Here we will discuss about the *Theorem of Joint Variation* with the detailed explanation.

The theorem of joint variation can be established by stating the relationship between three variables which are separately in direct variation with each other.

** 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.*

**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 that means the value of K doesn’t change for any value of X and Y.

Again, x ∝ z when y is constant.

or, ky ∝ z when y is constant (By putting ky in place of x we get).

or, k ∝ z (y is constant).

or, k = mz where m is a constant which is independent to the changes of k and z that means the value of m doesn’t change for any value 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.

So in this theorem we use the principle of direct variation to prove that how joint variation works for to establish a correlation among more than two variables.

For solving a problems related to the theory of joint variation first we need to solve by following steps.

**1.** Build the correct equation by adding a constant and relate the variables.

**2.** We need to determine the value of the constant from the given data.

**3.** Substitute the value of the constant in the equation.

**4.** Put the values of variables for required situation and determine the answer.

Now we will see some problems and solutions related to the theorem of joint variation:

**1. The variable x is in joint
variation with y and z. When the values of y and z are 2 and 3, x is 16.
What is the value of x when y = 8 and z =12?**

The equation for the given problem of joint variation is

x = Kyz where K is the constant.

For the given data

16 = K × 2 × 3

or, K = \(\frac{8}{3}\)

So substituting the value of K the equation becomes

x = \(\frac{8yz}{3}\)

Now for the required condition

x = \(\frac{8 × 8 × 12}{3}\) = 256

Hence the value of x will be 256.

**2. A is in joint variation with B
and square of C. When A = 144, B = 4 and C = 3. Then what is the value of
A when B = 6 and C = 4?**

From the given problem equation for the joint variation is

A = KBC^{2}

From the given data value of the constant K is

K = \(\frac{BC^{2}}{A}\)

K = \(\frac{4 × 3^{2}}{144}\) = \(\frac{36}{144}\) = \(\frac{1}{4}\).

Substituting the value of K in the equation

A = \(\frac{BC^{2}}{4}\)

A = \(\frac{6 × 4^{2}}{4}\) = 24

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.

Summarisation:

(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.

**●** **Variation**

**What is Variation?****Direct Variation****Inverse Variation****Joint Variation****Theorem of Joint Variation****Worked out Examples on Variation****Problems on Variation**

**11 and 12 Grade Math**** ****From Theorem of Joint Variation to HOME PAGE**

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