AMC 10 · 2011 · #12
Grade 7 algebraPick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Everything is tied to the number of two-point shots: the free throws are one more than it, and the equal-points rule lets me write the three-point total without ever counting three-point shots. So I name that one number, turn each clue into an expression in dollars of points, and solve a single equation.
Name the number of two-point shots
Let t be the number of two-point shots. Each is worth 2 points, so the two-point score is 2t.
Pin down the one quantity every other clue depends on, and the rest can be written in terms of it.
6.EE.B.6Introduce A VariableUse the equal-points rule for threes
Equal points means the three-point score is also 2t — I never have to count the three-point shots.
Counting points instead of shots turns 'equal points' into a copy of a value I already have.
6.EE.B.6Change Focus Count The ComplementWrite the free-throw points
Free throws number t + 1, and at 1 point each their score is also t + 1.
A one-point shot means its point total is the same number as its count.
6.EE.B.6Introduce A VariableAdd the parts to 61
All three scoring sources sum to the total: 2t + 2t + (t + 1) = 61, so 5t + 1 = 61.
Adding up every scoring channel must land on the known grand total, which gives one equation.
Adding up every scoring channel must land on the known grand total.
▸ Why?
The score is exactly its parts added, so nothing scored can sit outside the sum.
▸ Why?
Each channel is a count of equal-value shots, so its points are that count times its value.
Solve and read off the free throws
5t = 60 gives t = 12, so the free throws number t + 1 = 13, which is choice (A).
Undo the equation step by step to free the variable, then translate back to the thing asked for.
7.EE.B.4Convert To AlgebraAnchor everything to one unknown, use 'equal points' to reuse a value you already wrote, then add all the scoring to the total and solve.
- Name the number of two-point shots
- Use the equal-points rule for threes
- Write the free-throw points
- Add the parts to 61
- Solve and read off the free throws