Ratio And Proportion

 

 Proportion :

Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.Proportions are denoted by the symbol  ‘::’ or ‘=’.

The proportion can be classified into the following categories, such as:

  • Direct Proportion
  • Inverse Proportion
  • Continued Proportion

Direct Proportion

The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b.

Inverse Proportion

The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).

Continued Proportion

Consider two ratios to be a: b and c: d.

Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.

For the given ratio, the LCM of b & c will be bc.

Thus, multiplying the first ratio by c and the second ratio by b, we have

First ratio- ca:bc

Second ratio- bc: bd

Thus, the continued proportion can be written in the form of ca: bc: bd

Ratio Meaning

 The comparison of two quantities by the method of division is very efficient. We can say that the comparison or simplified form of two quantities of the same kind is referred to as a ratio. This relation gives us how many times one quantity is equal to the other quantity.

The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. The sign used to denote a ratio is ‘:’.

 The ratio can be represented in three different forms, such as:

  • a to b
  • a : b
  • a/b

Ratio Formula

Assume that, we have two quantities (or two numbers or two entities) and we have to find the ratio of these two, then the formula for ratio is defined as;

a: b ⇒ a/b

where a and b could be any two quantities.

Here, “a” is called the first term or antecedent, and “b” is called the second term or consequent.

Proportion Formula

Now, let us assume that, in proportion, the two ratios are a:b & c:d. The two terms ‘b’ and ‘c’ are called ‘means or mean term,’ whereas the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’

a/b = c/d or  a : b :: c : d


Important Properties of Proportion

The following are the important properties of proportion:

  • Addendo – If a : b = c : d, then a + c : b + d
  • Subtrahendo – If a : b = c : d, then a – c : b – d
  • Dividendo – If a : b = c : d, then a – b : b = c – d : d
  • Componendo – If a : b = c : d, then a + b : b = c+d : d
  • Alternendo – If a : b = c : d, then a : c = b: d
  • Invertendo – If a : b = c : d, then b : a = d : c
  • Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

Properties of Ratio

  • Ratio of nimbers a and b is written as a:b or a/b is called as the predecessor(forst term) and b is called as successor (second term).
  • In the ratio of two numbers, if the second term is 100 then it is known as a percentage.
  • The ratio remains unchanged , if its term::s are multiplied or divided by non-zero number.           eg.. 3:4 = 6:8 =9:12,  similiarly 2:3:5  = 8:12:20 If k is a non-zero number, then a:b = ak :bk        a:b:c  = ak:bk:ck.
  • The quantities taken in the ratio must be expressed in the same unit.
  • The ratio of two quantities is unitless. for example   The ratio of 2 kg ang 300 g is not 2:300, but it is 2000:300 as  (2 kg = 2000 gm)  i.e. 20:3

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