Miranda has some marbles. She gave 42 of them to Joyce and 3/5 of the remainder to Michelle. After that, she found that she still had 1/7 of her marbles left. How many marbles did Miranda have at first?

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.

Learn how to solve Miranda's marbles problem step-by-step. Understand fractions, equations, and word problems with this detailed explanation. Suitable for students in Grades 5-7.

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