The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. Then, the largest number in the original set of three numbers is
Question:
The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. Then, the largest number in the original set of three numbers is
The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. Then, the largest number in the original set of three numbers is
Options
Answer: 70
Explanation:
Let the three numbers be x < y < z. Step 1: Original average: (x + y + z)/3 = 28 → x + y + z = 84 Step 2: After changes: smallest +7, largest −10 New mean = middle + 2 → (x + 7 + y + z − 10)/3 = y + 2 Simplify: x + z = 2y + 9 Step 3: Original sum: x + z = 84 − y → 84 − y = 2y + 9 → y = 25 Step 4: Difference after change: (z − 10) − (x + 7) = 64 → z − x = 81 Also x + z = 84 − 25 = 59 Solve: z = 70, x = −11 Step 5: Largest number in original set = 70
Explanation:
Let the three numbers be x < y < z. Step 1: Original average: (x + y + z)/3 = 28 → x + y + z = 84 Step 2: After changes: smallest +7, largest −10 New mean = middle + 2 → (x + 7 + y + z − 10)/3 = y + 2 Simplify: x + z = 2y + 9 Step 3: Original sum: x + z = 84 − y → 84 − y = 2y + 9 → y = 25 Step 4: Difference after change: (z − 10) − (x + 7) = 64 → z − x = 81 Also x + z = 84 − 25 = 59 Solve: z = 70, x = −11 Step 5: Largest number in original set = 70
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