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?
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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.
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.