Question

Some amount of monthly expenditure varies directly as the income and remaining part of the expenditure directly varies as the square of the income. If the total expenditure is Rs 3500 then the income is Rs 5000 and if total expenditure is Rs 6300 then income is Rs 7000. If the total income is Rs 3000 what will be the expenditure?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{total}\mathrm{monthly}\mathrm{expenditure}bedenotedby\mathrm{E},\mathrm{some}\mathrm{part}\mathrm{of}\mathrm{the}\mathrm{monthly}\mathrm{expenditure}\mathrm{be}\mathrm{denoted}\mathrm{by}xandthe\mathrm{total}\mathrm{income}\mathrm{be}\mathrm{denoted}\mathrm{by}\mathrm{I}. \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question},wehave: \\ x\alpha \mathrm{I} \\ \Rightarrow x=k_1\mathrm{I}...\left(1\right) \\ And, \\ \left(\mathrm{E}-x\right)\alpha {\mathrm{I}}^2 \\ \Rightarrow \mathrm{E}-x=k_2{\mathrm{I}}^2 \\ \Rightarrow \mathrm{E}-k_1\mathrm{I}=k_2{\mathrm{I}}^2\left(\mathrm{From}1\right) \\ \Rightarrow \mathrm{E}=k_2{\mathrm{I}}^2+k_1\mathrm{I} \\ \Rightarrow \mathrm{E}=I\left(k_2\mathrm{I}+k_1\right) \\ \mathrm{We}\mathrm{have}: \\ 3500=5000\left(k_2\times 5000+k_1\right)\left(\because \mathrm{E}=\mathrm{Rs}3500\mathrm{when}\mathrm{I}=\mathrm{Rs}5000\right) \\ \Rightarrow \frac{7}{10}=5000k_2+k_1 \\ \Rightarrow k_1=\frac{7}{10}-5000k_2...\left(2\right) \\ \therefore 6300=7000\left(k_2\times 7000+k_1\right)\left(\because \mathrm{E}=\mathrm{Rs}6300\mathrm{when}\mathrm{I}=\mathrm{Rs}7000\right) \\ \Rightarrow \frac{9}{10}=7000k_2+k_1 \\ \Rightarrow k_1=\frac{9}{10}-7000k_2...\left(3\right) \end{gathered}$$

$$\begin{gathered} \mathrm{From}\left(2\right)\mathrm{and}\left(3\right),weget: \\ \therefore \frac{7}{10}-5000k_2=\frac{9}{10}-7000k_2 \\ \Rightarrow 2000k_2=\frac{2}{10} \\ \Rightarrow k_2=\frac{1}{10000}...\left(4\right) \\ Putting\left(4\right)\mathrm{in}\left(2\right),weget: \\ {k}_1=\frac{7}{10}-5000\times \frac{1}{10000} \\ \Rightarrow k_1=\frac{2}{10} \\ \Rightarrow k_1=\frac{1}{5}...\left(5\right) \\ E\mathrm{quation}\mathrm{of}\mathrm{variation}: \\ \therefore \mathrm{E}=I\left(\frac{1}{10000}I+\frac{1}{5}\right)[F\mathrm{rom}(4)\mathrm{and}(5\left)\right] \\ \Rightarrow \mathrm{E}=3000\left(\frac{1}{10000}\times 3000+\frac{1}{5}\right)\left[\because \mathrm{I}=\mathrm{Rs}3000\right] \\ \Rightarrow \mathrm{E}=3000\left(\frac{3+2}{10}\right) \\ \Rightarrow \mathrm{E}=3000\times \frac{1}{2} \\ \Rightarrow \mathrm{E}=\mathrm{Rs}1500 \end{gathered}$$