# Problem on Change the Subject of a Formula

We will solve different types of problems on change the subject of a formula.

The subject of a formula is a variable whose relation with other variables of the context is sought and the formula is written in such a way that subject is expressed in terms of the other variables.

For example, in the formula A = $$\frac{1}{2}$$bh, A is the subject which in terms of the other variables b and h.

By knowing the values of the variables b and h, the value of the subject A can be easily calculated. For example, if the base of a triangle is 6 cm and the height is 4 cm, its area

A = $$\frac{1}{2}$$bh = A = $$\frac{1}{2}$$ × 6 × 4 cm2 = 12 cm2

When a formula involving certain variables is known, we can change the subject of the formula.

Solved examples to change the subject of a formula:

1. In the formula S = $$\frac{n}{2}$$[2a + (n - 1) d], S is the subject. Write the formula with d as the subject.

Solution:

Given S = $$\frac{n}{2}$$[2a + (n - 1) d]

⟹ 2S = 2an + n(n -1)d

⟹ 2S – 2an = n(n - 1)d

⟹ n(n - 1)d = 2(S - an)

⟹ d = $$\frac{2(S - an)}{n(n - 1)}$$. Here, d is the subject.

2. If a = 2b + $$\sqrt{b^{2} + m}$$, express m in terms of a and b.

Solution:

Here, a = 2b + $$\sqrt{b^{2} + m}$$

⟹ a - 2b = $$\sqrt{b^{2} + m}$$

Squaring the both sides we get,

⟹ (a - 2b)2 = b2 + m

⟹ (a - 2b)2 - b2 = m

⟹ {(a - 2b) + b}{(a - 2b) - b} = m

⟹ (a - b)(a - 3b) = m

⟹ m =(a - b)(a - 3b)

3. Make u the subject of the formula f = $$\frac{uv}{u + v}$$.

Solution:

Give, f = $$\frac{uv}{u + v}$$

⟹ $$\frac{1}{f}$$ =  $$\frac{u + v}{uv}$$

⟹ $$\frac{1}{f}$$  = $$\frac{1}{u}$$ + $$\frac{1}{ v}$$

⟹ $$\frac{1}{u}$$ = $$\frac{1}{f}$$ - $$\frac{1}{v}$$

⟹ $$\frac{1}{u}$$ = $$\frac{v - f}{fv}$$

⟹ u = $$\frac{fv}{v - f}$$. Here, u is the subject.

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