Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #12
Grade 6 rate-ratioPick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 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.
Pick the easiest 11 to 4 record
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.
A ratio 11:4 is a Grade 6 "for every 11 wins, 4 losses" statement. Taking exactly one copy of that block is the simplest case that still obeys the rule.
No matter how many games the team actually played, it lost the same share of them — 4 out of every 15 — so the answer can be found using the smallest matching schedule of 11 wins and 4 losses.
▸ Why?
A ratio of 11 wins to 4 losses means the season is built from identical blocks, each holding 11 wins and 4 losses, so k such blocks give 11k wins and 4k losses.
▸ Why?
With no ties, every game is either a win or a loss, so the total is the wins and losses put together: 11k + 4k = 15k.
▸ Why?
The loss share 4k/15k is worth exactly 4/15: its top and bottom are just 4 and 15 each multiplied by the same factor k, and scaling a fraction's top and bottom by the same nonzero number leaves its value unchanged.
Write the loss fraction
Draw 15 equal blocks: 11 marked W (win), 4 marked L (loss). The loss fraction is the shaded part, of the bar.
Splitting the bar into 15 equal pieces and counting 4 of them is the Grade 3 definition of 4/15 as 4 copies of 1/15.
3.NF.A.1Draw A DiagramConvert the fraction to a percent
Turn the fraction into a percent: × 100% = %, and 400 ÷ 15 = 26.6%.
Percent means "per 100". Multiplying the fraction by 100 rescales the same ratio to a per-100 form, which is the Grade 6 percent move.
6.RP.A.3Solve An Easier Related ProblemRound to the nearest percent
Round 26.6% up — the first decimal is 6 (≥ 5), so it becomes 27% → (B).
Grade 5 rounding rule: look at the next digit; 6 ≥ 5, so the integer part goes up by one.
5.NBT.A.4Solve An Easier Related ProblemWhen 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.
- Pick the easiest 11 to 4 record
- Write the loss fraction
- Convert the fraction to a percent
- Round to the nearest percent
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