AMC 8 · 2007 · #20

Grade 6 algebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State

percentageratio-proportionlinear-equations-one-varsystems-of-equations convert-to-algebra ↑ Prerequisites: percentagelinear-equations-one-var

📏 Short solution 💡 2 insights

Problem

Before district play the Unicorns had won 45% of their games. During district play they won 6 more and lost 2, ending the season at exactly 50% wins. Find the total number of games played in the whole season.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

The problem already hands us five candidate totals, so Tool #6 (Guess and Check) is the most direct route — far simpler than setting up algebra. For each choice we know the games-before count (choice - 8), and we just check whether 45% of that is a whole number AND whether adding 6 wins puts us at exactly half the season total. Tool #3 (Eliminate Possibilities) sharpens the check: 45% = $\frac{9}{20}$ , so the games-before count must be a multiple of 20, which knocks out almost every choice instantly.

1STEP 1

Let T be the season total: games before district play = T - 8, wins before = 45% of that, and half of T = (wins before) + 6.

games before = T - 8, wins before = 0.45(T - 8),

\frac{T}{2}

= 0.45(T - 8) + 6

2STEP 2

Since 45% = $\frac{9}{20}$ , wins-before is whole only when T - 8 is a multiple of 20. Among the choices, only T - 8 = 40 qualifies.

T - 8 ∈ {40, 42, 44, 46, 52} → multiple of 20: only 40

3STEP 3

Check T = 48: games before = 40, wins before = 0.45 × 40 = 18, so 18 + 6 = 24 out of 48 — exactly 50%.

T = 48: wins before = 0.45 × 40 = 18, wins after = 18 + 6 = 24,

\frac{24}{48}

\frac{1}{2}

✓

4STEP 4

The rest fail the whole-number test: 0.45 × 42, 44, 46, 52 give 18.9, 19.8, 20.7, 23.4 — none whole. Only (A) survives.

0.45 × 42 = 18.9, 0.45 × 44 = 19.8, 0.45 × 46 = 20.7, 0.45 × 52 = 23.4 — none are whole

Answer

Replay the season with T = 48: the team starts 18-22 (18 wins out of 40 games, exactly 45%), then adds 6 wins and 2 losses in district play to finish 24-24 — half wins, half losses. Both percentage conditions land exactly, and all counts are whole numbers. The win rate climbed because the team's district-play rate (6 out of 8 = 75%) is well above its earlier 45%, which is consistent with the rate rising toward 50%.

💡Key takeaway

Since 45% = $\frac{9}{20}$ , the games-before count has to be a multiple of 20 — that one fact narrows five answer choices to one. The Unicorns played 48 games in all.

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