AMC 10 · 2011 · #6
Grade 7 algebraPick an answer.
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
One letter names every count.
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 equal points, not equal 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
The third contribution is one more than a count.
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
The three add to a single equation.
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, which gives one equation.
▸ Why?
The score is exactly its parts added together, with nothing else contributing.
▸ Why?
Each kind of shot is worth the same fixed amount every time, so its points are a count times a value.
Solve and read off the free throws
Solving gives 13, 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