AMC 10 · 2010 · #19
Grade 8 algebraPick an answer.
The problem hides three unknowns inside two number patterns. Naming the shared first-quarter score a, the geometric ratio r, and the arithmetic step d turns the story into expressions we can compare. The single equation 'won by one point' then links them, and a short bounded search finds the one whole-number solution.
Model both teams as sequences
Two sequences describe the whole game.
Each team follows one simple rule every quarter, so a single letter for the rule captures all four numbers.
5.OA.B.3Convert To AlgebraWrite each team's total
Each team's total is one short formula.
A total is just the four terms summed, and writing it with letters lets us compare the two teams directly.
6.EE.A.2Introduce A VariablePin down the ratio
Whole numbers force the factor to be 2.
Doubling each quarter is the gentlest whole-number growth that still beats an arithmetic sequence, keeping totals under the cap.
Doubling is the gentlest whole-number growth that still outruns a steady step, so it keeps the total under the cap.
▸ Why?
Each quarter multiplies the previous one by the same fixed number, which is what makes the run geometric.
▸ Why?
Any larger ratio would push the four-quarter total past the allowed ceiling on its own.
Use 'won by one point'
The one-point margin is a single equation.
The one-point margin is a fact we can write as an equation, tying the ratio side to the step side.
7.EE.B.4Introduce A VariableSearch for the whole-number fit
Only one starting score fits whole numbers.
Only one first-quarter score in the allowed range makes the step come out to a whole number.
8.EE.C.7Guess And CheckCheck and total the first half
The first half adds to 34, choice (E).
Once the one valid scenario is found, the first-half sum is a direct addition of the first two quarters.
6.EE.B.5Eliminate PossibilitiesName the hidden numbers with letters, turn each clue into one equation, then check the few values that fit.
- Model both teams as sequences
- Write each team's total
- Pin down the ratio
- Use 'won by one point'
- Search for the whole-number fit
- Check and total the first half