unit 1 ratio & proportion

Ratio:

A ratio is simply a fraction.   The following notations all express the ratio of x to y
=>  $x:y$ , $x÷y$ , or $\frac{x}{y}$.
In the ratio x : y, we call a as the first term or antecedent and b, the second term or consequent.
Writing two numbers as a ratio provides a convenient way to compare their sizes.   For example, since $\frac{3}{\pi }<1$, we know that 3 is less than π.
A ratio compares two numbers.  Just as you cannot compare apples and oranges, so the numbers you are comparing must have the same units. 
For example, you cannot form the ratio of 2 feet to 4 meters because the two numbers are expressed in different units—feet vs. meters.

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Example 1: What is the ratio of 2 feet to 4 yards?
(A)  1 : 2        (B)  1 : 8          (C)  1 : 7        (D)  1 : 6            (E)  1 : 5       
The ratio cannot be formed until the numbers are expressed in the same units.  Let’s turn the yards into feet.
Since there are 3 feet in a yard, 4 yards = 4 * 3 feet = 12 feet . 
Forming the ratio yields $\frac{\text{2 feet}}{\text{12 feet}}=\frac{1}{6}$ or $1:6$
The answer is (D).
Note. Taking the reciprocal of a fraction usually changes its size.  For example,
$\frac{3}{4}\ne \frac{4}{3}$
So order is important in a ratio=> 3:4 ≠  4:3.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Ex. 4 : 5 = 8 : 10 = 12 : 15.
Also, 4 : 6 = 2 : 3

Proportion:

The equality of two ratios (fractions) is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
$$\text{Product of means = Product of extremes}$$
Thus, $$a:b::c:d\iff \left(b*c\right)=\left(a*d\right)$$

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Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

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Third Proportional:

a : b = c : d, then c is called the third proportional to a and b.

Mean Proportional:

Mean proportional between a and b is ab.

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Comparison of Ratios:

We say that $\left(a:b\right)>\left(c:d\right)\iff \frac{a}{b}>\frac{c}{d}$

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Compounded Ratio:

The compounded ratio of the ratios: $\left(a:b\right),\left(c:d\right),\left(e:f\right)is\left(ace:bdf\right)$

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Duplicate Ratios:

Duplicate ratio of $\left(a:b\right)$ is $\left(a^2:b^2\right)$
Sub-duplicate ratio of $\left(a:b\right)$ is $\left(a^{\frac{1}{2}}:b^{\frac{1}{2}}\right)$
Triplicate ratio of $\left(a:b\right)$ is $\left(a^3:b^3\right)$
Sub-triplicate ratio of $\left(a:b\right)$ is $\left(a^{\frac{1}{3}}:b^{\frac{1}{3}}\right)$
If $\frac{a}{b}=\frac{c}{d}$ then, $\frac{a+b}{a-b}=\frac{c+d}{c–d}$ [Componendo and Dividendo]

Variations:

We say that x is directly proportional to y,
if x = ky for some constant k and we write, $x\propto y$
We say that x is inversely proportional to y,
if xy = k for some constant k and we write, x∝1