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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

ab = cd

ba = dc

⟹ 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

ab = cd

ab   bc = cd  bc

ac = bd

⟹ 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

ab = cd

Adding 1 to both sides of ab = cd, we get

ab  + 1 = cd + 1

a+bb = c+dd

⟹ (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

ab = cd

Subtracting 1 from both sides,

ab  - 1 = cd - 1

abb = cdd

⟹ (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

ab = cd ............................... (i)

ab  - 1 = cd - 1

abb = cdd ............................... (ii)

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

ababb=cdcdd

aab = ccd

⟹ 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

ab = cd

ab  + 1 = cd + 1 and ab  - 1 = cd - 1

a+bb = c+dd and abb = cdd

Dividing  the corresponding sides,

a+bbabb=c+ddcdd

a+bab = c+dcd

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

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

ab = cd ⟹ (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, ab = cd = ef  = k (k ≠ 0).

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

Now, a+c+eb+d+f = bk+dk+fkb+d+f = k(b+d+f)b+d+f = k

Therefore, ab = cd = ef  = a+c+eb+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 am+cn+epbm+dn+fp where m, n, p may be non zero number.]

In general, ab = cd = ef  =  ..................... = a+c+e+..................b+d+f+..................

 

As, 23 = 69 = 812 = 2+6+83+9+12 = 1624 = 23



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, ab = cd  = k (k ≠ 0).

Therefore, a = bk, c = dk.

Now, a±cb±d = bk±dkb±d = k(b±db±d = k = ab = cd  .

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

Algebraically, the property gives the following.

ab = cdab = cd = a+cb+d = acbd

Similarly, we can prove that

ab = cdab = cd = pa+qcpb+qd

ab = cd = efab = cd = ef = a+c+eb+d+f = ap+cq+erbp+dq+fr


For example:

1. ab = cd = ab = cd = 2a+3c2b+3d = ab+cdb2+d2, etc.

2. ab = cd = efab = cd = ef = a+2c+3eb+2d+3f = 4a3c+9e4b3d+9f, etc.

 

● Ratio and proportion






10th Grade Math

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