It is obvious that one does not have to go through all the intermediate calculations. One must try to find the greatest common divisor of the two numbers.

Example :

 

We see immediately that 27 is divisible by 3 and 36. But we also see that these two numbers are divisible by 9 ; from where :

 

We have just seen that we do not change the value of a fraction by dividing its two terms by the same number. This is also true for the multiplication of these two terms, but always by the same non-zero number.

Example :

From the rule : We do not change the value of a fraction by multiplying or dividing each of its terms by the same non-zero number.

2 - One can, to simplify the writing during the reduction to the same denominator, not to pose all the calculations as one did it but to carry out the operations mentally.

In the following example, the arrows indicate the products to be performed.

Example :

On the other hand, since we know that the denominator will be the same for both fractions, we can register it only once, which becomes :

5. 3. 4. - SUBSTRACTION

Rule : To remove two fractions from each other, you must :

1) Reduce them to the same denominator ;

2) Reduce them to the same denominator ;

3) Keep the common denominator.

In the following example we go, to lighten the writing, proceed as just said in the previous remark :

5. 3. 5. - POWERS

Rule : To raise a fraction to a power, we raise each term of the fraction to the power (we should say "exponent").

Example :

5. 3. 6. - ROOTS

Rule : the root of a fraction is equal to the root of each of its terms.

Example :

 5. 4. - OPERATIONS ON FRACTIONS AND ENTIRE NUMBERS

5. 4. 1 - Multiplication

rule : To multiply a fraction by a number (or a number by a fraction), either multiply the numerator by that number or divide the denominator by that number.

5. 4. 2. - TRANSFORMATION OF A FRACTION INTO ENTIRE NUMBER (OR DECIMAL) AND INVERSION

As has been said, a fraction is in the form of a quotient. If we perform this quotient, we obtain an integer or decimal equal to the fraction.

example :

Remark : In the case where the result of the quotient is an integer, the fraction is also called ratio. If instead, we have an integer, or decimal, to convert to a fraction we will operate as follows :

1 - Integer number : multiply this number by the desired denominator.

Examples : We want to transform the number 3 into a number of thirds, quarter, fifth, etc. ...

2. - Decimal number : Multiply this number by the power of 10 which will transform the decimal number into Integer.

Examples : 

5. 5. - FRACTIONAL NUMBER

The above operations are dedicated to children and those they want to learn as well as adults as below.

Definition : A fractional number is an integer followed by a fraction.

These fractional numbers can be converted to decimal numbers.

Examples :

Fractional numbers can also be converted to a fraction.

Examples :

Summary :

  6. - REPORTS

Definition : The ratio of two numbers "a" and "b" is the exact quotient of these two numbers..

This we had seen by examining the fractions.

We also define the ratio of two numbers as the number by which we must multiply the second to obtain the first.

Thus from a / b = c, we can write c x b = a. This is now obvious. Indeed, we recognize the manner of performing the proof of division..

6. 1. - PROPERTIES OF REPORTS

As a report is expressed as a fraction, the rules examined about fractions all apply to reports, especially sums, products, and quotients.

A report is therefore subject to the same rules and subject to the same simplifications as a fraction.

In the following study of properties, we will simply recall those we have already seen with fractions.

First property : We do not change the value of a report by multiplying or dividing its two terms by the same number :

Second property : To add two or more ratios, we reduce them to the same denominator, then add the numerators and keep the common denominator.

Third property : To multiply between them two or more ratios, we multiply the numerators between them and the denominators between them :

Fourth property : To divide two relations between them, we multiply the dividend ratio by the inverted divisor ratio :

Fifth property : In a series of equal ratios, the ratio obtained, taking as numerator the sum of the numerators, and as the denominator the sum of the denominators, is a ratio equal to the previous ones.

Example :

Sixth property : In a series of equal ratios, the ratio obtained by taking as numerator the difference of the numerators, and as the denominator the difference of the denominators, is a ratio equal to the preceding ones.

7. - PROPORTIONS

Definitions : The ratio is the equality of two ratios.

• - a, b, c and d are the terms of the proportion ;

• - a and d are the extreme terms, more simply called ("extremes") ;

• - b and c are the intermediate terms, called "means".

7. 1. - PROPORTIONS PROPERTIES

First property : In a proportion, the product of the extremes is equal to the product of the means. Is :

We immediately see that : 2 X 6 = 3 X 4

And more generally :

Second property : In a proportion, we can switch :

      either the extremes ;

      the means ;

      or the extremes and the means.

Let's take again the literal equality found previously :

Let's divide the two members of this equality by the product ab :

Let's divide both members by the product cd :

Let's finally divide this equality by the product ac :

 

Third property : In a proportion, we can replace each report by its inverse :

7. 2. - FOURTH PROPORTIONAL

Definition : Four is called proportional to the three numbers a, b and c the number x such that :

By doing the equality between the product of extremes and means, we find : ax = bc

And, dividing the two terms by a :

7. 3. - AVERAGE PROPORTIONAL

Definition : It is said that the number x is proportional way between a and b. If :

Let's produce extremes = product means :

Remarks :

• 1) a and b are both positive or both negative.

• 2) The sign ± reads "more or less". It is necessary in our example because a square always has two opposite square roots. Indeed, (+2)² = 4 and (-2)² = 4 (Application of the sign rule).

7. 4. - SUMMARY OF PROPERTIES OF PROPORTION REPORTS

To fully understand the electronic lessons, we will continue the maths to understand and know the graphical representation.

With graphics, we have all the values under the eyes and the calculation is reduced to a simple observation, supplemented at most by some graphical operations.

---

Nombre de pages vues, à partir de cette date : le 27 Décembre 2019

Mon audience Xiti

---

Send an email to Corporate Webmaster for any questions or comments about this Web Site.

Web Site Version : 11. 5. 12 - Web Site optimization 1280 x 1024 pixels - Faculty of Nanterre - Last modification : JANUARY 02, 2020.

This Web Site was Created on, 12 JUNE 2019 and has Remodeled, in JANUARY 2020.

---