**Proportion:**

Proportion is the equality of two fractions, that is a : b = c : d. This can be represented as a : b :: c : d,

Where a and d are called extremes and b and c are called the mean of the ratio.

**Fourth, Third and mean proportions:**

In a : b :: c : d,

d is the fourth proportion of a, b and c

and c is the third proportion of a and b.

The mean proportion of a and b is √ab

**Invertendo:**

If a : b = c : d then

b : a :: d : c or b/a = d/c

**Alterando:**

If a : b= c : d then

a : c :: b : d or 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 & Dividendo:**

If a : b = c : d then

(a + b)/(a - b) = (c + d)/(c - d)

**Properties of proportion:**

Product of extremes is equal to the product of means

If a : b :: c : d, then

ad = bc

If three quantities are in proportion as in a : b :: b: c, then,

ac = b^{2}

Where b is the mean proportion to a and b

C is the third proportion to a and b

• If three quantities are in proportion as in a : b :: b : c, then,

a ∶ c = a^{2} : b^{2}

**Variation: **

**Direct variation:** If two quantities a and b vary directly that is if a increases then b also increases and vice versa, and if there is a non-zero constant m, then,

a = mb

Their relation is denoted as a ∝ b

**Inverse variation:** If two quantities a and b vary inversely that is if a increases then b decreases and vice versa, and if there is a non-zero constant m, then,

a = m/b

Their relation is denoted as a ∝ 1/b

**Learn more on Ratios & its properties:**