2x articles cost ₹ (5x + 54) and (x + 2) similar articles cost ₹(10x - 4); find x .

2x articles cost ₹(5x + 54)

So, cost of each article = ₹$\Big(\dfrac{5x + 54}{2x}\Big)$

Similar (x + 2) articles cost ₹(10x - 4)

So, cost of each article = ₹$\Big(\dfrac{10x - 4}{x + 2}\Big)$

$\therefore \dfrac{5x + 54}{2x} = \dfrac{10x - 4}{x + 2} \\[1em] \Rightarrow (5x + 54)(x + 2) = 2x(10x - 4) \\[1em] \Rightarrow 5x^2 + 10x + 54x + 108 = 20x^2 - 8x \\[1em] \Rightarrow 5x^2 - 20x^2 + 64x + 8x + 108 = 0 \\[1em] \Rightarrow -15x^2 + 72x + 108 = 0 \\[1em] \Rightarrow 15x^2 - 72x - 108 = 0 \text{ (On multiplying equation by -1) }\\[1em] \Rightarrow 3(5x^2 - 24x - 36) = 0 \\[1em] \Rightarrow 5x^2 - 24x - 36 = 0 \\[1em] \Rightarrow 5x^2 - 30x + 6x - 36 = 0 \\[1em] \Rightarrow 5x(x - 6) + 6(x - 6) = 0 \\[1em] \Rightarrow (5x + 6)(x - 6) = 0 \\[1em] \Rightarrow (5x + 6) = 0 \text{ or } x - 6 = 0 \\[1em] x = -\dfrac{6}{5} \text{ or } x = 6$

Value of x cannot be negative and in fraction as that will make number of articles in fraction which is not possible hence, x ≠ -$\dfrac{6}{5}$

∴ x = 6

Hence , value of x is 6.