Emma and Mie had some beads. Emma lost $\frac{2}{5}$ of her beads to Mie. Mie then lost $\frac{1}{6}$ of her beads to Emma. Emma now has twice as many beads as Mie. If they both have a total of $600$ beads, how many beads did Emma have at first?

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Q. Emma and Mie had some beads. Emma lost

\frac{2}{5}

of her beads to Mie. Mie then lost

\frac{1}{6}

of her beads to Emma. Emma now has twice as many beads as Mie. If they both have a total of

600

beads, how many beads did Emma have at first?

Define Variables: Let's call the number of beads Emma had at first $E$ and the number of beads Mie had at first $M$ .
Emma Gives Beads to Mie: Emma gives away $\frac{2}{5}$ of her beads to Mie, so she's left with $\frac{3}{5}$ of $E$ . Mie receives $\frac{2}{5}$ of $E$ from Emma, so now Mie has $M + \frac{2}{5} E$ .
Mie Gives Beads to Emma: Mie then gives away $\frac{1}{6}$ of her total beads to Emma, which is $\frac{1}{6}$ of $(M + \frac{2}{5} E)$ . Emma receives $\frac{1}{6}$ of $(M + \frac{2}{5} E)$ from Mie, so now Emma has $\frac{3}{5} E + \frac{1}{6} (M + \frac{2}{5} E)$ .
Equation $1$ : Emma has Twice as Many Beads: After the exchanges, Emma has twice as many beads as Mie. So, we can write the equation: $\newline$ $\frac{3}{5} E + \frac{1}{6} (M + \frac{2}{5} E) = 2(M - \frac{1}{6} (M + \frac{2}{5} E))$
Equation $2$ : Total Number of Beads: We also know that together they have $600$ beads, so we can write another equation: $\newline$ $\frac{3}{5} E + M = 600$
Solve for $M$ : Now we have two equations with two variables. We can solve these equations simultaneously to find the values of $E$ and $M$ .
Substitute $M$ in Equation $1$ : Let's solve the second equation for $M$ : $M = 600 - \frac{3}{5} E$
Simplify Equation: Substitute $M$ in the first equation: $\newline$ $\frac{3}{5} E + \frac{1}{6} (600 - \frac{3}{5} E + \frac{2}{5} E) = 2(600 - \frac{3}{5} E - \frac{1}{6} (600 - \frac{3}{5} E + \frac{2}{5} E))$
Combine Like Terms: Simplify the equation: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{6} (\frac{3}{5} E) + \frac{1}{6} (\frac{2}{5} E) = 1200 - \frac{6}{5} E - 100 + \frac{1}{6} (\frac{3}{5} E) - \frac{1}{6} (\frac{2}{5} E)$
Solve for E: Combine like terms and solve for E: $\frac{3}{5} E + 100 - \frac{1}{6} \left(\frac{1}{5} E\right) = 1200 - \frac{6}{5} E - 100 + \frac{1}{6} \left(\frac{1}{5} E\right)$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$ Combine like terms: $\newline$ $\frac{3}{5} E - \frac{1}{30} E + \frac{6}{5} E - \frac{1}{30} E = 1100 - 100$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$ Combine like terms: $\newline$ $\frac{3}{5} E - \frac{1}{30} E + \frac{6}{5} E - \frac{1}{30} E = 1100 - 100$ Simplify the equation: $\newline$ $(\frac{18}{30} E - \frac{1}{30} E + \frac{36}{30} E - \frac{1}{30} E) = 1000$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$ Combine like terms: $\newline$ $\frac{3}{5} E - \frac{1}{30} E + \frac{6}{5} E - \frac{1}{30} E = 1100 - 100$ Simplify the equation: $\newline$ $(\frac{18}{30} E - \frac{1}{30} E + \frac{36}{30} E - \frac{1}{30} E) = 1000$ Combine the E terms: $\newline$ $(\frac{52}{30} E) = 1000$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$ Combine like terms: $\newline$ $\frac{3}{5} E - \frac{1}{30} E + \frac{6}{5} E - \frac{1}{30} E = 1100 - 100$ Simplify the equation: $\newline$ $(\frac{18}{30} E - \frac{1}{30} E + \frac{36}{30} E - \frac{1}{30} E) = 1000$ Combine the E terms: $\newline$ $(\frac{52}{30} E) = 1000$ Solve for E: $\newline$ $E = 1000 \times (\frac{30}{52})$
Calculate E: This simplifies to: $\newline$ $\frac{3}{5} E + 100 - \frac{1}{30} E = 1100 - \frac{6}{5} E + \frac{1}{30} E$ Combine like terms: $\newline$ $\frac{3}{5} E - \frac{1}{30} E + \frac{6}{5} E - \frac{1}{30} E = 1100 - 100$ Simplify the equation: $\newline$ $(\frac{18}{30} E - \frac{1}{30} E + \frac{36}{30} E - \frac{1}{30} E) = 1000$ Combine the E terms: $\newline$ $(\frac{52}{30} E) = 1000$ Solve for E: $\newline$ $E = 1000 \times (\frac{30}{52})$ Calculate E: $\newline$ $E = 1000 \times (\frac{30}{52}) = 576.92$