Fractions are common and decimal. When the student learns about the existence of the latter, he begins to convert everything possible into decimal form at every opportunity, even if this is not required.
Oddly enough, high school students and students have different preferences, because it is easier to perform many arithmetic operations with ordinary fractions. And the values that graduates deal with can sometimes be simply impossible to convert to a decimal form without loss. As a result, both types of fractions are, one way or another, adapted to the case and have their own advantages and disadvantages. Let's see how to work with them.
Definition
Fractions are the same fractions. If there are ten slices in an orange, and you were given one, then you have 1/10 of the fruit in your hand. With such a notation, as in the previous sentence, the fraction will be called an ordinary fraction. If you write the same as 0, 1 is decimal. Both options are equal, but have their own advantages. The first option is more convenient when multiplying anddivision, the second - for addition, subtraction and in a number of other cases.
How to convert a fraction to another form
Suppose you have a common fraction and you want to convert it to a decimal. What needs to be done for this?
By the way, you need to decide in advance that not any number can be written in decimal form without problems. Sometimes you have to round the result, losing a certain number of decimal places, and in many areas - for example, in the exact sciences - this is a completely unaffordable luxury. At the same time, actions with decimal and ordinary fractions in the 5th grade allow such a transfer from one form to another without interference, at least as a practice.
If you can get a multiple of 10 from the denominator by multiplying or dividing by an integer, the transfer will pass without any difficulties: ¾ becomes 0.75, 13/20 becomes 0.65.
The inverse procedure is even easier, because from a decimal fraction you can always get an ordinary one without loss of accuracy. For example, 0.2 becomes 1/5 and 0.08 becomes 4/25.
Internal transformations
Before performing joint actions with ordinary fractions, you need to prepare numbers for possible mathematical operations.
First of all, you need to bring all the fractions in the example to one common form. They must be either ordinary or decimal. Let's make a reservation right away that it is more convenient to perform multiplication and division with the first ones.
In preparing numbers for further actions, you will be helped by a rule known as the basic property of a fraction and used both in the early years of studying the subject and in higher mathematics, which is studied at universities.
Properties of fractions
Suppose you have some value. Let's say 2/3. What happens if you multiply the numerator and denominator by 3? Get 6/9. What if it's a million? 2000000/3000000. But wait, because the number does not change qualitatively at all - 2/3 remain equal to 2000000/3000000. Only the form changes, not the content. The same thing happens when both parts are divided by the same value. This is the main property of the fraction, which will repeatedly help you perform actions with decimal and ordinary fractions on tests and exams.
Multiplying the numerator and denominator by the same number is called fraction expansion, and division is called reduction. I must say that crossing out the same numbers at the top and bottom when multiplying and dividing fractions is a surprisingly pleasant procedure (as part of a math lesson, of course). It seems that the answer is close and the example is almost solved.
Irregular fractions
An improper fraction is one in which the numerator is greater than or equal to the denominator. In other words, if it has an integer part, it falls under this definition.
If such a number (greater than or equal to one) is represented as an ordinary fraction, it will be calledwrong. And if the numerator is less than the denominator - correct. Both types are equally convenient in the implementation of possible actions with ordinary fractions. They can be freely multiplied and divided, added and subtracted.
If an integer part is selected at the same time and there is a remainder in the form of a fraction, the resulting number will be called mixed. In the future, you will come across various ways of combining such structures with variables, as well as solving equations where this knowledge is required.
Arithmetic operations
If everything is clear with the basic property of a fraction, then how to behave when multiplying fractions? Actions with ordinary fractions in the 5th grade involve all kinds of arithmetic operations that are performed in two different ways.
Multiplication and division are very easy. In the first case, the numerators and denominators of two fractions are simply multiplied. In the second - the same thing, only crosswise. Thus, the numerator of the first fraction is multiplied by the denominator of the second, and vice versa.
To perform addition and subtraction, you need to perform an additional action - bring all components of the expression to a common denominator. This means that the lower parts of the fractions must be changed to the same value - a multiple of both available denominators. For example, for 2 and 5 it will be 10. For 3 and 6 - 6. But then what to do with the top? We cannot leave it as it was if we changed the bottom one. According to the basic property of a fraction, we multiply the numerator by the same number,which is the denominator. This operation must be performed on each of the numbers that we will be adding or subtracting. However, such actions with ordinary fractions in the 6th grade are already performed “on the machine”, and difficulties arise only at the initial stage of studying the topic.
Comparison
If two fractions have the same denominator, then the one with the larger numerator will be larger. If the upper parts are the same, then the one with the smaller denominator will be larger. It should be borne in mind that such successful situations for comparison rarely occur. Most likely, both the upper and lower parts of the expressions will not match. Then you need to remember about the possible actions with ordinary fractions and use the technique used in addition and subtraction. Also, remember that if we are talking about negative numbers, then the larger fraction will be smaller.
Advantages of common fractions
It happens that teachers tell children one phrase, the content of which can be expressed as follows: the more information is given when formulating the task, the easier the solution will be. Does it sound weird? But really: with a large number of known values, you can use almost any formula, but if only a couple of numbers are provided, additional reflections may be required, you will have to remember and prove theorems, give arguments in favor of your being right…
What are we doing this for? And besides, ordinary fractions, for all their cumbersomeness, can greatly simplify life.student, allowing for multiplication and division to reduce entire lines of values, and when calculating the sum and difference, take out common arguments and, again, reduce them.
When it is required to perform joint actions with ordinary and decimal fractions, transformations are carried out in favor of the first: how do you convert 3/17 to decimal form? Only with loss of information, not otherwise. But 0, 1 can be represented as 1/10, and then as 17/170. And then the two resulting numbers can be added or subtracted: 30/170 + 17/170=47/170.
The benefits of decimals
If operations with ordinary fractions are more convenient, then writing everything with their help is extremely inconvenient, decimals have a significant advantage here. Compare: 1748/10000 and 0.1748. This is the same value presented in two different versions. Of course, the second way is easier!
Also, decimals are easier to represent because all the data has a common base that differs only by orders of magnitude. Let's say we can easily recognize a 30% discount and even evaluate it as significant. Will you immediately understand which is more - 30% or 137/379? Thus, decimal fractions provide standardization of calculations.
In high school students solve quadratic equations. It is already extremely problematic to perform actions with ordinary fractions here, since the formula for calculating the values \u200b\u200bof the variable contains the square root of the sum. In the presence of a fraction that is not reducible to a decimal, the solution becomes so complicated thatit becomes almost impossible to calculate the exact answer without a calculator.
So each way of representing fractions has its own advantages in its respective context.
Entry Forms
There are two ways to write actions with ordinary fractions: through a horizontal line, into two “tiers”, and through a slash (aka “slash”) - into a line. When a student writes in a notebook, the first option is usually more convenient, and therefore more common. The distribution of a number of numbers into cells contributes to the development of attentiveness in calculations and transformations. When writing to a string, you can inadvertently mix up the order of actions, lose any data - that is, make a mistake.
Quite often in our time there is a need to print numbers on a computer. You can separate fractions with a traditional horizontal bar using a function in Microsoft Word 2010 and later. The fact is that in these versions of the software there is an option called "formula". It displays a rectangular transformable field within which you can combine any mathematical symbols, make up both two- and “four-story” fractions. In the denominator and numerator, you can use brackets, operation signs. As a result, you will be able to write down any joint actions with ordinary and decimal fractions in the traditional form, i.e., as they are taught to do in school.
If you use the standard Notepad text editor, then everythingfractional expressions will need to be written through a slash. Unfortunately, there is no other way here.
Conclusion
So we have considered all the basic actions with ordinary fractions, which, it turns out, are not so many.
If at first it may seem that this is a difficult section of mathematics, then this is only a temporary impression - remember, once you thought so about the multiplication table, and even earlier about ordinary copybooks and counting from one to ten.
It is important to understand that fractions are used everywhere in everyday life. You will deal with money and engineering calculations, information technology and musical literacy, and everywhere - everywhere! - fractional numbers will appear. Therefore, do not be lazy and study this topic thoroughly - especially since it is not so difficult.
