If A is `(1)/(6)` of C, and B is twice of A, and the average of A,B and C is 30, then the difference between A and C is

Options

A. 60
B. 40
C. 80
D. 50

Answer

To solve the problem step by step, we will define the relationships given in the question and then find the required difference between A and C.

### Step 1: Define the relationships
We know from the problem:
- A = (1/6)C
- B = 2A

### Step 2: Express B in terms of C
Since A = (1/6)C, we can substitute this into the equation for B:
- B = 2A = 2 * (1/6)C = (2/6)C = (1/3)C

### Step 3: Find the average of A, B, and C
The average of A, B, and C is given as 30. We can express this mathematically:
\[ \text{Average} = \frac{A + B + C}{3} = 30 \]

### Step 4: Substitute A and B in terms of C
Now we substitute A and B in terms of C into the average formula:
\[ \frac{(1/6)C + (1/3)C + C}{3} = 30 \]

### Step 5: Combine the terms in the numerator
To combine the terms, we need a common denominator:
\[ \frac{(1/6)C + (2/6)C + (6/6)C}{3} = \frac{(1 + 2 + 6)C/6}{3} = \frac{9C/6}{3} = \frac{3C}{6} = \frac{C}{2} \]

### Step 6: Set up the equation
Now we set the equation:
\[ \frac{C}{2} = 30 \]

### Step 7: Solve for C
To find C, we multiply both sides by 2:
\[ C = 30 * 2 = 60 \]

### Step 8: Find A
Now that we have C, we can find A:
\[ A = \frac{1}{6}C = \frac{1}{6} * 60 = 10 \]

### Step 9: Find the difference between A and C
Finally, we find the difference between C and A:
\[ \text{Difference} = C - A = 60 - 10 = 50 \]

### Conclusion
The difference between A and C is **50**.

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