Question

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"explain the properties of ratio"
here's the question ​

Answer 1

• If a:b = c:d is a proportion, then

• Product of extremes = product of means i.e., ad = bc

• a, b, c, d,…. are in continued proportion means, a:b = b:c = c:d

• a:b = b:c then b is called mean proportional and b2 = ac

• The third proportional of two numbers, a and b, is c, such that, a:b = b:c

• d is fourth proportional to numbers a, b, c if a:b = c:d

Answer 2

Step-by-step explanation:

1. 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

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

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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

⟹ a−bb = c−dd

⟹ (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

⟹ a−bb = c−dd ............................... (ii)

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

⟹ aba−bb=cdc−dd

⟹ aa−b = cc−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

⟹ ab = cd

⟹ ab + 1 = cd + 1 and ab - 1 = cd - 1

⟹ a+bb = c+dd and a−bb = c−dd

Dividing the corresponding sides,

⟹ a+bba−bb=c+ddc−dd

⟹ a+ba−b = c+dc−d

⟹ (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.]

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 = cd ⟹ ab = cd = a+cb+d = a−cb−d

Similarly, we can prove that

ab = cd ⟹ ab = cd = pa+qcpb+qd

ab = cd = ef ⟹ ab = 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 = ef ⟹ ab = cd = ef = a+2c+3eb+2d+3f = 4a–3c+9e4b–3d+9f, etc.