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Created it, 05/10/15
Update it, 05/12/24
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MATHEMATICS “3rd Part”
5. 1 - GENERAL
An unspecified object can be divided (or split) into a certain number of pieces.
For example, a wafer can be cut out in a certain number of shares. Each share is a fraction of wafer.
The meter was divided into a certain number of “pieces”. These “pieces” are fractions of the meter: the decimetre, the centimetre, the millimetre.
If we cross, or split, the wafer in 8 equal shares and that we take one of them, it will be said that we took a wafer eighth, and this fraction of wafer will be written in figure : 1 / 8
This notation points out division : numerator - denominator.
The figure above the bar of fraction is called numerator, that which is below is called denominator.
The denominator indicates in how much equal fractions one divided, split a unit. The numerator indicates how much fractions which one considers.
A rule measures 30 centimetres. That wants to say that one divided the meter into 100 equal parts (one obtained centimetres) and that the length of the rule is equal to 30 of these equal parts.
The numerator and the denominator are the terms of the fraction.
We now will state the various rules which govern the operations on the fractions. We invite you to retain them (in particular for the children and others).
Regulate : To multiply 2 or several fractions between them, one multiplies the numerators between them, and the denominators between them.
Example :
Regulate : To divide two fractions between them, one multiplies the divisor fraction by the dividend fraction reversed.
Example :
Note: If the numerator and the denominator of the dividend fraction are divisible respectively by the numerator and the denominator of the divisor fraction, one divides the numerators between them and the denominators between them.
Example :
5. 3. 3. - ADDITION
Regulate : To add several fractions between them, it is necessary :
Example :
Reduction with the same denominator: To reduce two fractions to the same denominator, one multiplies the two terms of the first by the denominator of the second and the two terms of the second by the denominator of the first.
Example :
If there is more than two fractions, one can proceed by successive stages.
Example :
1 - The first two following fractions are considered :
2 - One considers now this new fraction with 3rd.
and, possibly, so on.
Note: It will be necessary to simplify the fractions each time that is possible, i.e. front, during and after calculations.
One simplifies a fraction by dividing each one of his two terms by the same number. The new fraction obtained is equal to the first.
Example :
It is quite obvious that one is not obliged to pass by all intermediate calculations. It is necessary to try to find highest common factor of the two numbers.
Example :
It is seen immediately that 27 is divisible by 3 like 36. But one also realizes that these two numbers are divisible by 9 ; from where :
We have just seen that one does not change the value of a fraction by dividing his two terms by the same number. This is also true for the multiplication of these two terms, but always by the same number not no one.
Example :
From where the rule : One does not change the value of a fraction while multiplying or by dividing each one of his terms by the same number not no one.
2 - One can, to simplify the writing during the reduction to the same denominator, not to pose all calculations as one did but to carry out the operations mentally.
In the example which follows, the arrows indicate the products to be carried out.
Example :
In addition, since it is known that the denominator will be the same one for the two fractions, it can register it only only once, which becomes:
Regulate : To cut off two fractions between them, it is necessary :
1) To reduce to the same denominator ;
2) To withdraw the numerators ;
3) To keep the common denominator.
In the example which follows we go, to reduce the writing, to proceed as it has just been known as in the preceding remark :
Regulate : To raise a fraction with a power, one raises each term of the fraction to the power (one should say “exhibitor”).
Example :
Regulate : the root of a fraction is equal to the root of each one of its terms.
Example :
5.
4. - OPERATIONS
ON THE FRACTIONS AND THE INTEGERS
5. 4. 1 - Multiplication
regulate : To multiply a fraction by a number (or a number by a fraction), one multiplies the numerator by this number or one divides the denominator by this number.
5. 4. 2. - TRANSFORMATION Of a FRACTION INTO INTEGER (OR DECIMAL) AND CONVERSELY
Like he was said, a fraction appears himself as a quotient. If this quotient is carried out, one obtains an integer or decimal equal to the fraction.
example :
Note : If the result of the quotient is an integer, the fraction is also called report/ratio. If on the contrary, we have an integer, or decimal, to convert into fraction we operate as follows :
1 - Integer : it is necessary to multiply this number by the desired denominator.
Examples: One wishes to transform figure 3 into a certain number of thirds, quarter, fifth, etc…
2. - Decimal Number: It is necessary to multiply this number by the power of 10 which will transform the decimal number into integer.
Examples :
The operations above are dedicated to the children and to those which they want to learn thus that below.
Definition : A fractional number is an integer followed by a fraction.
One can convert these fractional numbers of decimal numbers.
Examples :
One can also convert the fractional numbers into fraction.
Examples :
6. - REPORTS/RATIOS
Definition : One calls ratio of two numbers “a” and “b” the exact quotient of these two numbers.
It is what we had seen by examining the fractions.
One also defines the ratio of two numbers as being the number per which it is necessary to multiply the second to obtain the first.
Thus of a / b = c, one can write c x b = a. That is now obvious. Indeed, we recognize the manner of carrying out the proof of division.
As a report/ratio is expressed in the form of a fraction, the rules examined about the fractions apply all to the reports/ratios, and in particular the sums, products and quotients.
A report/ratio is thus fixed with the same rules and likely of same simplifications as a fraction.
In the study of the properties which follows, we will recall simply those which we already saw with the fractions.
First property : One does not change the value of a report/ratio while multiplying or by dividing his two terms by the same number :
Second property : To add two or several reports/ratios, one reduces them to the same denominator, then one adds the numerators and one preserves the common denominator.
Third property : To multiply between them two or several reports/ratios, one multiplies the numerators between them and the denominators between them :
Fourth property : To divide two relationship between them, one multiplies the dividend report/ratio by the reversed dividing report/ratio :
Fifth property : In a succession of equal reports/ratios, the report/ratio obtained, by taking as numerator the sum of the numerators, and as denominator the sum of the denominators, is a report/ratio equal to the precedents.
Example :
Sixth property : In a succession of equal reports/ratios, the report/ratio obtained by taking as numerator the difference of the numerators, and as denominator the difference of the denominators, is a report/ratio equal to the precedents.
7. - PROPORTIONS
Definitions : One calls proportion the equality of two reports/ratios.
First property : In a proportion, the product of the extremes is equal to the product of the means. That is to say :
We see immediately that : 2 x 6 = 3 x 4
And more generally :
Second property : In a proportion, one can permute :
that is to say extremes ;
that is to say means ;
or extremes and means.
Let us take again the literal equality found previously :
Let us divide the two members of this equality by the product ab :
Let us divide this time the two members by the product cd :
Let us divide finally this equality by the product ac :
Third property : In a proportion, one can replace each report/ratio by his reverse :
Definition : One calls fourth proportional to the three numbers a, b and c number x such as :
By making the equality between the product of the extremes and the means, one finds : ax = bc
And, by dividing the two terms by a :
Definition : It is said that number x is average proportional between a and b. If :
Let us make produces extremes = produced means :
Note :
For including/understanding the lessons of electronic well, we will continue the maths in order to know and to know the chart.
With the graphs, we have all the values under the eyes and calculation is reduced to a simple observation, supplemented at most by some graphic operations.
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