# Ratio Proportion

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## Ratio

Ratio of two quantities of the same kind P and Q is written as P/Q or P:Q.
P/Q is known as fractional form whereas P:Q is known as linear form.
In a ratio first term is known as the antecedent and the second term is known as the consequent.
Ratio does not has any units. As ratio is just a number so multiplying both the antecedent and the consequent by the same number does not change the fraction.

## Proportion

Proportion is the equality of two ratios. If A, B, C, D are in a proportion then A:B :: C:D.
A:B :: C:D can even be written as A:B = C:D.
A and D are known as extremes terms whereas B and C are mean terms.
A:B = C:D and AD = BC i.e. Product of extremes = Product of means

### Fourth Proportional

D is the fourth proportional of A, B, C if A:B = C:D.

Example: Find the 4th proportional to 3, 7 and 9.
Solution:

3:7::9:x
3/7 = 9/x
x = 21

### Third Proportional

C is the third proportional of two numbers A, B if A:B = B:C.

Example: Find the 3rd proportional to 16 and 32.
Solution:

16:32::32:x
16/32 = 32/x
x = 64

### Mean Proportion

C is the mean proportion of two numbers A, B if C2 = AB.

Example: Find the mean proportional to 4 and 25.
Solution:

4:x::x:25
x2 = 100
x = 10

### Direct Proportion:

If P is directly proportional to Q then P = KQ where K is a constant of proportionality.

### Inverse Proportion:

If P is inversely proportional to Q then PQ = K where K is a constant of proportionality.
If a/b = c/d then
1)Invertendo, b/a = d/c
2)Alternendo, a/c = b/d
3)Componendo, (a + b)/b = (c + d)/d
4)Dividendo, (a - b)/b = (c - d)/d
5)Componendo, (a + b)/b = (c + d)/d
6)Componendo and Dividendo, (a + b)/(a - b) = (c + d)/(c - d)
7)If a/b = c/d = e/f = ... = k, then (a + c + e +...)/(b + d + f +...) = k
(Pa + Qc + Re +...)/(Pb + Qd + Rf +...) = k where P, Q, R,... are constants.