Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #20
Grade 6 algebraPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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% = 9/20, so the games-before count must be a multiple of 20, which knocks out almost every choice instantly.
Translate the win conditions
Let T be the season total: games before district play = T - 8, wins before = 45% of that, and half of T = (wins before) + 6.
Write everything in terms of the season total T so each guess becomes a single arithmetic check.
6.RP.A.3Guess And CheckNarrow the candidates
Since 45% = , wins-before is whole only when T - 8 is a multiple of 20. Among the choices, only T - 8 = 40 qualifies.
Win counts must be whole numbers, and 45% forces a multiple of 20 — most choices fail this test before any deeper check.
The team must have played a number of games that is a multiple of 20 before district play, since only then could they have won exactly 45% of them.
▸ Why?
Winning 45% of the games means the number of wins equals 9/20 of the games played before district play.
▸ Why?
45% is 45 out of every 100, which names the same share as 9 out of 20.
▸ Why?
9 out of 20 and 45 out of 100 are equal because scaling both the 9 and the 20 by 5 is multiplying the fraction by 5/5, which is just 1.
▸ Why?
For 9/20 of the games to come out as an exact number of wins, the games played must be a multiple of 20.
▸ Why?
Each win pairs with exactly one game the team won, so wins are counted in whole units and can never be a fraction of a game.
▸ Why?
Since 9 and 20 share no common prime factor, the 9 on top can cancel none of the 20 underneath, so 9/20 of a number is a whole number only when the entire 20 divides that number.
▸ Why?
9 breaks into primes as 3 × 3 and 20 breaks into 2 × 2 × 5, and those two prime lists share nothing, so every prime factor of the 20 must instead come from the number of games itself.
Verify the survivor
Check T = 48: games before = 40, wins before = 0.45 × 40 = 18, so 18 + 6 = 24 out of 48 — exactly 50%.
Wins go from 18 out of 40 to 24 out of 48 — a clean 45% → 50% jump powered by 6 extra wins in 8 extra games.
6.RP.A.3Guess And CheckRule out the other choices
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.
Wins are countable objects; a fractional win count means the choice can't match a real season record. Answer: (A) 48.
6.RP.A.3Eliminate PossibilitiesSince 45% = , 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.
- Translate the win conditions
- Narrow the candidates
- Verify the survivor
- Rule out the other choices
A parent dashboard for the family lives at sensimlab.com.