Question

Which number should be subtracted from each of the numbers 13, 25 and 55 so that the resulting numbers would be in continued proportion.

Answer 1

$$\begin{gathered} \mathrm{Let}x\mathrm{be}\mathrm{the}\mathrm{number}\mathrm{that}\mathrm{should}\mathrm{be}\mathrm{subtracted}\mathrm{from}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{numbers}13,25\mathrm{and}55\mathrm{so}\mathrm{that}\mathrm{the}\mathrm{resulting}\mathrm{numbers}\mathrm{are}\mathrm{in}\mathrm{continued}\mathrm{proportion}. \\ \mathrm{Now},\mathrm{the}\mathrm{new}\mathrm{numbers}\mathrm{are}\left(13-x\right),\left(25-x\right)\mathrm{and}\left(55-x\right);\mathrm{these}\mathrm{are}\mathrm{in}\mathrm{continued}\mathrm{proprtion}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \frac{\left(13-x\right)}{\left(25-x\right)}=\frac{\left(25-x\right)}{\left(55-x\right)} \\ \Rightarrow \left(25-x\right)\times \left(25-x\right)=\left(13-x\right)\times \left(55-x\right) \\ \Rightarrow 625+x^2-50x=715-13x-55x+x^2 \\ \Rightarrow -50x+68x=715-625 \\ \Rightarrow 18x=90 \\ \Rightarrow x=5 \end{gathered}$$

Hence, 5 should be subtracted from 13, 25 and 55 so that the resulting numbers are in continued proportion.