Question

Find a number such that when it is subtracted from each of 19, 26, 35 and 50 so that remainders would be in proportion.

Answer 1

$$\begin{gathered} \mathrm{Let}x\mathrm{be}\mathrm{the}\mathrm{number}\mathrm{subtracted}\mathrm{from}19,26,35\mathrm{and}50\mathrm{such}\mathrm{that}\mathrm{the}\mathrm{remainders}\mathrm{are}\mathrm{in}\mathrm{proportion}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \frac{19-x}{26-x}=\frac{35-x}{50-x} \\ \Rightarrow \left(19-x\right)\left(50-x\right)=\left(35-x\right)\left(26-x\right) \\ \Rightarrow 950-19x-50x-x^2=910-35x-26x-x^2 \\ \Rightarrow -69x+61x=910-950 \\ \Rightarrow -8x=-40 \\ \Rightarrow x=5 \end{gathered}$$

Thus, by subtracting 5 from 19, 26, 35 and 50, we get the resulting numbers, i.e., 14, 21, 30 and 45, in proportion.