**Theorem of Joint Variation**

If x ∝ y when z is constant and x ∝ z when y is constant, then x ∝ yz when both y and z vary. If one variable varies as the product of other variables, it is called joint variation. The phrase “y varies jointly as x and z” is translated in two ways.

**Proof**

Since x ∝ y when z is constant Therefore x = ky where k = constant of variation and is independent to the changes of x and y.

Again, x ∝ z when y is constant.

or, ky ∝ z when y is constant (since, x = ky).

or, k ∝ z (y is constant).

or, k = mz where m is a constant which is independent to the changes of k and z.

Now, the value of k is independent to the changes of x and y. Hence, the value of m is independent to the changes of x, y and z.

Therefore x = ky = myz (since, k = mz)

where m is a constant whose value does not depend on x, y and z.

Therefore x ∝ yz when both y and z vary.

Note: (i) The above theorem can be extended for a longer number of variables. For example, if A ∝ B when C and D are constants, A ∝ C when B and D are constants and A ∝ D when B and C are constants, thee A ∝ BCD when B, C and D all vary.

(ii) If x ∝ y when z is constant and x ∝ 1/Z when y is constant, then x ∝ y when both y and z vary.

**Some Useful Results**

Theorem of Joint Variation

(i) If A ∝ B, then B ∝ A.

(ii) If A ∝ B and B∝ C, then A ∝ C.

(iii) If A ∝ B, then Aᵇ ∝ Bᵐ where m is a constant.

(iv) If A ∝ BC, then B ∝ A/C and C ∝ A/B.

(v) If A ∝ C and B ∝ C, then A + B ∝ C and AB ∝ C²

(vi) If A ∝ B and C ∝ D, then AC ∝ BD and A/C ∝ B/D

Now we are going to proof the useful results with step-by-step detailed explanation

**Proof: (i)** If A ∝ B, then B ∝ A.

Since, A ∝ B Therefore A = kB, where k = constant.

or, B = 1/K ∙ A Therefore B ∝ A. (since,1/K = constant)

**Proof: (ii)** If A ∝ B and B ∝ C, then A ∝ C.

Since, A ∝ B Therefore A = mB where, m = constant

Again, B ∝ C Therefore B = nC where n= constant.

Therefore A= mB = mnC = kC where k = mn = constant, as m and n are both Constants.

Therefore A ∝ C.

**Proof: (iii)** If A ∝ B, then Aᵇ ∝ Bᵐ where m is a constant.

Since A ∝ B Therefore A = kB where k= constant.

Aᵐ = KᵐBᵐ = n ∙ Bᵐ where n = kᵐ = constant, as k and m are both constants.

Therefore Aᵐ ∝ Bᵐ.

Results (iv), (v) and (vi) can be deduced by similar procedure.

**Summarization:**

(i) If A varies directly as B, then A ∝ B or, A = kB where k is the constant of variation. Conversely, if A = kB i.e., A/B = k where k is a constant, then A varies directly as B.

(ii) If A varies inversely as B, then A ∝ 1/B or, A= m ∙ 1/B or, AB= m where m = constant of variation. Conversely, if AB = k (a constant), then A varies inversely as B.

(iii) If A varies jointly as B and C, then A ∝ BC or A = kBC where k = constant of variation.

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