What quantity must be added to each term of the ratio x : y so that it may become equal to c : d ?

Let the number to be added be a

$\Rightarrow \dfrac{x + a}{y + a} = \dfrac{c}{d} \\[1em] \Rightarrow d(x + a) = c(y + a) \\[1em] \Rightarrow dx + da = cy + ca \\[1em] \Rightarrow da - ca = cy - dx \\[1em] \Rightarrow a(d - c) = cy - dx \\[1em] \Rightarrow a = \dfrac{cy - dx}{d - c}.$

Hence, number to be added = $\dfrac{cy - dx}{d - c}.$