864 is divided among A, B and C such that 8 times A's share is equal to 12 times B's share and also equal to 6 times C's share. How much did B get?

Options

A. 399
B. 192
C. 288
D. 72

Answer

To solve the problem, we need to express the shares of A, B, and C in terms of a common variable and then find out how much B received.

### Step 1: Set up the equations based on the given conditions.
We know from the problem that:
- \( 8A = 12B \)
- \( 8A = 6C \)

### Step 2: Express A, B, and C in terms of a common variable.
Let \( k \) be the common variable such that:
- \( 8A = k \)
- \( 12B = k \)
- \( 6C = k \)

From these equations, we can express A, B, and C as:
- \( A = \frac{k}{8} \)
- \( B = \frac{k}{12} \)
- \( C = \frac{k}{6} \)

### Step 3: Find a common value for k.
To find a common value for \( k \), we need to find the least common multiple (LCM) of the denominators 8, 12, and 6.
- The LCM of 8, 12, and 6 is 24.

### Step 4: Substitute k with the LCM.
Now we substitute \( k \) with 24:
- \( A = \frac{24}{8} = 3 \)
- \( B = \frac{24}{12} = 2 \)
- \( C = \frac{24}{6} = 4 \)

### Step 5: Calculate the total shares.
Now, we can express the shares in terms of the total amount:
- Total shares = \( A + B + C = 3 + 2 + 4 = 9 \)

### Step 6: Find the actual amounts for A, B, and C.
Since the total amount is 864, we can find the share for B:
- Each share is worth \( \frac{864}{9} = 96 \)

Now we can find B's share:
- \( B = 2 \times 96 = 192 \)

### Final Answer:
B received **192**.

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