whole of the stack separately and proper fractions.And then to summarize these two results.In practice, it is much easier, it is worth just a little exercise.For example, in the task required to lay down such mixed numbers: 11/3 and 42/5.To do this, first fold 1 and 4 - 5 will then summarize the 1/3 and 2/5, using the methods of reduction to the lowest common denominator.The solution is to 11/15.A final answer - it is 511/15.The school notebook it will look much shorter:
11/3 + 42/5 = (1 + 4) + (1/3 + 2/5) = 5+ 5/15 + 6/15 = 5+ 11/15 = 511/15.
Addition decimal
addition of fractions, decimals and there.They are, incidentally, are much more likely to occur in life.For example, the price in the store often looks like this: 20.3 rubles.It is precisely the fraction.Of course, these add a lot easier than ordinary.Basically, you just need to add up the number 2 common, most importantly, in the right place to put a comma.This is where the difficulties arise.For example
required to lay down such decimals 2.5 and 0.56.To do this correctly, you need to finish first at the end of zero, and everything will be fine.
2,50 + 0,56 = 3,06.
is important to know that any decimal fraction can be converted into a simple, but not any simple fraction can be written as a decimal.So, in our example 2.5 = 0.56 = 21/2 and 14/25.But this fraction is 1/6, is only approximately equal to 0.16667.The same situation is similar with other numbers - 2/7, 1/9 and so on.
Conclusion
Many students do not understand the practical side of operations with fractions, refer to this topic in a slipshod manner.However, in the more senior classes this basic knowledge will allow snapping like peanuts complex examples with logarithms and finding derivatives.So it is once well understand operations with fractions, so you do not bite your elbows in frustration.It is hardly a teacher in high school will come back to this already passed, subject.Any high school student should be able to carry out similar exercises.
