We are given that the ratio of red marbles to green marbles to blue marbles is 2:3:5. So, if we let the number of red marbles be $2k$, the number of green marbles be $3k$, and the number of blue marbles be $5k$, we can use this to solve both parts of the problem.

Part (a):

We are asked to find the percentage change in the number of green marbles, given that the number of red marbles increases by 20%, the number of blue marbles increases by 10%, and the total number of marbles remains the same.

Let's first calculate the initial total number of marbles: $\text{Initial total} = 2k + 3k + 5k = 10k$

After the changes:

• The number of red marbles increases by 20%, so the new number of red marbles is: $\text{New red marbles} = 2k \times 1.20 = 2.4k$
• The number of blue marbles increases by 10%, so the new number of blue marbles is: $\text{New blue marbles} = 5k \times 1.10 = 5.5k$

Let the new number of green marbles be $x$. Since the total number of marbles remains unchanged, we have the equation: $2.4k + x + 5.5k = 10k$ Simplifying this: $7.9k + x = 10k$ $x = 10k - 7.9k = 2.1k$

Thus, the new number of green marbles is $2.1k$, which means it has decreased from $3k$ to $2.1k$.

To find the percentage change: $\text{Percentage change} = \frac{3k - 2.1k}{3k} \times 100 = \frac{0.9k}{3k} \times 100 = 30\% \text{ decrease}$

Part (b):

We are given that the number of red marbles is now 6 more than the number of green marbles. From part (a), we know the number of red marbles is $2.4k$ and the number of green marbles is $2.1k$. So, we have the equation: $2.4k = 2.1k + 6$ Solving for $k$: $2.4k - 2.1k = 6$ $0.3k = 6$ $k = \frac{6}{0.3} = 20$

Now, we can find the total number of marbles. The total number is always $10k$, so: $\text{Total number of marbles} = 10 \times 20 = 200$

Final answers:

• (a) The percentage change in the number of green marbles is a 30% decrease.
• (b) The total number of marbles in the box is 200.

Let me know if you need further clarification or have any questions!

Here are 5 related questions:

1. What would the new number of blue marbles be if the increase was 15% instead of 10%?
2. If the number of red marbles increased by 30%, how would the total number of marbles change?
3. How would the solution change if the ratio of red to green to blue marbles was 3:4:5 instead?
4. What is the percentage increase in the number of red marbles after the change?
5. If the number of green marbles after the decrease was 21, what was the initial total number of marbles?

Tip: Always double-check your percentage calculations by verifying that the final total matches the unchanged overall total in these types of problems.