Properties Of Ratio

A ratio is a comparison of two similar quantities, hence it does not have any units. It will remain unchanged if both the antecedent and the consequent are multiplied and divided by the same non-zero multiplier. The properties of ratio are essential to solve ratio and proportion problems in CAT quantitative aptitude.

  • Ratios of the squares of p & q = p2 : q2 (Duplicate Ratio)
  • Ratios of the square roots of p & q = √p : √q (Sub Duplicate Ratio)

Some important properties of ratio and proportion are componendo , invertendo , dividendo, alternendo, componendo-dividendo , convertendo, and equivalent ratio. These properties of ratio are explained below with examples.

Componendo

Suppose there are 4 numbers a, b, c, d. if a : b = c : d then (a + b) : b :: (c + d) : d.

If a : b=c : d then (a+b) : b=(c+d) : d

Example:

8 : 10 = 16 : 20

So, (8 + 10) : 10 = 18 : 10 = 18 : 10

= (8 + 10) : 10

Dividendo

If a/b=c/d then (a-b)/b=(c-d)/d

Example:

10 : 8 = 20 : 16

Therefore, (10 – 8) : 8 = 2 : 8 = (20 – 16) : 16 = 1 : 4

Componendo-Dividendo

If a : b :: c : d then (a + b) : (a – b) :: (c + d) : (c – d).

Example:

7 : 3 = 14 : 6

(7 + 3) : ( 7 – 3) = 10 : 4 = 5 : 2

Again, (14 + 6) : (14 – 6) = 20 : 8 = 5 : 2

Therefore, ( 7 + 3) : ( 7 – 3) = ( 14 + 6) : ( 14 – 6)

Invertendo

For 4 numbers a, b, c, d if a : b = c : d, then b : a = d : c; i.e., if 2 ratios are equal, then their inverse ratios are also equal.

If a : b :: c : d then b : a :: d : c.

Example:

6 : 10 = 9 : 15

So, 10 : 6 = 5 : 3 = 15 : 9

Alterando

If a : b :: c : d then a : c :: b : d

Example:

If 3 : 5 = 6 : 10 then 3 : 6 = 1 : 2 = 5 : 10

If 3 : 5 = 6 : 10 then 3 : 6 = 1 : 2 = 5 : 10

Continued Proportion

In this property of ratio and proportion a,b and c are said to be in continued proportion if a/b=b/c or b2=ac. Also b=√ac.

so, a : b : : b : c

This means that b is the geometric mean of a and c

Examples:

What is the mean proportional between 9 and 25?

Let x be the mean proportional between 9 and 25.⇒ x 2 = 9 x 25 ⇒ x 2 = 225 ⇒ x = 15

Sum rule

a/b=c/d=(a+c)/(b+d)

In general, if a/b=c/d=e/f=⋯..K (Some constant)

Then a/b=c/d=e/f= ………..K = (a+c+e+⋯)/(b+d+f+⋯..) i.e.

Each ratio = (Sum of numerators)/(Sum of Denominators)

Given that a/b,c/d,e/fare all unequal ratios, the value of (a+c+e+⋯)/(b+d+f+⋯) lies between the minimum and maximum of all these ratios

Power Rule

Given that a/b=c/d=e/f=⋯, then each ratio is equal

to [(map+ ncp+ …..)/(mbp+ ndp+ ….)](1/p) where m,n,p – all non zeroes.

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