There was a box of stamps. Grace took $\frac{1}{4}$ of the stamps and an additional $5$ stamps. Melissa took $\frac{2}{5}$ of the remainder and an additional $3$ stamps. There were $84$ stamps left in the box. How many stamps were in the box at first?

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Q. There was a box of stamps. Grace took

\frac{1}{4}

of the stamps and an additional

5

stamps. Melissa took

\frac{2}{5}

of the remainder and an additional

3

stamps. There were

84

stamps left in the box. How many stamps were in the box at first?

Denote Stamps as $S$ : Let's denote the total number of stamps at first as $S$ . Grace took $\frac{1}{4}$ of the stamps and an additional $5$ stamps. So, the stamps taken by Grace can be represented as $\frac{1}{4}S + 5$ .
Grace's Stamps Calculation: After Grace took her share, the remainder of the stamps is $S - \left(\frac{1}{4}S + 5\right)$ . This simplifies to $\frac{3}{4}S - 5$ .
Remaining Stamps After Grace: Melissa then took $(\frac{2}{5})$ of the remainder and an additional $3$ stamps. The stamps taken by Melissa can be represented as $(\frac{2}{5})((\frac{3}{4})S - 5) + 3$ .
Melissa's Stamps Calculation: After Melissa took her share, there were $84$ stamps left. So, the equation representing the situation after Melissa took her share is: $\newline$ $\left(\frac{3}{4}\right)S - 5 - \left[\left(\frac{2}{5}\right)\left(\left(\frac{3}{4}\right)S - 5\right) + 3\right] = 84.$
Equation After Melissa: Let's simplify the equation step by step. First, distribute the $(\frac{2}{5})$ across the terms inside the brackets: $(\frac{3}{4})S - 5 - (\frac{2}{5})(\frac{3}{4})S + (\frac{2}{5})(5) - 3 = 84$ .
Distribute $(\frac{2}{5})$ : Simplify the terms: (\frac{$3$}{$4$})S - \(5 - (\frac{ $3$ }{ $10$ })S + $2$ - $3$ = $84$ .
Combine Like Terms: Combine like terms: (\frac{$3$}{$4$})S - (\frac{$3$}{$10$})S - \(5 + $2$ - $3$ = $84$ \.
Combine S and Constants: Further simplify by combining the S terms and the constant terms: $(\frac{30}{40})S - (\frac{12}{40})S - 6 = 84$ .
Simplify S Terms: Simplify the S terms: $(\frac{18}{40})S - 6 = 84$ .
Simplify Fraction: Simplify the fraction $(\frac{18}{40})$ to $(\frac{9}{20})$ : $(\frac{9}{20})S - 6 = 84.$
Add $6$ to Isolate S: Add $6$ to both sides of the equation to isolate the term with S: $\newline$ $(\frac{9}{20})S = 84 + 6$ .
Simplify Right Side: Simplify the right side of the equation: $(\frac{9}{20})S = 90$ .
Multiply by Reciprocal: Multiply both sides by the reciprocal of $(\frac{9}{20})$ to solve for $S$ : $S = 90 \times \left(\frac{20}{9}\right).$
Simplify Multiplication: Simplify the multiplication: $S = 10 \times 20$ .
Final Value of S: Calculate the final value of S: $S = 200$ .