One of the most difficult things for a student to understand are different actions with simple fractions. This is due to the fact that it is still difficult for children to think abstractly, and fractions, in fact, look just like that for them. Therefore, when presenting the material, teachers often resort to analogies and explain the subtraction and addition of fractions literally on the fingers. Although not a single lesson of school mathematics can do without rules and definitions.
Basic concepts
Before you start any actions with fractions, it is advisable to learn a few basic definitions and rules. Initially, it is important to understand what a fraction is. By it is meant a number representing one or more fractions of a unit. For example, if you cut a loaf into 8 parts and put 3 slices of them on a plate, then 3/8 will be a fraction. Moreover, in this writing it will be a simple fraction, where the number above the line is the numerator, and below it is the denominator. But if it is written as 0.375, it will already be a decimal fraction.
In addition, simple fractions are divided into proper, improper and mixed. The former include all those whose numerator is less thandenominator. If, on the contrary, the denominator is less than the numerator, it will already be an improper fraction. If there is an integer in front of the correct one, they speak of mixed numbers. Thus, the fraction 1/2 is correct, but 7/2 is not. And if you write it in this form: 31/2, then it will become mixed.
To make it easier to understand what addition of fractions is, and to perform it with ease, it is also important to remember the basic property of a fraction. Its essence is as follows. If the numerator and denominator are multiplied by the same number, the fraction will not change. It is this property that allows you to perform the simplest actions with ordinary and other fractions. In fact, this means that 1/15 and 3/45 are, in fact, the same number.
Adding fractions with the same denominators
This action is usually easy to perform. The addition of fractions in this case is very much like a similar action with integers. The denominator remains unchanged, and the numerators are simply added together. For example, if you need to add fractions 2/7 and 3/7, then the solution to a school problem in a notebook will be like this:
2/7 + 3/7=(2+3)/7=5/7.
Besides, such addition of fractions can be explained with a simple example. Take an ordinary apple and cut, for example, into 8 parts. Lay out separately first 3 parts, and then add 2 more to them. And as a result, 5/8 of a whole apple will lie in the cup. The arithmetic problem itself is written as shown below:
3/8 + 2/8=(3+2)/8=5/8.
Additionfractions with different denominators
But often there are more difficult problems, where you need to add together, for example, 5/9 and 3/5. This is where the first difficulties arise in actions with fractions. After all, adding such numbers will require additional knowledge. Now you will need to fully recall their main property. To add the fractions from the example, first they need to be reduced to one common denominator. To do this, simply multiply 9 and 5 among themselves, multiply the numerator "5" by 5, and "3", respectively, by 9. Thus, such fractions are already added: 25/45 and 27/45. Now it only remains to add the numerators and get the answer 52/45. On a piece of paper, an example would look like this:
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 adding fractions with such denominators does not always require a simple multiplication of numbers under the line. First look for the lowest common denominator. For example, as for fractions 2/3 and 5/6. For them, this will be the number 6. But the answer is not always obvious. In this case, it is worth remembering the rule for finding the least common multiple (abbreviated LCM) of two numbers.
It is understood as the least common factor of two integers. To find it, decompose each into prime factors. Now write out those of them that appear at least once in each number. Multiply them together and get the same denominator. In fact, everything looks a little simpler.
For example, you needadd the fractions 4/15 and 1/6. So, 15 is obtained by multiplying the simple numbers 3 and 5, and six - two and three. This means that the LCM for them will be 5 x 3 x 2=30. Now, dividing 30 by the denominator of the first fraction, we get a factor for its numerator - 2. And for the second fraction it will be the number 5. Thus, it remains to add ordinary fractions 8/30 and 5/30 and get a response on 13/30. Everything is extremely simple. In the notebook, this task should be written as follows:
4/15 + 1/6=(4 x 2)/(15 x 2) + (1 x 5)/(6 x 5)=8/30 + 5/30=13/30.
NOK (15, 6)=30.
Add mixed numbers
Now, knowing all the basic tricks in adding simple fractions, you can try your hand at more complex examples. And these will be mixed numbers, which means a fraction of this kind: 22/3. Here, the integer part is written before the proper fraction. And many get confused when performing actions with such numbers. In fact, the same rules apply here.
To add mixed numbers together, add the whole parts and proper fractions separately. And then these 2 results are already summed up. In practice, everything is much simpler, you just need to practice a little. For example, in a problem you need to add the following mixed numbers: 11/3 and 42/ 5. To do this, first add 1 and 4 to get 5. Then add 1/3 and 2/5 using the least common denominator technique. The decision will be 11/15. And the final answer is 511/15. In a school notebook it will look muchin short:
11/3 + 42/5 =(1 + 4) + (1/3 + 2/5)=5 + 5/15 + 6/15=5 + 11/15=511/ 15.
Adding decimals
In addition to ordinary fractions, there are also decimals. By the way, they are much more common in life. For example, the price in a store often looks like this: 20.3 rubles. This is the same fraction. Of course, these are much easier to fold than ordinary ones. In principle, you just need to add 2 ordinary numbers, most importantly, put a comma in the right place. This is where the difficulty comes in.
For example, you need to add such decimal fractions 2, 5 and 0, 56. To do this correctly, you need to add zero to the first at the end, and everything will be fine.
2, 50 + 0, 56=3, 06.
It is important to know that any decimal fraction can be converted to a simple fraction, but not every simple fraction can be written as a decimal. So, from our example 2, 5=21/2 and 0, 56=14/25. But a fraction like 1/6 will only be approximately equal to 0.16667. The same situation will be with other similar numbers - 2/7, 1/9 and so on.
Conclusion
Many schoolchildren, not understanding the practical side of actions with fractions, treat this topic carelessly. However, in older grades, this basic knowledge will allow you to click like nuts on complex examples with logarithms and finding derivatives. And therefore, it is worth once to understand well the actions with fractions, so that later you do not bite your elbows out of annoyance. After all, hardly a teacher in high schoolwill return to this, already passed, topic. Any high school student should be able to do these exercises.
