In this week’s gift-themed Pre-Algebra Problem of the Week, students had to figure out an original cost given some percentages of the original cost. Specifically, they knew that five children contributed to pay for one big gift for their mother. Two of the (relatively broke) children paid \$75 each. Of the remaining three, one paid 20% of the cost, one paid 25% of the cost, and the last paid 1/3 of the cost. Then the question was, how much did the whole thing cost? (A more realistic question might have been, “if the whole thing cost \$692.31, did the last person to pay really contribute 1/3 of the cost like she planned to?” but it’s more algebraic reasoning this way).

Anyway, for me, the one reading the student solutions to the problem, the interesting challenge was to figure out what went wrong when students got answers other than \$692.31. Were there conceptual struggles? Problems thinking up methods to get to the answer? Or problems executing the chosen paths? I always look for common wrong answers, because those usually show me conceptual struggles. This week, the common wrong answer was \$681.82 (or sometimes just \$680, or \$681). I thought, “huh, I wonder why students are getting that answer…” Why do you think they might be?Here are some answers from students who got \$681 for the original price of the gift:

I know:
·Mark will pay 20%
·Sorin will pay \$75
·Raymond will pay 25%
·Stephanie will pay \$75
·Kristin will pay 33%

I figured out that Mark, Raymond, Kristin take up 78% of the cost which
means that Sorin and Stephanie will pay 22% of the cost. Sorin and Stephanie’s payment totals to \$150 which is 22%. This means that \$75 is 11%. This means that \$6.82 is 1% and you can just multiply the percent you need to find by this number.
Since Mark is paying is 20% then you multiply that by \$6.82 and get \$136.40. Raymond is paying 25% so you multiply that by \$6.82 and get \$170.46. Kristin is paying 33% so the amount of money she is paying is \$225.02. If you add all these amounts together you get \$681.82 for your total.
What I Know :
Mark: Contributes 20%
Sorin: Contributes \$75
Stephanie: Contributes \$75
Ray: Contributes 25%
Kristen: Contributes 1/3

Mark             Sorin             Stephanie             Raymond             Kristen
20%               \$75                  \$75                       25%                   33.3%
\$136               11%                 11%                      \$170                  \$224

10% = \$68   20% = \$136
3% = \$20   30% = \$204

Do you think those students had a conceptual error? Or is something else going on? How would you help them check their work and revise their solutions?