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In certain situations, comparison by division makes it better than doing comparison by taking the difference. The comparison by division is called the ratio. In this chapter we will introduce the concept of ratios & proportions.

Ratios —                                                                   

A ratio is a fraction or comparison of two or more quantities of same type by division. i.e, If a and b are two quantities of the same kind then the fraction a/b is called the ratio a is to b.

Here, a is called the first term (or antecedent) and b is called the second term (or consequent).

Example: Suppose a class has 12 Girls and 20 Boys. Then ratio of girls to boys is 12/20 => 3/5

Ratios & Proportions   

Properties of Ratios

1. Ratio a/b has no unit and can be return as a:b
2. If both the terms of a ratio are multiplied or divided by the same non-zero number, the ratio remains unchanged.
3. The ratio must always be expressed in its lowest terms.
4. Ratios a:b and b:a cannot be equal unless a=b
5. Ratio is taken only between positive quantities.

Composition of Ratios

 

Compound Ratio:

When two or more ratios are multiplied together, then the resulting ratio is called as compounded ratio.

     Example: if a/b and c/d are two ratios, then ac/bd is their compounded ratio.

Duplicate Ratio:

When a ratio is compounded with itself, then the resulting ratio is called the duplicate ratio.                                              So, a² : b² is duplicate ratio of a:b 

     Example: Duplicate ratio of 2:4 is 2² : 4² => 4 : 16

Triplicate Ratio: 

If a ratio is compounded three times with itself, then the resulting ratio is called the triplicate ratio. So, a³ : b³ is triplicate ratio of a:b

     Example: Triplicate ratio of 2:3 is 2³ : 3³ => 8 : 27

Subduplicate Ratio: 

If square root is applied on a ratio, then the resulting ratio is called the subduplicate ratio. So, if a : b  is a ratio, then the subduplicate ratio is √a : √b

     Example: subduplicate ratio of 4:16 is √4 : √16 => 2 : 4

Subtriplicate Ratio: 

If  cube root is applied on a ratio, then the resulting ratio is called the Subtriplicate ratio. So, if a : b is a ratio, then its subtriplicate ratio is ³√a : ³√b  

     Example: Subtriplicate ratio of 8:27 is ³√8 : ³√27   => 2 : 3

Reciprocal Ratio: 

If  a : b is a ratio, then its reciprocal ratio is 1/a : 1/b 

     Example: reciprocal ratio of 3 : 4 is 1/3 : 1/4 => 4:3

 

Proportions —

An equality of two ratios is called a Proportion. Like if a, b, c, and d are four quantities of same kind, then a:b :: c:d is called as the proportion.

which means a:b = c:d  or  a/b = c/d  or  ad = bc

Notes

1. In a : b = c : d,
   1. a, b, c and d are called the terms of the proportion; where a = first term, b = second term, c = third term and d = fourth term.
   2. ‘a’ and ‘d’ are called extremes (end-terms) whereas ‘b’ and ‘c’ are means (middle terms).
2. a : b = c : d  ⇒  a/b = c/d  ⇒  a×d = b×c  ⇒  Product of extremes = product of means.
3. In a : b = c : d, the fourth term ‘d’ is fourth proportional.
4. In a : b = c : d, quantities a and b must be of the same kind with the same units, whereas; c and d may separately be of the same kind with the same units.

Properties of Proportions

Invertendo 

If  a : b = c : d then b : a :: d : c or if a, b, c, d are in proportion, then their reciprocals are in proportion as well.

a/b = c/d  ⇒ b/a = d/c

Alternendo

If  a : b = c : d, then a : c :: b : d

a/b = c/d  ⇒ a/c = b/d

Componendo (Adding the denominator)

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

or a/b = c/d  ⇒ (a + b)/b = (c + d)/d

Dividendo (Subtracting the denominator)

If  a : b = c : d, then a : (a – b) ::  c : (c – d)

or a/b = c/d  ⇒ a/(a – b) = c/(c – d)

Componendo  and Dividendo 

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

or  a/b = c/d  ⇒ (a + b)/(a – b) = (c + d)/(c – d)

 

So this is all about Ratios & Proportions. Catch up all related examples here