Question

What quantity must be added to each term of the ratio $$a : b$$ to make it $$(c : d)$$?

Answer 1

Let the required quantity to be added be $$x$$.

According to the given condition: $$\frac{a+x}{b+x} = \frac{c}{d}$$

Cross-multiplying both sides: $$d(a+x) = c(b+x)$$

$$ad + dx = bc + cx$$

Rearranging the terms containing $$x$$ to one side: $$dx - cx = bc - ad$$

$$x(d - c) = bc - ad$$

Solving for $$x$$: $$x = \frac{bc - ad}{d - c} \quad \text{or} \quad x = \frac{ad - bc}{c - d}$$

Therefore, the quantity to be added is $$\frac{bc - ad}{d - c}$$.