avenus wrote:Nice one, give it a shot.

Jim's only source of income comes from his job as a question writer. In this capacity, Jim earns a flat salary of $200 per week plus a fee of $9 for every question that he writes. Every year, Jim takes exactly two weeks of unpaid vacation to visit his granny in Antarctica to get some fresh air. If a regular year consists of 52 weeks and the number of questions that Jim wrote in each of the past 5 years was an odd number greater than 20, which of the following could be Jim's median annual income over the past 5 years?

(A) $22,474
(B) $25,673
(C) $27,318
(D) $28,423
(E) $31,227

Tricksy!

Jim's salary per year is 10000 + 9n (n = # of questions written that year).

Since Jim writes an odd # of questions (the greater than 20 part is irrelevant give these answer choices), his annual salary will always be odd. Adding up 5 odd numbers gives us an odd total. When we divide that odd total by 5, we'll get an odd (or non-integer) average annual salary: eliminate (A) and (C).

Next, let's knock off the $10k flat salary he receives each year to determine what his annual income is from writing questions. The remaining choices become:

(B) 15673
(D) 18423
(E) 21227

Since this part of his salary is "9n", we know the number must be divisible by 9. Of the 3 choices, only 18423 is divisible by 9: choose (D).

[note: to quickly determine if a number is divisible by 9, add up the digits; if the sum of the digits is a multiple of 9, so is the original number. 1+8+4+2+3 = 18, therefore 18423 is divisible by 9!]

If we had started with step 2 (divisibility by 9), we could have narrowed it down to:

(a) 12474 (1+2+4+7+4 = 18); and
(d) 18423

and then used oddness/evenness to make our final choice.

Answer is right, although with a little bit of luck. There's one little mistake. The problem talks about median annual income, not average income :!: :!: A similar approach shows that the median must be an odd number and thus answer is the one given by Stuart.

avenus wrote:Answer is right, although with a little bit of luck. There's one little mistake. The problem talks about median annual income, not average income :!: :!: A similar approach shows that the median must be an odd number and thus answer is the one given by Stuart.

The mean is equal to the median in an evenly spaced set of numbers. Since it does not specify that he made up a different number of odd problems every year or the same number of odd problems every year, I think it is safe to assume an evenly spaced set for the number of problems he made up for each year.

avenus wrote:Answer is right, although with a little bit of luck. There's one little mistake. The problem talks about median annual income, not average income :!: :!: A similar approach shows that the median must be an odd number and thus answer is the one given by Stuart.

Whoops! Misread that part.

Actually, if I had noticed it was median instead of mean, it's even easier, since we just need to show that it will be odd each year instead of worrying about dividing by 5.