Step 1: Understanding the Concept:
This problem involves calculating the average of numbers and using algebraic ratios to establish relationships between individual values.
Step 2: Key Formula or Approach:
1. Average of \( n \) numbers \( = \frac{\text{Sum of numbers}}{n} \).
2. Establish algebraic equations based on the relationships given between the three numbers.
Step 3: Detailed Explanation:
Let the three numbers be \( a, b, \) and \( c \).
According to the problem, half of the average of these three numbers is 20:
\[ \frac{1}{2} \times \left( \frac{a + b + c}{3} \right) = 20 \] \[ \frac{a + b + c}{6} = 20 \] \[ a + b + c = 120 \] Now, we define the relationships between the numbers:
1. The second number \( (b) \) is thrice the third number \( (c) \): \( b = 3c \).
2. The first number \( (a) \) is twice the second number \( (b) \): \( a = 2b = 2(3c) = 6c \).
Substitute these values into the sum equation:
\[ 6c + 3c + c = 120 \] \[ 10c = 120 \] \[ c = 12 \] Finding the other numbers:
\[ b = 3 \times 12 = 36 \] \[ a = 6 \times 12 = 72 \] The smallest number is \( c = 12 \) and the largest number is \( a = 72 \).
The sum of the smallest and largest number is:
\[ 12 + 72 = 84 \]
Step 4: Final Answer:
The sum of the smallest and largest number is 84.