Four numbers are such that the mean of the first two exceeds the first number by 1; the mean of the first three exceeds the mean of the first two by 2; and the mean of all four exceeds the mean of the first three by 3. Find the difference between the largest and the smallest of the four numbers.
Answer and solution
Correct answer: 15
Solution
Answer: 15
1. Let the first number be a. The mean of two numbers is a+1, so their sum is 2a+2 and the second number is a+2.
2. The mean of three is a+3, giving total 3a+9. Subtract 2a+2 to get third number a+7.
3. The mean of four is a+6, giving total 4a+24. Subtract 3a+9 to get fourth number a+15.
4. They increase in order a, a+2, a+7, a+15. Largest−smallest=15.
Why this method works:
An average is a total divided by a count. Whenever a new number is added, recover its value by subtracting the earlier total from the new total; the increase in the average is not itself the new number.
Independent check and common trap:
Setting the first number to 0 gives numbers 0,2,7,15. Their successive means are 1,3,6, with increments 1,2,3 as required. Adding any common constant shifts all numbers equally and leaves the range 15 unchanged.
Apply: Shift the first number to 0 without changing the range. Running averages become 0,1,3,6; running totals 0,2,9,24. Differences give 0,2,7,15.
Check: Setting the first number to 0 gives numbers 0,2,7,15. Their successive means are 1,3,6, with increments 1,2,3 as required. Adding any common constant shifts all numbers equally and leaves the range 15 unchanged.
1. Let the first number be a. The mean of two numbers is a+1, so their sum is 2a+2 and the second number is a+2.
2. The mean of three is a+3, giving total 3a+9. Subtract 2a+2 to get third number a+7.
3. The mean of four is a+6, giving total 4a+24. Subtract 3a+9 to get fourth number a+15.
4. They increase in order a, a+2, a+7, a+15. Largest−smallest=15.
Why this method works:
An average is a total divided by a count. Whenever a new number is added, recover its value by subtracting the earlier total from the new total; the increase in the average is not itself the new number.
Independent check and common trap:
Setting the first number to 0 gives numbers 0,2,7,15. Their successive means are 1,3,6, with increments 1,2,3 as required. Adding any common constant shifts all numbers equally and leaves the range 15 unchanged.
G Strategy shortcut
Spot: Averages.Apply: Shift the first number to 0 without changing the range. Running averages become 0,1,3,6; running totals 0,2,9,24. Differences give 0,2,7,15.
Check: Setting the first number to 0 gives numbers 0,2,7,15. Their successive means are 1,3,6, with increments 1,2,3 as required. Adding any common constant shifts all numbers equally and leaves the range 15 unchanged.
G Strategy: Spot: Averages. Apply: Shift the first number to 0 without changing the range. Running averages become 0,1,3,6; running totals 0,2,9,24. Differences give 0,2,7,15. Check: Setting the first number to 0 gives numbers 0,2,7,15. Their successive means are 1,3,6, with increments 1,2,3 as required. Adding any common constant shifts all numbers equally and leaves the range 15 unchanged.