Ratio and Proportion

Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. 

Ratio Meaning

In certain situations, 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. In simple words, the ratio is the number that can be used to express one quantity as a fraction of the other ones.

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 ‘:’.

A ratio can be written as a fraction, say 5/9.

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

  • a to b
  • a : b
  • a/b

Remember: 

  • The ratio should exist between the quantities of the same kind
  • While comparing two things, the units should be similar
  • There should be significant order of terms
  • The comparison of two ratios can be performed, if the ratios are equivalent like the fractions

Definition of 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.

For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.

Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be in proportion. In simple words, it compares two ratios. 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 and Proportion Formula

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.

Example: In ratio 4:9, is represented by 4/9, where 4 is antecedent and 9 is consequent.

If we multiply and divide each term of ratio by the same number (non-zero), it doesn’t affect the ratio.

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

Difference Between Ratio and Proportion

To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.

S.No Ratio Proportion
1 The ratio is used to compare the size of two things with the same unit The proportion is used to express the relation of two ratios
2 It is expressed using a colon (:), slash (/) It is expressed using the double colon (::) or equal to the symbol (=)
3 It is an expression It is an equation
4 Keyword to identify ratio in a problem is “to every” Keyword to identify proportion in a problem is “out of”

Fourth, Third and Mean Proportional

If a : b = c : d, then:

  • d is called the fourth proportional to a, b, c.
  • c is called the third proportion to a and b.
  • Mean proportional between a and b is √(ab).

Comparison of Ratios

If (a:b)>(c:d) = (a/b>c/d)

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios

If a:b is a ratio, then:

  • a2:b2 is a duplicate ratio
  • √a:√b is the sub-duplicate ratio
  • a3:b3 is a triplicate ratio

Ratio and Proportion Tricks

Let us learn here some rules and tricks to solve problems based on ratio and proportion topics.

  • If u/v = x/y, then uy = vx
  • If u/v = x/y, then u/x = v/y
  • If u/v = x/y, then v/u = y/x
  • If u/v = x/y, then (u+v)/v = (x+y)/y
  • If u/v = x/y, then (u-v)/v = (x-y)/y
  • If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as componendo -Dividendo Rule
  • If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c