Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1996 AHSME Problems/Problem 2"
 
Line 43: Line 43:
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=1|num-a=3}}
 
{{AHSME box|year=1996|num-b=1|num-a=3}}
 +
{{MAA Notice}}

Latest revision as of 13:06, 5 July 2013

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.

Multiply the first equation by $3$ to get:

$3b + 3w = 30$

$3b + 5w = 36$

Subtract the first equation from the second equation to get:

$5w - 3w = 36 - 30$

$2w = 6$

$w = 3$

Thus, he does his chores well $3$ days, and the answer is $\boxed{A}$.

See also

1996 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1996_AHSME_Problems/Problem_2&oldid=56469"