boomtangboy wrote:
The pages of a report are numbered consecutively from 1 to 10. If the sum of the page numbers up to and including page number x of the report is equal to one more than the sum of the page numbers following page number x, then x =
A. 4
B. 5
C. 6
D. 7
E. 8
Solution:The sum of all the page numbers is 1 + 2 + … + 10 = 55. The sum of the page numbers from 1 to x, inclusive, is (1 + x)/2 * x = x(1 + x)/2. Therefore, the sum of the page numbers from x + 1 to 10, inclusive, is 55 - x(1 + x)/2. From the information given in the problem, we can create the equation:
x(1 + x)/2 = 55 - x(1 + x)/2 + 1
x(1 + x) = 56
x^2 + x - 56 = 0
(x + 8)(x - 7) = 0
x = -8 or x = 7
Since x can’t be negative, x must be 7.
Alternate solution:If you have difficulty solving this problem algebraically, you can just check the given choices numerically. Let’s start with 6 in choice C; if it works, then that is the answer and if it doesn’t, we can then decide to move forward to 7 or move backward to 5.
If x = 6, we have the sum of the numbers 1 to 6, inclusive, as 1 + 2 + … + 6 = 21 and the sum of the numbers 7 to 10, inclusive, as 7 + 8 + 9 + 10 = 34. We see that x is not 6 and since 21 is less than 34, we should increase the value of x.
So let x = 7. We have the sum of the numbers 1 to 7, inclusive, as 1 + 2 + … + 7 = 28 and the sum of the numbers 8 to 10, inclusive, as 8 + 9 + 10 = 27. We see that 28 is exactly 1 more than 27, therefore, x must be 7.
Answer: D _________________
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