Proportions

If two ratios are equal, we say that they are in proportion. The symbol ‘: :’ or ‘=’ is used to equate the two ratios.

E.g. a : b :: b : c OR a : b :: c : d

Normal Proportion

Let us consider Normal Proportion of Four quantities.

Four quantities are said to be in normal proportion (or just proportion), if the ratio of the first and the second quantities is equal to the ratio of the third and the fourth quantities.

a : b = c : d or a : b : : c : d (Normal Proportion)

Here first and fourth terms are known as extreme/outer terms (a and d). Second and third terms are known as middle/mean terms (b and c).

So, a/b = c/d
Or ad = bc
That is, Product of extremes (ad) = Product of means (bc)

Properties of Normal Proportion

If four quantities a, b, c and d are in normal proportion then:

Invertendo

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

Alternando

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

Componendo

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

Dividendo

If a/b = c/d, then (a − b)/b = (c − d)/d

Componendo and dividendo

If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d)
The converse of this is also true - (a + b) / (a - b) = (c + d)/(c - d), then we canconclude that a/b = c/d.

A Special Property

a/b = c/d = e/f……. = a$k_1$ + c$k_2$ + e$k_3$ +…../b$k_1$ + d$k_2$ + f$k_3$ +…..

(Where $k_1$, $k_2$, $k_3$… are real numbers, such that all of them cannot be zero simultaneously)

A special relation:
When $k_1$ = $k_2$ = $k_3$… = 1
Then, a/b = c/d = e/f……. = a + c + e +…../b + d + f +…..

Continued Proportion

Continued Proportion of three quantities

Three quantities are said to be in continued proportion, if the ratio of the first and the second quantities is equal to the ratio of the second and the third quantities.

a : b = b : c or a : b : : b : c

Here first and third terms are known as extreme/outer terms (a and c). Second term is known as middle/mean term (b).

So, a/b = b/c
Or $b^2$ = ac.
(b is said to be the mean proportional of a and c.)

a, b, c are in Geometric Progression, e.g. 1, 4, 16 ($4^2$ = 1 × 16).

Continued Proportion of four quantities

Four quantities are said to be in continued proportion, if: a : b = b : c = c : d

So, a/b = b/c = c/d
Or $b^2$ = ac and $c^2$ = bd

a, b, c, d are in Geometric Progression, e.g. 1, 4, 16, 64 ($4^2$ = 1 × 16; $16^2$ = 4 × 64).

We can find a : d by the multiplying these three ratios.
a/d = a/b × b/c × c/d

Four Proportions

If a/b = b/c, then

• a = $\frac{b^2}{c}$ (first proportional)

• b = √ac (second proportional, or mean proportional, or geometric mean)

• c = $\frac{b^2}{a}$ (third proportional)

If a/b = c/d, then

• d = bc/a (fourth proportional)

Q. By how much is the fourth proportional of 11, 121 and 36 more than the third proportional of 6 and 24?

(a) 300   (b) 396    (c) 96   (d) 192

Explanation:

Q. If M is the mean proportional between 18 and 8, and N is the mean proportional between 9 and 1, then what is the mean proportional between M and N?

(a) 16   (b) 6    (c) 64   (d) 8

Explanation: