A, B and C are boxes with marbles in the ratio 2 : 3 : 4. The total number of marbles in the boxes is 360. The above ratio can be changed 3 : 8 : 7 by which of the following?
(i) By transferring 20 marbles from A to B and 20 marbles from C to B.
(ii) By transferring 20 marbles from A to B and 10 marbles from C to B.
(iii) By transferring 40 marbles from A to B and 20 marbles from C to B.
(iv) By transferring 30 marbles from A to B and 30 marbles from C to B.

Options

A. (ii)
B. (iv)
C. (i)
D. (iii)

Answer

To solve the problem step by step, let's follow the outlined process:

### Step 1: Understand the initial ratio and total marbles
The initial ratio of marbles in boxes A, B, and C is given as 2:3:4. The total number of marbles is 360.

### Step 2: Calculate the value of each box in terms of x
Let the number of marbles in boxes A, B, and C be represented as:
- A = 2x
- B = 3x
- C = 4x

The total number of marbles can be expressed as:
\[ 2x + 3x + 4x = 360 \]
\[ 9x = 360 \]

### Step 3: Solve for x
Now, we can solve for x:
\[ x = \frac{360}{9} = 40 \]

### Step 4: Calculate the number of marbles in each box
Now we can find the number of marbles in each box:
- A = 2x = 2 * 40 = 80
- B = 3x = 3 * 40 = 120
- C = 4x = 4 * 40 = 160

### Step 5: Determine the new ratio after transfers
The new ratio is given as 3:8:7. Let’s express this in terms of y:
- A' = 3y
- B' = 8y
- C' = 7y

The total number of marbles remains the same:
\[ 3y + 8y + 7y = 360 \]
\[ 18y = 360 \]

### Step 6: Solve for y
Now we can solve for y:
\[ y = \frac{360}{18} = 20 \]

### Step 7: Calculate the number of marbles in each box after the new ratio
Now we can find the number of marbles in each box after the new ratio:
- A' = 3y = 3 * 20 = 60
- B' = 8y = 8 * 20 = 160
- C' = 7y = 7 * 20 = 140

### Step 8: Analyze the options for transferring marbles
Now, we will analyze the options given for transferring marbles to see which one achieves the new ratio of 3:8:7.

1. **Option (i)**: Transfer 20 marbles from A to B and 20 marbles from C to B.
- A = 80 - 20 = 60
- B = 120 + 20 + 20 = 160
- C = 160 - 20 = 140
- New ratio: 60:160:140 = 3:8:7 (Correct)

2. **Option (ii)**: Transfer 20 marbles from A to B and 10 marbles from C to B.
- A = 80 - 20 = 60
- B = 120 + 20 + 10 = 150
- C = 160 - 10 = 150
- New ratio: 60:150:150 = 2:5:5 (Incorrect)

3. **Option (iii)**: Transfer 40 marbles from A to B and 20 marbles from C to B.
- A = 80 - 40 = 40
- B = 120 + 40 + 20 = 180
- C = 160 - 20 = 140
- New ratio: 40:180:140 = 2:9:7 (Incorrect)

4. **Option (iv)**: Transfer 30 marbles from A to B and 30 marbles from C to B.
- A = 80 - 30 = 50
- B = 120 + 30 + 30 = 180
- C = 160 - 30 = 130
- New ratio: 50:180:130 = 5:18:13 (Incorrect)

### Conclusion
The only option that results in the new ratio of 3:8:7 is option (i).