geometric ratio
The geometric ratio is the ratio of two numbers. The first number is called history, and the second, consistent.
The reason can be presented in two ways:
· a: b: b where a is the antecedent the consequent.
·: in the form of broken or fraction where the numerator would be the antecedent and consequent denominator.
And it reads as follows: it is.
Examples:
The ratio of 6 and 3 is 2 (6: 3 = 2). It reads 6 is 3.
The reason 18 and 6 is 3 (18: 6 = 3). It reads 18 is to 6.
The ratio of 5 and 10 is 0.5 (5: 10 = 0.5). Reads 5 is 10.
Proportionality.
proportions is said that two quantities are proportional or out of proportion when the growth of one affects the growth of another.
If the relationship is positive, it grows and grows one another, decreasing one and decreases the other, we speak of direct proportionality (space and time, purchases and expenses, etc.). Otherwise, we have the inverse proportionality, in which the growth of a magnitude implies the decrease of the other, and vice versa (workers and time taken to do something, myopia and light, food and hunger, etc.).
A proportion is an equality between two reasons. These two reasons should be proportional.
The above is a ratio, which equals two reasons as a fraction.
The proportions are 4 terms: the first (numerator of the first fraction) and fourth (denominator of the second) are called extreme, and the second (denominator of the first fraction) and third (numerator of the second) are called media.
In the above example, the media (in red) and end (in blue) would be these:
For a proportion is correct, you must meet the media product is equal to the product of extremes. This rule is called the Fundamental Property of Proportions, and serves to solve and check if the proportions are correct, as discussed below.
In the example, this property is true, since 12 × 32 = 64 × 6 = 384.
The problem is usually reduced proportions to find an unfamiliar term that has to be a fraction fellow what his counterpart in the other fraction (if the No stranger is a numerator, its counterpart is the other number, if No stranger is the denominator, the counterpart must also be) is to their respective co fraction:
In above ratio, we have to find the word?, which must be a 4 which is 7 to 2.
How to solve?
If we divide 7 by 2, we get 3.5. As the two factions must be proportional, the result of the first fraction should be equal to the second. So? between 4 must also give 3.5. But
? is the dividend (antecedent) of the division? 4. Therefore, it must satisfy the ratio property, which says that
D = (d × c) + r. Applying this property in our case, we have
? = (4 x 3.5) + 0, and therefore
? = 14.
So in the end, we obtain the following equation:
How can I check?
But how do you know if it is well done? At first view is that it's okay, because the two are equivalent fractions (numerator and denominator of the first fraction was divided by two and you get the second fraction). And when two fractions are equivalent are also proportional.
However, when the proportionality is not as clear can be checked by multiplying the means by ends. Resolution by equations
This method of means and ends is also used to find unfamiliar terms, but then must use equations.
The proportion with we have worked it would solve this (the x replaces the term?):
2 • x = 7.4 =>
2x = 28 =>
x = 28 / 2 => x = 14
proportions Applications The proportions have multiple uses and applications. The most important are those of the rule of three, the percentages and the mean proportional.
Rule of Three simple
The simple rule of thumb is a ratio, ie equality of two reasons.
The simple rule of thumb can be direct or inverse, depending on how is the relationship of proportionality between the quantities that comprise them.
. If direct, it resolves to a normal rate:
Example: If twelve children eat 7 cakes, pies eat 6 How many children?
12 (2 °) -> 7 (3 °)
6 (1 °) -> x (?)
should multiply the opposite of the unknown (1) with which it is above or below the X (the unknown) (2). The product of these two numbers should be divided among the remaining number (3) and thus obtains x.
x = (6 × 7) / 12 = 3.5
. If reverse is exchanged half with one end and proceed as in any proportion.
Example: If six children make a mural in 24 minutes, how long a child alone?
6 -> 24
1 -> x
changed half a point:
6 -> x
1 -> 24
and operate as in any proportion.
x = (6 × 24) / 1 = 144 minutes
Percentage
To calculate a percentage or percent (%) should make a rule of three direct. Of the four quantities that are, always know of three: the amount you want to transform as a percentage, the total that is compared to the total with which compares the percent (100).
Example:
In a class but today there are 34 children have missed 13. What percentage of the total has been absent from school?
13 -> 34
X -> 100
13 is to 34 what X (the percentage) is 100. This would be the right approach to these problems, in which the unknown is the percentage of one hundred.
Al resolve it as a normal rate, we should
X = (100 × 13) / 34 = 38.23% approximately. Media
proportional.
The mean proportional of two numbers is the root of the product of the means of a proportion whose ends are the two numbers.
< Ejemplo:
What is the median age of 3 and 18?
Applying the Fundamental Property of Ratios (see above), for which the product Media is the product of extremes, we obtain:
3.18 = x · x =>
54 = x ² =>
x = 54 = 3.748 ...
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