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.
● Ratio and proportion
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