1996 AHSME Problems/Problem 2

Problem

Each day Walter gets dollars for doing his chores or dollars for doing them exceptionally well. After days of doing his chores daily, Walter has received a total of dollars. On how many days did Walter do them exceptionally well?

Solution 1

If Walter had done his chores for days without doing any of them well, he would have earned dollars. He got dollars more than this.

He gets a dollar bonus every day he does his chores well. Thus, he did his chores exceptionally well days, and the answer is .

Solution 2

If Walter had done his chores for days exceptionally well, he would have earned dollars. He got dollars less than this.

He gets dollars docked from his pay if he doesn't do his chores well. Therefore, he didn't do his chores well on days. The other days, he did them exceptionally well. Therefore, the answer is .

Solution 3

Let be the number of days Walter does his chores but doesn't do them well, and let be the number of days he does his chores exceptionally well.

since there are days Walter does chores.

since is the amount he earns from doing his chores not well, and is the amount he earnes from doing his chores exceptionally well, and those two sum to dollars.

Multiply the first equation by to get:

Subtract the first equation from the second equation to get:

Thus, he does his chores well days, and the answer is .