A school with 345 students had a sports day. The students chose between football, swimming, and bike riding. Twice as many students chose basketball as bicycling, and there are 30 fewer students who chose swimming than football. 120 students went swimming. How many students chose football? How many chose bicycling?

1) Example of “algebraic” solution:

M + S + C = 372

4C = S

M +18 = C

(C – 18) + 4C + C = 372

=>6C -18 = 372

=>6C = 390

=>C = 65

4(65) = S

=>260 = S

M + 18 = 65

=>M = 47

(This problem could be solved arithmetically with a guess and check method)

2) Example of “arithmetic” solution:

F = 120 + 30 = 150

B = 150 / 2 = 75

(Guess and check would also serve as a arithmetic solution strategy for this problem)

Example of “algebraic” solution:

120 + B + F = 345

2B = F

F – 30 = 120

120 + B + 2B = 345

=>3B = 225

=>B = 75

When solving word problems, some students automatically apply algebraic methods that are unnecessarily cumbersome. Other students refuse to use algebraic methods (relying solely on arithmetic calculations) because they do not see the utility of using algebra. These problems help assess student’s ability to flexibly apply arithmetic and algebraic solution strategies.