A certain sum of money was divided among A, B and C in a certain way. C got half as much as A and B together got. A got one third of what B and C together got. What is the ratio of A’s share to that of C’s share?

Options

A. `1 : 4`
B. ` 3 : 4`
C. ` 4 : 1`
D. ` 3 : 5`

Answer

To solve the problem step by step, we will define the shares of A, B, and C using variables and then set up equations based on the information given in the question.

### Step 1: Define the variables
Let:
- A's share = \( a \)
- B's share = \( b \)
- C's share = \( c \)

### Step 2: Set up the equations based on the problem statement
1. **C got half as much as A and B together**:
\[ c = \frac{1}{2}(a + b) \]

2. **A got one third of what B and C together got**:
\[ a = \frac{1}{3}(b + c) \]

### Step 3: Substitute the first equation into the second
From the first equation, we can express \( b \) in terms of \( a \) and \( c \):
\[ c = \frac{1}{2}(a + b) \implies b = 2c - a \]

Now substitute \( b \) into the second equation:
\[ a = \frac{1}{3}((2c - a) + c) \]
This simplifies to:
\[ a = \frac{1}{3}(3c - a) \]

### Step 4: Solve for \( a \)
Multiply both sides by 3 to eliminate the fraction:
\[ 3a = 3c - a \]
Rearranging gives:
\[ 3a + a = 3c \implies 4a = 3c \]

### Step 5: Find the ratio of A's share to C's share
Now, we can express the ratio of A's share to C's share:
\[ \frac{a}{c} = \frac{3}{4} \]

### Conclusion
Thus, the ratio of A's share to C's share is:
\[ \text{Ratio of A's share to C's share} = 3:4 \]

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