sanju09 wrote:Babette was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, she interchanged the two digits, say p and q, in one of these ten integers. As a result, her answer for the arithmetic mean was 1.8 more than what it should have been. Then q - p equals
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Fun question!
It's always good to begin by jotting down any common formulae that apply; in this question he have the average formula:
Average = sum of terms / # of terms
In this question, we can apply the formula to the difference between the average we got and the average we were supposed to get:
1.8 = extra sum of terms/10
18 = extra sum of terms
So, the difference between pq and and qp is 18.
At this point, I'd just experiment to find two numbers that fit; based on the choices, we know that the numbers can be at most 5 digits apart.
35 and 53 are 18 apart.. there we go! 5-3=2, so choose B.
There are actually a lot of different numbers we could have chosen (e.g. 24/42, 79/97...), but of course they always have the same difference.
If we wanted to do the last step with logic, that works too; since the units digit of the difference is 8, the units digit of the "reverse difference" will be 10-8 = 2.
In other words, to get a difference that ends in 8, our units digits have to be x and x-8 (going in cycles of units digits). Our units digits could have been:
9/1
8/0
7/9
6/8
5/7
4/6
3/5
2/4
1/3
When we reverse the order, we always get a difference that ends in 2.
