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Ratio and Proportion

Ratio(:)-
Ratio is a tool for comparing two quantities.Ratio of a and b is written as a:b or a/b where is antecedent and b is consequent.Ratio is a pure number it does not have a unit .the two numbers used in a ratio are known as terms.

Inverse ratio of a:b will be b:a

Compound ratio of a:b:c:d and e:f is ace:bdf

Duplicate ratio of a:b is a^2:b^2

Subduplicate ratio of a:b is √a:√b

If ratio of two quantities are given by
a:b:c

then actual quantity is given by
ak, bk, ck

Where k is constant

Eg: if there are 20 men and 30 boys then we can compare the no. of men and women as follows:

no. of men/no. of women = 20/30 = ⅔
no. of mens :no. of boys=2:3
no. of men’s = ⅔ x no. of womens
no. of women’s = ⅔ x no. of mens

Proportion( :: )-
Equality two ratio is known as proportionality.
If two ratios a:b and c:d are equal , then it can be written as a:b::c:d. then a, b, c, d are in proportion.
a:b::c:d=>ad-bc

If ratio b/w A,B and C is p:q:r
A=pk ,
B=qk ,
C= rk where k is constant

If a:b :: b:c then c is called 3rd proportion of a.
Rules: If a:b::c:d then
(a+b)/b = (c+d)/d (Componendo )
(a-b)/b = (c-d)/d (Dividendo)
(a+b)/(a-b)=(c+d)/(c-d)(Componendo & Dividendo )
Mean Proportion of a and b=√ab

If a is divided in the ratio x:y , then first part = x /(x+y)*a and
Second part = y/(x+y)* b

If a is divided in the ratio x:y:z ,then first part =x/ (x+y+z)*a ,
Second part = y / (x+y+z )*b and Third part =z/(x+y+z)* c

Eg: The ratio of the current ages of Anita and Sunita is 4:3. eight years hence, the ratio of their ages will be 6:5 then their current ages are?

Explanation: Let this ages of Anita and Sunita be 4x and 3x then
thier ages hence will be 4x+8 and 3x+8 hence of their ages 8 years hence will be (4x+8)/(3x+8) = 6/5

Solving we get 20x+40=18x+48
2x=8
x=4
Hence their current ages will be 16 and 12 years respectively