Question Number 176215 by MathsFan last updated on 15/Sep/22
$$ \\ $$A certain amount of money is distributed among A, B and C in the ratio of 2:5:3 and another amount of money among B, D and E is also distributed in the same ratio. If the amount distributed among A, B and C is ⅖ of the amount distributed among B, D and E, what is the ratio in which the amount is distributed among A, C and E?
Answered by Rasheed.Sindhi last updated on 15/Sep/22
$${Let}\:{the}\:{amount}\:{distributed}\:{among} \\ $$$${B},{D}\:\&\:{E}\:{is}\:{x}. \\ $$$${B}\:{receives}\:\frac{\mathrm{2}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{5}} \\ $$$${D}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{5}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{2}} \\ $$$${E}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{3}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{3}{x}}{\mathrm{10}} \\ $$$${Amount}\:{distributed}\:{among}\:{A},{B}\:\& \\ $$$${C}\:{is}\:{then}\:\left(\mathrm{2}/\mathrm{5}\right){x} \\ $$$${A}\:{receives}\:\frac{\mathrm{2}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{4}{x}}{\mathrm{5}×\mathrm{10}}=\frac{\mathrm{2}{x}}{\mathrm{25}} \\ $$$${B}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{5}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{5}} \\ $$$${C}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{3}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{6}{x}}{\mathrm{5}×\mathrm{10}}=\frac{\mathrm{3}{x}}{\mathrm{25}} \\ $$$$ \\ $$$$\:\: \\ $$$$\begin{array}{|c|c|c|c|}{{A}}&\hline{:}&\hline{{C}}&\hline{:}&\hline{{E}}\\{\frac{\mathrm{2}{x}}{\mathrm{25}}}&\hline{:}&\hline{\frac{\mathrm{3}{x}}{\mathrm{25}}}&\hline{:}&\hline{\frac{\mathrm{3}{x}}{\mathrm{10}}}\\{\frac{\mathrm{50}}{{x}}×\frac{\mathrm{2}{x}}{\mathrm{25}}}&\hline{:}&\hline{\frac{\mathrm{50}}{{x}}×\frac{\mathrm{3}{x}}{\mathrm{25}}}&\hline{:}&\hline{\frac{\mathrm{50}}{{x}}×\frac{\mathrm{3}{x}}{\mathrm{10}}}\\{\mathrm{4}}&\hline{:}&\hline{\mathrm{6}}&\hline{:}&\hline{\mathrm{15}}\\\hline\end{array}\: \\ $$$$\: \\ $$
Commented by MathsFan last updated on 15/Sep/22
$$\mathrm{but}\:\mathrm{sir},\:\mathrm{how}\:\mathrm{did}\:\mathrm{you}\:\mathrm{get}\:\frac{\mathrm{50}}{\mathrm{x}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{table} \\ $$
Commented by Rasheed.Sindhi last updated on 16/Sep/22
$${Simplifying}\:\:\frac{\mathrm{2}{x}}{\mathrm{25}}:\frac{\mathrm{3}{x}}{\mathrm{25}}:\frac{\mathrm{3}{x}}{\mathrm{10}}\:\:\:\:\:\:\: \\ $$$$\left({i}\right)\:\underset{\left({to}\:{get}\:{rid}\:\:{of}\:{denominator}\right)} {{Multiply}\:{by}\:\mathrm{LCM}\left(\mathrm{25},\mathrm{10}\right)=\mathrm{50}} \\ $$$$\:\:\:\:\:\:\mathrm{4}{x}:\mathrm{6}{x}:\mathrm{15}{x} \\ $$$$\left({ii}\right){Now}\:{divide}\:{by}\:{common}\:{factor}\:{x} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{4}:\mathrm{6}:\mathrm{15} \\ $$$$\mathcal{T}{he}\:{two}\:{steps}\:{at}\:{a}\:{time},\:{multiply}\:{by} \\ $$$$\:\:\:\frac{\mathrm{50}}{{x}} \\ $$
Commented by Tawa11 last updated on 18/Sep/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Answered by Rasheed.Sindhi last updated on 15/Sep/22
$$\begin{array}{|c|c|c|c|c|}{{A}}&\hline{{B}}&\hline{{C}}&\hline{{D}}&\hline{{E}}\\{}&\hline{\mathrm{2}}&\hline{}&\hline{\mathrm{5}}&\hline{\mathrm{3}}\\{\mathrm{2}}&\hline{\mathrm{5}}&\hline{\mathrm{3}}&\hline{}&\hline{}\\{\mathrm{4}}&\hline{\mathrm{10}}&\hline{\mathrm{6}}&\hline{\mathrm{25}}&\hline{\mathrm{15}}\\{\mathrm{4}}&\hline{}&\hline{\mathrm{6}}&\hline{}&\hline{\mathrm{15}}\\\hline\end{array} \\ $$
Commented by MathsFan last updated on 15/Sep/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir} \\ $$