aaron1981 wrote:The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?
(A) 23
(B) 25
(C) 32
(D) 43
(E) 63
This is essentially a problem about two overlapping groups, where every element belongs either to one group or to both groups.
Use the following formula:
Total = Group 1 + Group 2 - Both.
The big idea is to SUBTRACT THE OVERLAP.
In the problem above, there is an overlap between the SUM OF THE FIRST 4 NUMBERS and the SUM OF THE LAST 4 NUMBERS:
The 4th number is included in BOTH GROUPS.
Thus, when we calculate the sum of the first 4 numbers and the sum of the last 4 numbers, the OVERLAP -- the 4th number -- will be counted twice.
So that we don't double-count the 4th number, we must subtract it from the total.
Resulting equation:
Total of all 7 numbers = (sum of the first 4 numbers) + (sum of the last 4 numbers) - (4th number).
Sum = (number of numbers)(average of the numbers).
Thus:
Total of all 7 numbers = (number of numbers)(average of the 7 numbers) = 7*20 = 140.
Sum of the first 4 numbers = (number of numbers)(average of the first 4 numbers) = 4*19 = 76.
Sum of the last 4 numbers = (number of numbers)(average of the last 4 numbers) = 4*24 = 96.
Plugging these values into the blue equation above, we get:
140 = 76 + 96 - (4th number)
4th number = 76 + 96 - 140 = 32.
The correct answer is
C.
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