To solve the problem step by step, we will first determine the number of marbles in each box based on the given ratio and total number of marbles. Then, we will account for the transfers of marbles between the boxes and finally calculate the new ratio.
### Step 1: Determine the number of marbles in each box
The ratio of marbles in boxes A, B, and C is given as 3:5:7. Let the number of marbles in A, B, and C be represented as 3x, 5x, and 7x respectively.
Since the total number of marbles is 75, we can set up the equation:
\[
3x + 5x + 7x = 75
\]
### Step 2: Solve for x
Combine the terms:
\[
15x = 75
\]
Now, divide both sides by 15:
\[
x = \frac{75}{15} = 5
\]
### Step 3: Calculate the number of marbles in each box
Now that we have the value of x, we can find the number of marbles in each box:
- Number of marbles in A:
\[
3x = 3 \times 5 = 15
\]
- Number of marbles in B:
\[
5x = 5 \times 5 = 25
\]
- Number of marbles in C:
\[
7x = 7 \times 5 = 35
\]
### Step 4: Account for the transfers of marbles
1. **Transfer from B to A**: 3 marbles are transferred from B to A.
- New number of marbles in A:
\[
15 + 3 = 18
\]
- New number of marbles in B:
\[
25 - 3 = 22
\]
2. **Transfer from C to B**: 5 marbles are transferred from C to B.
- New number of marbles in B:
\[
22 + 5 = 27
\]
- New number of marbles in C:
\[
35 - 5 = 30
\]
### Step 5: Calculate the new ratio
Now we have the new number of marbles:
- A = 18
- B = 27
- C = 30
The new ratio of marbles in A, B, and C is:
\[
18 : 27 : 30
\]
### Step 6: Simplify the ratio
To simplify the ratio, we can divide each term by their greatest common divisor (GCD), which is 9:
\[
\frac{18}{9} : \frac{27}{9} : \frac{30}{9} = 2 : 3 : \frac{10}{3}
\]
However, to express the ratio in whole numbers, we can multiply each term by 3:
\[
2 \times 3 : 3 \times 3 : 10 = 6 : 9 : 10
\]
### Final Answer
The new ratio of the marbles in boxes A, B, and C is:
\[
6 : 9 : 10
\]
