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)



VII: Addendo Property:

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.

 






10th Grade Math

From  Properties of Ratio and Proportion to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.