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?\\n
Answered by Rasheed.Sindhi last updated on 15/Sep/22
$${Let}\:{the}\:{amount}\:{distributed}\:{among} \\ $$ $${B},{D}\:\&\:{E}\:{is}\:{x}. \\ $$ $${B}\:{receives}\:\frac{\mathrm{2}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{5}} \\ $$ $${D}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{5}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{2}} \\ $$ $${E}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{3}{x}}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{3}{x}}{\mathrm{10}} \\ $$ $${Amount}\:{distributed}\:{among}\:{A},{B}\:\& \\ $$ $${C}\:{is}\:{then}\:\left(\mathrm{2}/\mathrm{5}\right){x} \\ $$ $${A}\:{receives}\:\frac{\mathrm{2}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{4}{x}}{\mathrm{5}×\mathrm{10}}=\frac{\mathrm{2}{x}}{\mathrm{25}} \\ $$ $${B}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{5}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{{x}}{\mathrm{5}} \\ $$ $${C}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{3}\left(\frac{\mathrm{2}}{\mathrm{5}}{x}\right)}{\mathrm{2}+\mathrm{5}+\mathrm{3}}=\frac{\mathrm{6}{x}}{\mathrm{5}×\mathrm{10}}=\frac{\mathrm{3}{x}}{\mathrm{25}} \\ $$ $$ \\ $$ $$\:\: \\ $$ $$\begin{array}{|c|c|c|c|}{{A}}&\hline{:}&\hline{{C}}&\hline{:}&\hline{{E}}\\{\frac{\mathrm{2}{x}}{\mathrm{25}}}&\hline{\color{mathblue}{:}}&\hline{\frac{\mathrm{3}{x}}{\mathrm{25}}}&\hline{\color{mathblue}{:}}&\hline{\frac{\mathrm{3}{x}}{\mathrm{10}}}\\{\frac{\mathrm{\color{mathblue}{5}\color{mathblue}{0}}}{{\color{mathblue}{x}}}\color{mathblue}{×}\frac{\mathrm{2}{x}}{\mathrm{25}}}&\hline{\color{mathblue}{:}}&\hline{\frac{\mathrm{\color{mathblue}{5}\color{mathblue}{0}}}{{\color{mathblue}{x}}}\color{mathblue}{×}\frac{\mathrm{3}{x}}{\mathrm{25}}}&\hline{\color{mathblue}{:}}&\hline{\frac{\mathrm{\color{mathblue}{5}\color{mathblue}{0}}}{{\color{mathblue}{x}}}\color{mathblue}{×}\frac{\mathrm{3}{x}}{\mathrm{10}}}\\{\mathrm{\color{mathred}{4}}}&\hline{\color{mathblue}{:}}&\hline{\mathrm{\color{mathred}{6}}}&\hline{\color{mathblue}{:}}&\hline{\mathrm{\color{mathred}{1}\color{mathred}{5}}}\\\hline\end{array}\: \\ $$ $$\: \\ $$
Commented byMathsFan 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 byRasheed.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 byTawa11 last updated on 18/Sep/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Answered by Rasheed.Sindhi last updated on 15/Sep/22
$$\begin{array}{|c|c|c|c|c|}{{A}}&\hline{{B}}&\hline{{C}}&\hline{{D}}&\hline{{E}}\\{}&\hline{\mathrm{2}}&\hline{}&\hline{\mathrm{5}}&\hline{\mathrm{3}}\\{\mathrm{2}}&\hline{\mathrm{5}}&\hline{\mathrm{3}}&\hline{}&\hline{}\\{\mathrm{4}}&\hline{\mathrm{10}}&\hline{\mathrm{6}}&\hline{\mathrm{25}}&\hline{\mathrm{15}}\\{\mathrm{\color{mathred}{4}}}&\hline{}&\hline{\mathrm{\color{mathred}{6}}}&\hline{}&\hline{\mathrm{\color{mathred}{1}\color{mathred}{5}}}\\\hline\end{array} \\ $$
Commented byMathsFan last updated on 15/Sep/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir} \\ $$