Let \( d \) represent the number of dimes Rosalyn collected. According to the problem, the number of quarters she collected is \( d + 50 \).
The value of the dimes collected is \( 0.10d \) dollars, and the value of the quarters collected is \( 0.25(d + 50) \) dollars.
The total amount collected is given as $82.50. So we can write the equation:
\[ 0.10d + 0.25(d + 50) = 82.50 \]
Now, let's simplify the equation:
\[ 0.10d + 0.25d + 0.25 \times 50 = 82.50 \]
Calculating \( 0.25 \times 50 \):
\[ 0.25 \times 50 = 12.50 \]
So the equation now becomes:
\[ 0.10d + 0.25d + 12.50 = 82.50 \]
Combining the \( d \) terms:
\[ 0.35d + 12.50 = 82.50 \]
Now, subtract \( 12.50 \) from both sides:
\[ 0.35d = 82.50 - 12.50 \] \[ 0.35d = 70.00 \]
Now, divide both sides by \( 0.35 \):
\[ d = \frac{70.00}{0.35} \] \[ d = 200 \]
Thus, Rosalyn collected \( \boxed{200} \) dimes.
To confirm, we can calculate the number of quarters:
\[ d + 50 = 200 + 50 = 250 \]
Now, calculating the total amount collected:
Value of dimes:
\[ 200 \times 0.10 = 20.00 \text{ dollars} \]
Value of quarters:
\[ 250 \times 0.25 = 62.50 \text{ dollars} \]
Adding both values together:
\[ 20.00 + 62.50 = 82.50 \text{ dollars} \]
This matches the amount collected, confirming that the solution is correct. Rosalyn collected \( \boxed{200} \) dimes.
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