AMC 10 · 2015 · #5

Grade 7 rate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State
percentageratio-proportionlinear-equations-one-var complementary-countingextreme-principle ↑ Prerequisites: percentageratio-proportion
📏 Medium solution 💡 2 insights
Problem
After more games one side's win share must reach a very high percentage. Find the fewest extra games.

Pick an answer.

(A)
35
(B)
37
(C)
39
(D)
41
(E)
43
How to solve
Strategy Change Focus / Count the Complement

Tracking the Sharks' wins is awkward because that count keeps changing. Tracking the Tigers' wins is easy: the Tigers already have 2 wins locked in, and no later game can take them away. So switch focus to the complement — the Sharks hold at least 95% exactly when the Tigers hold at most 5% — and the whole problem becomes "how many games must be played before 2 is only 5% of them?" Two things then have to be settled, not one: that no smaller N can work, and that the winning N really is reachable by some actual set of results.

1STEP 1

Read the record so far

The record so far is fixed.

after 3 games: Tigers 2, Sharks 1
2STEP 2

Name the two unknowns

The condition is one plain fraction.

(1+s)/(3+N) ≥ 19/20, 0 ≤ s ≤ N
3STEP 3

Flip to the Tigers' share

Flipping sides makes it small.

(1+s)/(3+N) ≥ 19/20 ⇔ (2+(N-s))/(3+N) ≤ 1/20
4STEP 4

Push the Tiger count to its floor

The losing side already has 2 wins.

2 ≤ (3+N)/20 → 3+N ≥ 40 → N ≥ 37
5STEP 5

Check that 37 is reachable

That forces 37, and it is reachable, choice (B).

N=37: (1+37)/(3+37)=38/40=19/20=95%
Answer
37
Both halves of "minimum" were checked, which is what this question needs. Nothing below 37 works: the Tigers keep 2 wins forever, so the Sharks' share can never beat (1+N)/(3+N), and at N=35 that is 36/38=18/19≈ 94.7%, short of 95%. And 37 does work, with the Sharks sweeping all 37 to reach exactly 38 of 40. The answer sitting on the boundary is expected here — the wording is "at least 95%", so exact equality is allowed; had the problem said "more than 95%", 38 extra games would be needed instead (39/41≈ 95.1%). It is also worth noting where a shortcut can go wrong: solving (1+N)/(3+N)=19/20 as an equation happens to give the whole number 37 here, but the honest statement is the inequality N ≥ 37. With a different target the equation can land off the whole numbers — for a 4/7 target it gives N=5/3, and the real answer is the next whole number up, N=2. The inequality survives that; the equation does not.
💡Key takeaway

The Tigers' two wins can never be erased, so instead of chasing the Sharks' share, ask how many games it takes before 2 is only one twentieth of them.

  • Read the record so far
  • Name the two unknowns
  • Flip to the Tigers' share
  • Push the Tiger count to its floor
  • Check that 37 is reachable

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