There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than average of first two numbers, and average of first four numbers is 3 more than average of first three numbers. Then, the difference between the largest and the smallest numbers, is
Question:
There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than average of first two numbers, and average of first four numbers is 3 more than average of first three numbers. Then, the difference between the largest and the smallest numbers, is
There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than average of first two numbers, and average of first four numbers is 3 more than average of first three numbers. Then, the difference between the largest and the smallest numbers, is
Options
Answer: 15
Explanation:
Let the four numbers be a, b, c, d. Step 1: Translate the conditions into equations Average of first two numbers is 1 more than the first: (a + b)/2 = a + 1 → a + b = 2a + 2 → b = a + 2 Average of first three numbers is 2 more than average of first two: (a + b + c)/3 = (a + b)/2 + 2 Plug b = a + 2: a + b = 2a + 2 (a + b + c)/3 = (2a + 2 + c)/3 (a + b)/2 + 2 = (2a + 2)/2 + 2 = a + 3 Equation: (2a + 2 + c)/3 = a + 3 → c = a + 7 Average of first four numbers is 3 more than average of first three: (a + b + c + d)/4 = (a + b + c)/3 + 3 Plug b = a + 2, c = a + 7: a + b + c = 3a + 9 → average = a + 3 a + b + c + d = 3a + 9 + d → average = (3a + 9 + d)/4 Equation: (3a + 9 + d)/4 = a + 6 → d = a + 15 Step 2: Find difference between largest and smallest Smallest = a Largest = d = a + 15 Difference = 15
Explanation:
Let the four numbers be a, b, c, d. Step 1: Translate the conditions into equations Average of first two numbers is 1 more than the first: (a + b)/2 = a + 1 → a + b = 2a + 2 → b = a + 2 Average of first three numbers is 2 more than average of first two: (a + b + c)/3 = (a + b)/2 + 2 Plug b = a + 2: a + b = 2a + 2 (a + b + c)/3 = (2a + 2 + c)/3 (a + b)/2 + 2 = (2a + 2)/2 + 2 = a + 3 Equation: (2a + 2 + c)/3 = a + 3 → c = a + 7 Average of first four numbers is 3 more than average of first three: (a + b + c + d)/4 = (a + b + c)/3 + 3 Plug b = a + 2, c = a + 7: a + b + c = 3a + 9 → average = a + 3 a + b + c + d = 3a + 9 + d → average = (3a + 9 + d)/4 Equation: (3a + 9 + d)/4 = a + 6 → d = a + 15 Step 2: Find difference between largest and smallest Smallest = a Largest = d = a + 15 Difference = 15
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