Of the three numbers, the first number is twice of the second and the second is thrice of the third number. If the average of these 3 numbers is 20, then the sum of the largest and smallest numbers is:

Options

A. 24
B. 42
C. 54
D. 60

Answer

To solve the problem step by step, we need to define the three numbers based on the relationships provided in the question.

### Step 1: Define the Variables
Let:
- The third number be \( z \).
- The second number be \( y \), which is thrice the third number:
\[ y = 3z \]
- The first number be \( x \), which is twice the second number:
\[ x = 2y \]

### Step 2: Substitute the Variables
Now, substituting \( y \) in terms of \( z \) into the equation for \( x \):
\[ x = 2(3z) = 6z \]

### Step 3: Write the Average Equation
The average of the three numbers is given as 20. Therefore, we can write:
\[ \text{Average} = \frac{x + y + z}{3} = 20 \]
This implies:
\[ x + y + z = 60 \]

### Step 4: Substitute \( x \) and \( y \) in Terms of \( z \)
Now substitute \( x \) and \( y \) in the equation:
\[ 6z + 3z + z = 60 \]

### Step 5: Combine Like Terms
Combine the terms on the left:
\[ 10z = 60 \]

### Step 6: Solve for \( z \)
Now, divide both sides by 10:
\[ z = 6 \]

### Step 7: Find \( y \) and \( x \)
Now that we have \( z \), we can find \( y \) and \( x \):
\[ y = 3z = 3 \times 6 = 18 \]
\[ x = 6z = 6 \times 6 = 36 \]

### Step 8: Identify the Largest and Smallest Numbers
Now we have:
- First number \( x = 36 \)
- Second number \( y = 18 \)
- Third number \( z = 6 \)

The largest number is \( x = 36 \) and the smallest number is \( z = 6 \).

### Step 9: Calculate the Sum of the Largest and Smallest Numbers
Finally, we need to find the sum of the largest and smallest numbers:
\[ \text{Sum} = x + z = 36 + 6 = 42 \]

### Final Answer
The sum of the largest and smallest numbers is \( 42 \).

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