Re the answer about choosing $1,000 per year raise vs $300 every six months, every time I calculate it, if you chose the $300, at the end of the first year you’d have $10,900 vs $11,000, 2nd year you’d be ahead $12,100 vs $12,000, 3rd year $13,300 vs $13,000, 4th year $14,500 vs $14,000, and $15,700 vs $15,000 in the fifth year, not the 3rd year. Am I crazy?
LINK TO COLUMN: What’s better, a $1,000 raise each year, or a $300 raise every six months? - The Straight Dope
May as well show your numbers. Assume you only get paid once or twice a year (i.e. only count the raise money, not the base salary):
$300 every 6 mos. $1000 per year
0.5y +300 0
1.0y +600 +1000
1.5y +900 0
2.0y +1200 +2000
2.5y +1500 0
3.0y +1800 +3000
Totals: +6300 +6000
Assuming I undertstand the question right (and Cecil did, AND vos Savant did)…
If you choose to look at it as a simple rate-of-change problem, as Vos Savant apparently did, it’s pretty straightforward: 1000/y^2 versus 300/(0.5y)^2.
300/(0.5y)^2 = 300/.025y^2 = 1200/y^2 (Cecil’s $1200 annualized salary hike)
The fact that the increases are in discrete chunks muddies the waters a bit.
Of course, practically nobody would interpret an “$X raise” that way. As for the statement “at the end of one year you’d be ahead $300,” well, I’m damned if I know what that means.
At least I think that’s right. To be perfectly honest, I’m a bit squiffy at the moment.
A problem with most such logic or probability questions is that it often depends very much on the exact wording of the question. They’re typically not “realistic” day-to-day math problems, but special set-ups and special wording.
It looks like the first thing you are doing is assigning that first $300 for the first 6 month period. Cecil and Marilyn assume you don’t get paid that raise until the second period, with the first period at your base salary level.
I don’t know if you’re referring to Bryan or myself, but you’re right either way. With the $1000/year raise, you don’t get diddly the first year. That doesn’t affect my calculations, but it is why you’re ahead $300 after a year.
(Of course, the same would be true of the more intuative 300/0.5y^2 situation.)
It’s kind of a tortoise-and-hare race, but with a deceptively fast tortoise…who has a head start. 
I still don’t quite get this one. Is it like this?
$300 every 6 mos. $1000 per year
0.5y 0 0
1.0y +300 0
1.5y +600 0
2.0y +900 +1000
2.5y +1200 0
3.0y +1500 +2000
Totals: +4500 +3000
Yes. If you extend it a couple of years and keep annual totals for the semiannual rasies, you get Marilyn’s (and Cecil’s) numbers:
year $300/0.5y $1000/y
--------------------------------
0.5 0
1.0 +300 [+300] 0
1.5 +600
2.0 +900 [+1500] +1000
2.5 +1200
3.0 +1500 [+2700] +2000
3.5 +1800
4.0 +2100 [+3900] +3000
4.5 +2400
5.0 +2700 [+5100] +4000
Man, this one had me mystified forever.
Presented in terms of what you’re paid
year $300/0.5y $1000/y
salary total paid salary total paid
--------------------------------------------
0.5 5000 5000
1.0 5300 10,300 10,000 10,000
1.5 5600 15,900
2.0 5900 21,800 11,000 21,000
2.5 6200 28,000
3.0 6500 34,500 12,000 33,000
3.5 6800 41,300
4.0 7100 48,400 13,000 46,000
4.5 7400 55,800
5.0 7700 63,500 14,000 60,000