This trivial problem is in the spirit of one of my earlier posts in which I mull over how people are losing very basic mathematical skills. Here is the problem statement:
You have two job offers, from two different companies (Company A and Company B) with the same starting salary, $50,000 per year. Company A gives you an offer which says “We will increase your salary by $500 at the end of every six months” and Company B counters that offer by saying “We will increase your salary by $2000 at the end of every year“. Considering all other factors equal, which company would you choose?
Hint: Are these offers really different…or it’s just jumbling of the words to create an illusion?
There, jog your grey cells a bit with it. 
Don’t kill all the fun by using calculators or excel sheets. Do it mentally and make a note of how much time it takes. Btw, I have just reworded the problem from an old book of puzzles; so just in case you have the book, the answers may not be the same.
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Hm, that one actually looks pretty easy. Unless you meant $500 at the end of each quarter, or $1000 each year? I’m staring at this, trying to figure out the trick!
Ellen: don’t make this easier than it is
…or should I say - don’t make it harder than it is ? 
Try calculating salaries at 6 month intervals :)..there another hint.
Well, if you work there for longer than about 15 months, the $2K/year raise job outpaces the semiannual $500 raise job. So I’ll work with Company B, assuming I stay there longer than 15 months.
What did you take this problem from?
B, unless it’s an insufferable job, or unless I’m missing something…
You changed your layout! I was confused.
Oh wait, is the increase a monthly increase or an annual increase? As in, do you make $50K 500 500 in year one or do you make $50K 500(6) in year one?
(there are plusses in that last comment but you can’t see them for some reason)
Dimes I understand what you mean.
sorry about the confusion. I thought my readers might welcome a change after looking at the same old template for long. Do you like the new one?
Btw, I have asked the salary question for a reason :), some things are just not what they seem like at first glance.
The increase is after 6 months. So in this case, if your calculated annual salary is $50K, after six months the calculated annual salary will be $50,500.
Don’t worry, I prefer the new layout.
I don’t really get it, but I suppose that you’re hinting that A is the better offer. Running a spreadsheet would be cheating but (here’s a secret) I don’t know how to use Excel.
Don’t leave us hanging!
Sorry, couldn’t resist…
First offer:
0
25000 (now at 25500 half a year)
50500 (now at 26000 half a year)
76500 (now at 26500 half a year)
103000 (now at 27000 half a year)
130000 (now at 27500 half a year)
157500 (now at 28000 half a year)
Second offer:
0
25000
50000 (now at 52000 a year)
76000
102000 (now at 54000 a year)
129000
156000 (now at 56000 a year)
The first offer is better by $500 a year
Oh well, now I can go to bed.
1st offer:
0 1 2 3 4 5 6 7 .. N = ((N 1)*N))/2 = K*N
2nd offer:
0 0 0 4 0 8 0 12 .. (N-1)*2 = 0 4 8 12 .. (N-1)*2 = 4 * (0 1 2 3 .. ((N-1)/2)) = 4 * (((N-1)/2 1)*((N-1)/2))/2 = 2 * ((N-1 2)*(N-1))/4 = ((N 1)*(N-1))/2 = K*(N-1)
Assumptions:
a) 1=$500
b) N is the number of half year periods minus 1
Lag: K*N - K*(N-1) = K = (N 1)/2
(N 1)/2 = number of half year periods / 2 * $500 = number of years * $500
Example: N=5 (3 years)
Result: 6/2 = 3 * $1500
That doesn’t make sense to me. I’m thinking Company B.
Oh well, after spending so much time figuring this out I discovered that one of my assumptions was wrong. The $500 salary increase in the 1st offer should be spread over a year, not 6 months.
0
25000 (now at 25250 half a year)
50250 (now at 25500 half a year)
75750 (now at 25750 half a year)
101500 (now at 26000 half a year)
127500 (now at 26250 half a year)
153750 (now at 27000 half a year)
It sounds like 2nd offer out-paces 1st one sometime in the first half of second year.
Something about the question doesn’t seem right. I don’t see how the answer can be anything other than B.
Ah..now this is getting interesting.
OK, I won’t keep you guys hanging for long. The point is to show that it is not as straight forward to jump to a conclusion here….I think Yan’s confusion is valid.
Here is the original problem from the text:
“First Option (Offer B in our case): Initial salary $40,000 to be increased after each 12 months by $2000.
Second Option (Offer A in our case): Initial salary $40,000 to be increased after each 6 months by $500.
Which option should you choose?”
The book says Offer A is better because of this (Yan’s 1st solution):
- It assumes that the $500 is distributed over the next 6 months. So if you break salaries over 6 months they would look like this:
Offer A: 25000 (0 to 6 months) plus 25500 (6 to 12 months) plus 26000 (12 to 18 months) plus 26500 (18 to 24 months). Gives TOTAL salary for first year as $50,500 and for second year as $52,500.
Offer B: 25000 (0 to 6 months) plus 25000 (6 to 12 months) plus 26000 (12 to 18 months) plus 26000 (18 to 24 months). Gives TOTAL salary for first year as $50,000 and for second year as $52,000.
That’s interesting, but I don’t agree with the book. While it is quite possible that some companies do that. But in that case the appropriate offer should have been stated this way:
- $25,000 per 6 months, increased at $500 per every 6 months. This would effectively mean that $500 is distributed over 6 months.
What usually happens is companies talk in terms of “budgeted salary”. So when they give you a $500 raise, it is usually added to your “budgeted salary”. That means instead of $50,000 a year you will be getting at “the rate” of $50,500 a year. This is what makes Offer A look bad. Here is why:
- For the first six months you will be paid at the rate of $50,000 a year. For the next six months at the rate of $50,500 a year. The TOTAL salary in this case is NOT $50,500. It will be $50,250 (because $500 is distributed over 12 months, and not 6 months). You can calculate it in a complicated way like Yan did …or you can just take the average of the two rates.
- After one year the salary will increase to $51,000 per year and after 18 months it will increase to $51,500 per year. But your average salary for the second year is just $51,250 in this case. Now, this is easily beaten by Offer B.
Anyways, I think the point is established. Things are not as obvious on the first glance, especially when it comes to numbers…you need to stop and think for a while.
Another lesson is that with the second case of “budgeted salary” the Offer A is better in the first year…but not good in the second year (like Yan and Ellen mention). Watch out for such offers…always play around with numbers and calculate a few years down the line before you make a salary decision.
….That’s the longest comment I have ever made. I hate myself.
Thanks for participating guys…it was fun.
We don’t get this at all.
Offer b is clearly the best.
remove the 50k from both at the start:
6mo 12mo 18mo 24mo
a 500 1000 1500 2000 (running ‘extra’)
b 0 2000 2000 4000
Am I being stupid here?
Dav, here is what is happening; in the above table you gave, there is 2000 on 12 months and another 2000 on 18 months. That effectively makes it a $4000 raise in the second year. The correct way of presenting it would be:
Year 1 Year 2 Year 3
______ _________ _________
A: 0 500 1000 1500 2000 2500
B: 0 0 2000 0 4000 0
or, still better:
Year 1 Year 2 Year 3
______ _________ _________
A: 0 500 1000 1500 2000 2500
B: 0 0 1000 1000 2000 2000
Then, add the totals under Year 1 (A is better by $500), Year 2 (A is again better by $500)…and so on. This happens because $500 is distributed over 6 months, while $2000 is distributed over 12 months. (Check the later post titled “The Salary Problem, and Two Possible Solutions” for details)
The above was just one possible solution.
The second solution that puts A at disadvantage is the assumption that $500 is distributed over 12 months instead of 6 (this is mostly what happens in real life calculations)…and I think this is what you are thinking about. In this case, the distribution will look like this:
Year 1 Year 2 Year 3
______ _________ _________
A: 0 500 1000 0 1500 0
B: 0 0 2000 0 4000 0
Now here A certainly loses, but not in the first year. A will go behind B sometime in the first half of Year 2.
Hope that helps…or did I confuse you even more?
Its clear to me that offer B is clearly the best, if you believe you will be around substantially more then 3 years, (as a math problem, the second offer is better by a total of $500 at the end of 3 years).
In the real world, you can’t assume that either company can make any agreement that goes out three years unless its in writing. Also you’d expect your salary to be based on your performance after the second year, rather than the hiring agreement.
It wasn’t clearly stated whether the increase of $500 was over the annual salary (and hence would be prorated) or over 6 months.
so i read all those and i still dont get it. Both are worded the same, and it seems like it would just be 1000 added per year in A and 2000 in B. so you have to wait longer to get the extra money in B… that wouldnt matter to me. What am i missing?
haha just realized im reading a REALLY old post.
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