goyalsau wrote:The integers 1, 2......, 40 are written on a black board. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?
(1) 820
(2) 821
(3) 781
(4) 819
(5) 780
Step 1 of the Kaplan Method for PS: analyze the problem
Too many people dive right into math without understand what a question is asking. Especially when you have complicated word problems, it's essential to analyze the problem first.
Here, we see we have the numbers 1 through 40, then we're erasing two numbers and replacing them with the sum of the numbers minus 1.
Step 2 of the Kaplan Method for PS: state the task
Now we make sure we identify exactly what the question is asking. Here, we want the sum of the new numbers up on the board.
Well, each time we erase/replace, we're reducing the sum by 1 (if we replaced the pair with "a+b", there would be no change to the total sum; since we're replacing with "a+b-1", our sum is slowly being reduced).
Doing the operation 39 times means our final sum will be 39 less than the original sum.
So, the question is:
What number is 39 less than the sum of the integers from 1 to 40?
Step 3 of the Kaplan Method for PS: approach the problem strategically
The question asks for a sum, so backsolving won't work - we actually need to do some math.
To find the sum of a set of consecutive numbers, we use the formula:
sum = average of set * # of terms
and
average of a set of consecutive #s = (first term + last term)/2
So:
# of terms = 40
average = (1+40)/2 = 41/2 = 20.5
Sum = (20.5)(40) = 820
Answering the actual question:
820 - 39 = 781
Step 4 of the Kaplan Method for PS: confirm the answer makes sense
There are a lot of questions on the GMAT with twists at the end - we want to make sure that we've answered the correct question.
Here, we've remember to subtract 39, so we confidently choose (C).
