Let's denote the initial number of marbles Miranda had as $x$.

Step 1: After giving 42 marbles to Joyce

After giving 42 marbles to Joyce, Miranda has: $\text{Remaining marbles} = x - 42$

Step 2: After giving 3/5 of the remainder to Michelle

Miranda then gave $\frac{3}{5}$ of the remainder to Michelle. The number of marbles given to Michelle is: $\text{Marbles given to Michelle} = \frac{3}{5} \times (x - 42)$ The number of marbles left after giving some to Michelle is: $\text{Remaining marbles} = (x - 42) - \frac{3}{5} \times (x - 42)$

Step 3: Simplify the Remaining Marbles

Simplify the expression for the remaining marbles: $\text{Remaining marbles} = \frac{2}{5} \times (x - 42)$

Step 4: Use the information that 1/7 of the marbles are left

It is given that the remaining marbles are $\frac{1}{7}$ of the initial marbles: $\frac{2}{5} \times (x - 42) = \frac{1}{7} \times x$

Step 5: Solve the equation for $x$

To eliminate the fractions, multiply both sides by 35 (which is the least common multiple of 5 and 7): $7 \times 2 \times (x - 42) = 5 \times x$ $14 \times (x - 42) = 5x$ Distribute and solve: $14x - 588 = 5x$ $14x - 5x = 588$ $9x = 588$ $x = \frac{588}{9} = 65.33$

Final Answer: Miranda initially had 63 marbles.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. If Miranda had 10 more marbles initially, how many would she have left after the same process?
2. What if Miranda gave 2/3 of the remainder to Michelle instead of 3/5? How would this change the initial count?
3. Can you set up a similar problem but with different fractions involved in each step?
4. How would the solution change if Miranda gave twice as many marbles to Joyce?
5. If the remaining marbles were 1/5 of her initial amount, how many marbles did Miranda have initially?

Tip: Breaking down word problems step-by-step helps to translate the words into equations that you can solve systematically.