A proportion is defined as a statement of equality between two given ratios. Proportions are often expressed as equations. For example, \(\frac{a}{b} = \frac{c}{d} \,or\, a : b = c : d\), is a proportion.
In other words, two ratios forming a proportion are equivalent ratios.
We can determine whether two ratios form a proportion or not by comparing the values of each ratio. If the values are the same then the ratios are in proportion, otherwise the ratios are not proportional.
For example, let’s check for the ratios 9 : 15 and 15 : 25.
9 : 15 = \(\frac{9÷3}{15÷3} = \frac{3}{5}\)
15 : 25 = \(\frac{15÷5}{25÷5} = \frac{3}{5}\)
\(\frac{3}{5} = \frac{3}{5}\), so the two ratios form a proportion, that is, 9:15 = 15:25.
Types of Proportion:
The proportion can be categorized into the following types:
- Direct Proportion
- Inverse Proportion
- Continued Proportion
(i) Direct Proportion:
In proportion, if two sets of given numerals are rising or falling in the same ratio, then the ratios are considered to be directly proportional to each other. That is we can say that direct proportion illustrates the relationship between two portions wherein the gains in one there is a growth in the other quantity too. Likewise, if one quantity drops, the other also decreases.
Therefore, if “x” and “y” are two quantities, then the direction proportion is composed as x ∝ y.
(ii) Inverse Proportion:
The inverse proportion as the name outlines are in contrast to the direct one; where the relationship between two quantities is defined such that growth in one leads to a decline in the other quantity. Likewise, if there is a drop in one portion, there is an expansion in the other portion.
Accordingly, the inverse proportion of two quantities, say “p” and “q” is represented by p ∝ (1/q).
(iii) Continued Proportion:
If we assume two ratios to be p: q and r: s and we are interested in determining the continued proportion for the given ratio. Then we transform the means to a single term/digit. That is we will find the LCM of means.
- For the provided ratio, the LCM of q & r will be qr.
- The next step is to multiply the first ratio by r and the second ratio by q as shown:
- First ratio- rp : rq
- Second ratio- rq : qs
- Thus, the continued proportion can be documented in the form of rp : qr : qs.
Properties of Proportion:
- Addendo – If a : b = c : d, then value of each ratio is a + c : b + d
- Subtrahendo – If a : b = c : d, then value of each ratio is 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
