Among three numbers, the first is twice the second and thrice the third. If the average of the three numbers is 49.5, then the difference between the first and the third number is

Options

A. 54
B. 28
C. 39.5
D. 41.5

Answer

To solve the problem step by step, let's denote the three numbers as follows:

1. Let the second number be \( x \).
2. The first number is twice the second number, so it can be expressed as \( 2x \).
3. The first number is also thrice the third number, which can be expressed as \( 3y \). Thus, the third number can be expressed as \( y = \frac{2x}{3} \).

Now we have the three numbers:
- First number: \( 2x \)
- Second number: \( x \)
- Third number: \( \frac{2x}{3} \)

### Step 1: Calculate the average of the three numbers
The average of the three numbers is given by:
\[ \text{Average} = \frac{(2x + x + \frac{2x}{3})}{3} \]

### Step 2: Simplify the expression for the average
Combine the terms in the numerator:
\[ 2x + x = 3x \]
So, we rewrite the average as:
\[ \text{Average} = \frac{(3x + \frac{2x}{3})}{3} \]

To combine \( 3x \) and \( \frac{2x}{3} \), we need a common denominator:
\[ 3x = \frac{9x}{3} \]
Thus,
\[ \text{Average} = \frac{\left(\frac{9x}{3} + \frac{2x}{3}\right)}{3} = \frac{\frac{11x}{3}}{3} = \frac{11x}{9} \]

### Step 3: Set the average equal to 49.5
According to the problem, the average is 49.5:
\[ \frac{11x}{9} = 49.5 \]

### Step 4: Solve for \( x \)
Multiply both sides by 9 to eliminate the fraction:
\[ 11x = 49.5 \times 9 \]
Calculating \( 49.5 \times 9 \):
\[ 49.5 \times 9 = 445.5 \]
So we have:
\[ 11x = 445.5 \]
Now divide both sides by 11:
\[ x = \frac{445.5}{11} = 40.5 \]

### Step 5: Find the first and third numbers
Now that we have \( x \):
- First number \( = 2x = 2 \times 40.5 = 81 \)
- Third number \( = \frac{2x}{3} = \frac{2 \times 40.5}{3} = \frac{81}{3} = 27 \)

### Step 6: Calculate the difference between the first and third numbers
Now we find the difference:
\[ \text{Difference} = 81 - 27 = 54 \]

### Final Answer
The difference between the first and the third number is \( 54 \).

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