AMC 8 · 1999 · #12

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

ratio-proportionpercentagefraction-arithmetic ratio-proportionidentify-subproblems ↑ Prerequisites: fraction-arithmeticpercentage

📏 Medium solution 💡 3 insights

Problem

A team has wins-to-losses ratio 11:4 with no ties. Rounded to the nearest whole percent, what percent of all its games did the team lose?

Pick an answer.

(A)

24%

(B)

27%

(C)

36%

(D)

45%

(E)

73%

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Because the actual game count is not given but the ratio 11:4 is, the loss percent must be the same for every team with that ratio. Tool #9 (Solve an Easier Related Problem) says: pick the simplest team that fits — exactly 11 wins and 4 losses, for 15 games total. Then the loss fraction is $\frac{4}{15}$ , no variable needed. Tool #1 (Draw a Diagram) keeps it concrete: a bar of 15 equal blocks with 4 shaded as losses lets you see the fraction before converting to a percent.

1STEP 1

Pick the smallest schedule matching 11:4 — 11 wins and 4 losses, so 11 + 4 = 15 total games; scaling up keeps the same loss fraction.

wins = 11, losses = 4, total = 15

2STEP 2

Draw 15 equal blocks: 11 marked W (win), 4 marked L (loss). The loss fraction is the shaded part, $\frac{4}{15}$ of the bar.

loss fraction = losses/total =

\frac{4}{15}

3STEP 3

Turn the fraction into a percent: $\frac{4}{15}$ × 100% = $\frac{400}{15}$ %, and 400 ÷ 15 = 26.6%.

\frac{4}{15}

× 100% =

\frac{400}{15}

% = 26.6%

4STEP 4

Round 26.6% up — the first decimal is 6 (≥ 5), so it becomes 27% → (B).

26.6% ⟶ 27% → (B)

Answer

27%

Size check: more games were won than lost, so the loss percent must be under 50%. That kills (D) 45% as borderline and (E) 73% outright. Also

\frac{4}{15}

is close to

\frac{4}{16}

\frac{1}{4}

= 25%, so the answer should sit just above 25% — eliminating (A) 24% (below 25%) and (C) 36% (too far above). That leaves (B) 27%, matching the calculation. Cross-check with the general team of 11k wins and 4k losses: loss fraction =

\frac{4k}{15k}

\frac{4}{15}

, same answer for every k, confirming the easier-case shortcut was valid.

💡Key takeaway

When a ratio fixes the answer, swap the unknown game count for the smallest matching schedule (11 wins, 4 losses, 15 total). Then "4 out of 15" turns into about 27% with a quick divide-and-round.

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