# Properties of Ratio and Proportion

Some useful properties of ratio and proportion are invertendo property, alternendo property, componendo Property, dividendo property, convertendo property, componendo-dividendo property, addendo property and equivalent ratio property. These properties are explained below with examples.

I. Invertendo Property: For four numbers a, b, c, d if a : b = c : d, then  b : a = d : c; that is, if two ratios are equal, then their inverse ratios are also equal.

If a : b :: c : d then b : a :: d : c.

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$

⟹ $$\frac{b}{a}$$ = $$\frac{d}{c}$$

⟹ b : a :: d : c

Example: 6 : 10 = 9 : 15

Therefore, 10 : 6 = 5 : 3 = 15 : 9

II. Alternendo Property: For four numbers a, b, c, d if a : b = c : d, then  a : c = b : d; that is, if the second and third term interchange their places, then also the four terms are in proportion.

If a : b :: c : d then a : c :: b : d.

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$

⟹ $$\frac{a}{b}$$   $$\frac{b}{c}$$ = $$\frac{c}{d}$$  $$\frac{b}{c}$$

⟹ $$\frac{a}{c}$$ = $$\frac{b}{d}$$

⟹ a : c :: b : d

Example: If 3 : 5 = 6 : 10 then 3 : 6 = 1 : 2 = 5 : 10

III. Componendo Property: For four numbers a, b, c, d if a : b = c : d then (a + b) : b :: (c + d) : d.

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$

Adding 1 to both sides of $$\frac{a}{b}$$ = $$\frac{c}{d}$$, we get

⟹ $$\frac{a}{b}$$  + 1 = $$\frac{c}{d}$$ + 1

⟹ $$\frac{a + b}{b}$$ = $$\frac{c + d}{d}$$

⟹ (a + b) : b = (c + d) : d

Example: 4 : 5 = 8 : 10

Therefore, (4 + 5) : 5 = 9 : 5 = 18 : 10

= (8 + 10) : 10

IV:  Dividendo Property

If a : b :: c : d then (a - b) : b :: (c - d) : d.

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$

Subtracting 1 from both sides,

⟹ $$\frac{a}{b}$$  - 1 = $$\frac{c}{d}$$ - 1

⟹ $$\frac{a - b}{b}$$ = $$\frac{c - d}{d}$$

⟹ (a - b) : b :: (c - d) : d

Example: 5 : 4 = 10 : 8

Therefore, (5 - 4) : 4 = 1 : 4 = (10 - 8) : 8

V. Convertendo Property

If a : b :: c : d then a : (a - b) :: c : (c - d).

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$ ............................... (i)

⟹ $$\frac{a}{b}$$  - 1 = $$\frac{c}{d}$$ - 1

⟹ $$\frac{a - b}{b}$$ = $$\frac{c - d}{d}$$ ............................... (ii)

Dividing (i) by the corresponding sides of (ii),

⟹ $$\frac{\frac{a}{b}}{\frac{a - b}{b}} = \frac{\frac{c}{d}}{\frac{c - d}{d}}$$

⟹ $$\frac{a}{a - b}$$ = $$\frac{c}{c - d}$$

⟹ a : (a - b) :: c : (c - d).

VI. Componendo-Dividendo Property

If a : b :: c : d then (a + b) : (a - b) :: (c + d) : (c - d).

Proof:

a : b :: c : d

⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$

⟹ $$\frac{a}{b}$$  + 1 = $$\frac{c}{d}$$ + 1 and $$\frac{a}{b}$$  - 1 = $$\frac{c}{d}$$ - 1

⟹ $$\frac{a + b}{b}$$ = $$\frac{c + d}{d}$$ and $$\frac{a - b}{b}$$ = $$\frac{c - d}{d}$$

Dividing  the corresponding sides,

⟹ $$\frac{\frac{a + b}{b}}{\frac{a - b}{b}} = \frac{\frac{c + d}{d}}{\frac{c - d}{d}}$$

⟹ $$\frac{a + b}{a - b}$$ = $$\frac{c + d}{c - d}$$

⟹ (a + b) : (a - b) :: (c + d) : (c - d).

Writing in algebraic expressions, the componendo-dividendo property gives the following.

$$\frac{a}{b}$$ = $$\frac{c}{d}$$ ⟹ (a + b) : (a - b) :: (c + d) : (c - d)

Note: This property is frequently used in simplification.

Example: 7 : 3 = 14 : 6

(7 + 3) : ( 7 - 3) = 10 : 4 = 5 : 2

Again, (14 + 6) : (14 - 6) = 20 : 8 = 5 : 2

Therefore, ( 7 + 3) : ( 7 - 3) = ( 14 + 6) : ( 14 - 6)

If a : b = c : d = e : f, value of each ratio is (a + c + e) : (b + d + f)

Proof:

a : b = c : d = e : f

Let, $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$  = k (k ≠ 0).

Therefore, a = bk, c = dk, e = fk

Now, $$\frac{a + c + e}{b + d + f}$$ = $$\frac{bk + dk + fk}{b + d + f}$$ = $$\frac{k(b + d + f)}{b + d + f}$$ = k

Therefore, $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$  = $$\frac{a + c + e}{b + d + f}$$

That is, a : b = c : d = e : f, value of each ratio is (a + c + e) : (b + d + f)

Note: If a : b = c : d = e : f, then the value of each ratio will be $$\frac{am + cn + ep}{bm + dn + fp}$$ where m, n, p may be non zero number.]

In general, $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$  =  ..................... = $$\frac{a + c + e + .................. }{b + d + f + ..................}$$

As, $$\frac{2}{3}$$ = $$\frac{6}{9}$$ = $$\frac{8}{12}$$ = $$\frac{2 + 6 + 8}{3 + 9 + 12}$$ = $$\frac{16}{24}$$ = $$\frac{2}{3}$$

VIII: Equivalent ratio property

If a : b :: c : d then (a ± c) : (b ± d) : : a: b and (a ± c) : (b ± d) :: c : d

Proof:

a : b :: c : d

Let, $$\frac{a}{b}$$ = $$\frac{c}{d}$$  = k (k ≠ 0).

Therefore, a = bk, c = dk.

Now, $$\frac{a ± c}{b ± d}$$ = $$\frac{bk ± dk}{b ± d}$$ = $$\frac{k(b ± d}{b ± d}$$ = k = $$\frac{a}{b}$$ = $$\frac{c}{d}$$  .

Therefore, (a ± c) : (b ± d) : : a: b and (a ± c) : (b ± d) :: c : d.

Algebraically, the property gives the following.

$$\frac{a}{b}$$ = $$\frac{c}{d}$$ ⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{a + c}{b + d}$$ = $$\frac{a - c}{b - d}$$

Similarly, we can prove that

$$\frac{a}{b}$$ = $$\frac{c}{d}$$ ⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{pa + qc}{pb + qd}$$

$$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$ ⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$ = $$\frac{a + c + e}{b + d + f}$$ = $$\frac{ap + cq + er}{bp + dq + fr}$$

For example:

1. $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{2a + 3c}{2b + 3d}$$ = $$\frac{ab + cd}{b^{2} + d^{2}}$$, etc.

2. $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$ ⟹ $$\frac{a}{b}$$ = $$\frac{c}{d}$$ = $$\frac{e}{f}$$ = $$\frac{a + 2c + 3e}{b + 2d + 3f}$$ = $$\frac{4a – 3c + 9e}{4b – 3d + 9f}$$, etc.