Question

The half of the average value of three number is 20. The second number is thrice the third number and first number is twice the second number. What is the sum of smallest and largest number?

• A. 68
• B. 56
• C. 84
• D. 96

Hint:

In problems involving relationships between numbers, always start by defining one number as a variable (e.g., \(x\)) and expressing the other numbers in terms of \(x\). This simplifies the process of forming a single equation to solve.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the relationship between three numbers and the half of their average. We need to find the sum of the smallest and the largest of these three numbers.
Step 2: Key Formula or Approach:
The average of three numbers A, B, and C is given by the formula: \[ \text{Average} = \frac{A + B + C}{3} \] We will set up equations based on the given conditions to solve for the numbers.
Step 3: Detailed Explanation:
First, let's find the average of the three numbers. Given that half of the average value is 20. \[ \frac{1}{2} \times \text{Average} = 20 \] \[ \text{Average} = 20 \times 2 = 40 \] Let the three numbers be the first, second, and third number. The sum of these three numbers is: \[ \text{Sum of numbers} = \text{Average} \times 3 = 40 \times 3 = 120 \] Now, let's establish the relationship between the numbers. Let the third number be \(x\). According to the question: The second number is thrice the third number, so: \[ \text{Second number} = 3x \] The first number is twice the second number, so: \[ \text{First number} = 2 \times (3x) = 6x \] The three numbers are \(6x\), \(3x\), and \(x\). Their sum is 120. \[ 6x + 3x + x = 120 \] \[ 10x = 120 \] \[ x = \frac{120}{10} = 12 \] Now we can find the three numbers: Smallest number (the third one) = \(x = 12\).
Second number = \(3x = 3 \times 12 = 36\).
Largest number (the first one) = \(6x = 6 \times 12 = 72\).
The question asks for the sum of the smallest and largest numbers. \[ \text{Sum} = \text{Smallest number} + \text{Largest number} \] \[ \text{Sum} = 12 + 72 = 84 \]
Step 4: Final Answer:
The sum of the smallest and largest number is 84.