1996 AHSME Problems/Problem 2

Problem

Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?

$\text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7$

Solution 1

If Walter had done his chores for $10$ days without doing any of them well, he would have earned $3 \cdot 10 = 30$ dollars. He got $6$ dollars more than this.

He gets a $5 - 3 = 2$ dollar bonus every day he does his chores well. Thus, he did his chores exceptionally well $\frac{6}{2} = 3$ days, and the answer is $\boxed{A}$ .

Solution 2

If Walter had done his chores for $10$ days exceptionally well, he would have earned $5 \cdot 10 = 50$ dollars. He got $50 - 36 = 14$ dollars less than this.

He gets $2$ dollars docked from his pay if he doesn't do his chores well. Therefore, he didn't do his chores well on $\frac{14}{2} = 7$ days. The other $10 - 7 = 3$ days, he did them exceptionally well. Therefore, the answer is $\boxed{A}$ .

Solution 3

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

$b + w = 10$ since there are $10$ days Walter does chores.

$3b + 5w = 36$ since $3b$ is the amount he earns from doing his chores not well, and $5w$ is the amount he earnes from doing his chores exceptionally well, and those two sum to $36$ dollars.