Fractions are ordinary and decimal. When the student learns about the existence of the latter, he begins at every opportunity to translate everything that is possible into decimal, even if it is not required.
Oddly enough, in high school students and students, preferences change, because it is easier to perform many arithmetic operations with ordinary fractions. And the values that graduates deal with, to convert to decimal without loss is sometimes simply impossible. As a result, both types of fractions are, one way or another, adapted to the business and have their own advantages and disadvantages. Let's see how to work with them.
Definition
Fractions are the same shares. If there are ten slices in an orange, and you have been given one, then you have 1/10 of the fruit in your hand. With such a record, as in the previous sentence, the fraction will be called ordinary. If you write the same as 0.1 - decimal. Both options are equal, but they have their advantages. The first option is more convenient for multiplication and division, the second - for addition, subtraction, and in a number of other cases.
How to transfer a fraction to another type
Suppose you have an ordinary fraction, and you want to make it decimal. What do I need to do?
By the way, you need to decide in advance that not every number can be written in decimal form without any problems. Sometimes you have to round off the result, losing a certain number of decimal places, and in many areas - for example, in the exact sciences - this is an absolutely inadmissible luxury. At the same time, actions with decimal and ordinary fractions in the 5th grade allow such a transfer from one type to another without interference, at least as a training.
If, by multiplying or dividing by an integer, you can get a multiple of 10, the translation will go without any difficulties: ¾ turns into 0.75, 13/20 - into 0.65.
The reverse procedure is even simpler, because from the decimal fraction you can always get the ordinary without loss in 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 the numbers for possible mathematical operations.
First of all, you need to bring all the fractions available in the example to one general view. They must be either ordinary or decimal. Immediately make a reservation that multiplication and division is more convenient to perform with the former.
In preparing the numbers for further actions, you will be helped by the rule, known as the
main property of the fraction and used both in the early years of the study of the subject and in higher mathematics, which is studied at universities.
Fraction Properties
Suppose you have some meaning. Say 2/3. What will change if you multiply the numerator and denominator by 3? It will turn out 6/9. And if a million? 2000000/3000000. But wait a minute, because the number does not change qualitatively at all - 2/3 remain equal to 2,000,000/3,000,000. Only the form changes, but not the content. The same thing will happen when dividing both parts by the same value. This is the main property of the fraction, which will repeatedly help you to perform actions with decimal and ordinary fractions in control and exams.
Multiplying the numerator and denominator by the same number is called the expansion of the fraction, and division is called the reduction. It must be said that striking out the same numbers in the upper and lower parts when multiplying and dividing fractions is a surprisingly pleasant procedure (as part of a math lesson, of course). It seems that the answer is already close and the example is practically resolved.
Wrong fractions
An improper fraction is one whose numerator is greater than or equal to the denominator. In other words, if a whole part can be distinguished from her, she falls under this definition.
If such a number (greater than or equal to unity) is presented in the form of an ordinary fraction, it will be called incorrect. And if the numerator is less than the denominator - correct. Both types are equally convenient when carrying out possible actions with ordinary fractions. They can be freely multiplied and divided, added and subtracted.
If at the same time the whole part is selected and at the same time 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 main property of a fraction, then how to behave when multiplying fractions? Actions with ordinary fractions in the 5th grade mean all kinds of arithmetic operations that are performed in two different ways.
Multiplication and division are very simple. 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 upper part? We cannot leave it in its original form if we change the lower one. According to the main property of the fraction, we multiply the numerator by the same number as the denominator. This operation must be performed with each of the numbers that we will add or subtract. 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 there will be more of them, the numerator of which is greater. 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. In addition, remember that if we are talking about negative numbers, then the larger modulus fraction will be smaller.
The benefits of common fractions
It happens that teachers tell children one phrase, the contents of which can be expressed as follows: the more information is given in the wording of the assignment, the easier the decision will be. Does that sound weird? But really: with a large number of known quantities, almost any formulas can be used, but if only a couple of numbers are provided, additional thought may be required, you will have to remember and prove the theorems, give arguments in favor of your case ...
Why are we doing this? And besides, ordinary fractions, for all their bulkiness, can greatly simplify the life of the student, allowing them to multiply whole lines of values when multiplying and dividing, and when calculating the sum and difference, make general arguments and, again, reduce them.
When it is required to carry out joint actions with ordinary and decimal fractions, transformations are carried out in favor of the former: how do you translate 3/17 to decimal? 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.
Why decimal fractions are useful
If it is more convenient to carry out actions with ordinary fractions, then recording everything with their help is extremely inconvenient, decimals here have a significant advantage. Compare: 1748/10000 and 0.1748. This is the same meaning, presented in two different ways. Of course, the second way is easier!
In addition, decimal fractions are easier to imagine, since all data has a common basis, differing only by orders of magnitude. Say, we easily recognize the 30% discount and even rate it as significant. And do you immediately realize that more - 30% or 137/379? Thus, decimal fractions provide standardization of calculations.
In high school, students solve quadratic equations. Performing actions with ordinary fractions is already extremely problematic here, since the formula for calculating the variable values contains the square root of the sum. If there is a fraction not reducible to decimal, the solution becomes so complicated that it becomes almost impossible to calculate the exact answer without a calculator.
So, each way of representing fractions has its advantages in the corresponding context.
Recording Forms
There are two ways to record actions with ordinary fractions: through a horizontal line, into two “tiers”, and through an oblique line (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 across the cells contributes to the development of mindfulness in the calculations and the implementation of transformations. When writing to a string, you can inadvertently confuse 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 line using the function in Microsoft Word 2010 and later. The fact is that in these versions of software there is an option called "formula". It displays a rectangular transformable field, within the framework of which you can combine any mathematical symbols, make up two- and four-story fractions. In the denominator and numerator, you can use brackets, operation signs. As a result, you will be able to record any joint actions with ordinary and decimal fractions in the traditional form, that is, in the way they are taught to do in school.
If you use the standard text editor Notepad, then all fractional expressions will need to be written through the slash. Unfortunately, there is no other way.
Conclusion
So we examined 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 prescriptions and the count from one to ten.
It is important to understand that fractions are used in everyday life everywhere. 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 well - especially since it is not so complicated.