The following numbers, K + 3, K + 2, 3K − 7 and 2K − 3 are in proportion. Find the value of K.

Let,

∴ K + 3 : K + 2 :: 3K − 7 : 2K − 3

$\Rightarrow \dfrac{K + 3}{K + 2} = \dfrac{3K - 7}{2K - 3} \\[1em] \Rightarrow (K + 3)(2K - 3) = (3K - 7)(K + 2) \\[1em] \Rightarrow 2K^2 - 3K + 6k - 9 = 3K^2 + 6K - 7K - 14 \\[1em] \Rightarrow 2K^2 + 3K - 9 = 3K^2 - K - 14 \\[1em] \Rightarrow 0 = 3K^2 - K - 14 - 2K^2 - 3K + 9 \\[1em] \Rightarrow K^2 - 4K - 5 = 0 \\[1em] \Rightarrow K^2 + 1K - 5K - 5 = 0 \\[1em] \Rightarrow K(K + 1) - 5(K + 1) = 0 \\[1em] \Rightarrow (K - 5)(K + 1) = 0 \\[1em] \Rightarrow (K - 5) = 0 \text{ or }(K + 1) = 0 \text{[Using Zero - product rule]}\\[1em] \Rightarrow K = 5 \text{ or } K = - 1$

Hence, K = 5 or K = −1.