To solve this problem, let’s break it down step by step.

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Step 1: Define variables

Let:

• $r$ = number of red marbles at first,
• $g$ = number of green marbles at first.

Given that the ratio of red to green marbles was $3:7$, we have: $\frac{r}{g} = \frac{3}{7} \implies r = \frac{3}{7}g$

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Step 2: Relating the number of marbles after removing 26 green marbles

After removing 26 green marbles, the number of green marbles becomes $g - 26$. At this point, the number of red marbles is $\frac{4}{5}$ of the remaining green marbles: $r = \frac{4}{5}(g - 26)$

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Step 3: Set up the equations

We now have two equations:

1. $r = \frac{3}{7}g$ (from the initial ratio),
2. $r = \frac{4}{5}(g - 26)$ (after removing 26 green marbles).

Equating the two expressions for $r$: $\frac{3}{7}g = \frac{4}{5}(g - 26)$

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Step 4: Solve for $g$

Eliminate the fractions by multiplying through by $35$ (the least common multiple of 7 and 5): $35 \cdot \frac{3}{7}g = 35 \cdot \frac{4}{5}(g - 26)$ $15g = 28(g - 26)$ Expand and simplify: $15g = 28g - 728$ $728 = 28g - 15g$ $728 = 13g$ $g = \frac{728}{13} = 56$

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Step 5: Find $r$

Using $r = \frac{3}{7}g$: $r = \frac{3}{7}(56) = 24$

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Step 6: Find the difference

The difference between the initial number of green marbles and red marbles is: $g - r = 56 - 24 = 32$

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Final Answer:

There were 32 more green marbles than red marbles at first.

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Would you like further explanation or details on any part? Here are related questions to deepen understanding:

1. How can ratios be used in other real-world scenarios?
2. What happens if the ratios are reversed, e.g., $7:3$?
3. How do you handle fractional equations like $\frac{4}{5}(g - 26)$?
4. What is the importance of eliminating fractions in solving equations?
5. Can we solve this problem using trial-and-error or estimation?

Tip: When solving ratio problems, always express one variable in terms of the other to simplify the equations.