The only quibble I have with Cecil’s reasoning about this is the statement of the conditions. The set-up for this problem is the following: Suppose you’re hired for a job at a given salary. Let’s say it’s $10,000, just to give a number, although it’s not really relevant to the problem. Just to have a definite year for this, let’s say that your first year on the job is the year 2000. You’re given two options: You can have a $1000 raise every year, or you can have a $300 raise every six months. It’s clear that a $1000 raise every year means that you will get $10,000 in 2000, $11,000 in 2001, $12,000 in 2002, etc. What’s not clear is what a $300 raise every six months means. Does that mean that the amount you make *for that six months* is raised by $300 dollars? Or does it mean that the amount you make *per year* is raised by $300, so the amount you make for six months is raised by $150? To make the analysis that Cecil work right, you would have to assume that he means the first of those two, that the amount you make for that six months is raised by $300.

Here’s the figures: Suppose you ask for $1000 raise every year. Then the amount of money you make is as follows:

2000 (first six months): $5,000

2000 (second six months): $5,000

2000 (yearly total): $10,000

2001 (first six months): $5,500

2001 (second six months): $5,500

2001 (yearly total): $11,000

2002 (first six months): $6,000

2002 (second six months): $6,000

2002 (yearly total): $12,000

2003 (first six months): $6,500

2003 (second six months): $6,500

2003 (yearly total): $13,000

2004 (first six months): $7,000

2004 (second six months): $7,000

2004 (yearly total): $14,000

2005 (first six months): $7,500

2000 (second six months): $7,500

2000 (yearly total): $15,000

2005 (first six months): $8,000

2005 (second six months): $8,000

2005 (yearly total): $16,000

On the other hand, suppose that you ask for a $300 raise every six months, and we’ll assume that this means that you will get a $300 raise in the amount that you get for each six months. Then this works out to:

2000 (first six months): $5,000

2000 (second six months): $5,300

2000 (yearly total): $10,300

2001 (first six months): $5,600

2001 (second six months): $5,900

2001 (yearly total): $11,500

2002 (first six months): $6,200

2002 (second six months): $6,500

2002 (yearly total): $12,700

2003 (first six months): $6,800

2003 (second six months): $7,100

2003 (yearly total): $13,900

2004 (first six months): $7,400

2004 (second six months): $7,700

2004 (yearly total): $15,100

2005 (yearly total): $8,000

2000 (second six months): $8,300

2000 (yearly total): $16,300

2005 (first six months): $8,600

2005 (second six months): $8,900

2005 (yearly total): $17,500

Notice what has happened here. There’s two things going on. In effect, the $300 per six months option is equivalent to getting a $1200 per year raise, plus getting a $300 bonus each year. Clearly that’s better than getting a $1000 per year raise.

On the other hand, suppose that the condition that you get a $300 raise each six months means that you get a raise in your yearly salary of $300, so each six months you get a raise of $150. (This strikes me as just as reasonable an interpretation of the condition as the other one.) This works out as follows:

2000 (first six months): $5,000

2000 (second six months): $5,150

2000 (yearly total): $10,150

2001 (first six months): $5,300

2001 (second six months): $5,450

2001 (yearly total): $10,750

2002 (first six months): $5,600

2002 (second six months): $5,750

2002 (yearly total): $11,350

2003 (first six months): $5,900

2003 (second six months): $6,050

2003 (yearly total): $11,950

2004 (first six months): $6,200

2004 (second six months): $6,350

2004 (yearly total): $12,550

2005 (first six months): $6,500

2000 (second six months): $6,650

2000 (yearly total): $13,150

2005 (first six months): $6,800

2005 (second six months): $6,950

2005 (yearly total): $13,750

So this would be the equivalent of getting a $600 per year raise, plus a $150 bonus per year. After the first year, this is not as good as getting a $1000 raise per year.