Jars A, B, and C each contains 8 marbles. What is the minimum number of marbles that must be transferred among the jars so that the ratio of the number of marbles in jar A to the number in jar B to the number in jar C is 1:2:3?

Answer

To solve the problem, we need to determine the minimum number of marbles that must be transferred among jars A, B, and C so that the ratio of the number of marbles in jar A to jar B to jar C is 1:2:3.

### Step-by-Step Solution:

1. **Initial Setup**:
Each jar (A, B, and C) contains 8 marbles.
- Marbles in A = 8
- Marbles in B = 8
- Marbles in C = 8
- Total marbles = 8 + 8 + 8 = 24

2. **Final Ratio Requirement**:
We want the final ratio of marbles in jars A, B, and C to be 1:2:3.
Let’s denote the number of marbles in jar A as \( x \), in jar B as \( 2x \), and in jar C as \( 3x \).

3. **Setting Up the Equation**:
The total number of marbles must remain 24:
\[ x + 2x + 3x = 24 \]
Simplifying this gives:
\[ 6x = 24 \]
Dividing both sides by 6:
\[ x = 4 \]

4. **Calculating Final Counts**:
Now, we can find the number of marbles in each jar:
- Marbles in A = \( x = 4 \)
- Marbles in B = \( 2x = 2 \times 4 = 8 \)
- Marbles in C = \( 3x = 3 \times 4 = 12 \)

5. **Determining Transfers**:
- Initially, jar A had 8 marbles and needs to have 4 marbles. This means we need to remove 4 marbles from jar A.
- Jar B remains unchanged at 8 marbles.
- Jar C initially had 8 marbles but needs to have 12 marbles, which means we need to add 4 marbles to jar C.

6. **Final Transfer**:
To achieve the desired ratio:
- We take 4 marbles from jar A and transfer them to jar C.

### Conclusion:
The minimum number of marbles that must be transferred to achieve the ratio of 1:2:3 is **4 marbles**.

---