At the end of Week 5 of my basketball team's season, our record is 3 wins and 17 losses. Each week after that, we win 3 games and lose 1 game. At the end of which week will my team have won at least 65% of its games total for the first time in the season?
The answer is actually 30.
I agree with Anonymous. Solution:
Let $x$ be the number of weeks we play after week 5, at which point we have won $3+ 3x$ games (3 in the first 5 weeks plus 3 per week afterwards) out of $20+ 4x$ (20 in the first 5 weeks plus 4 per week afterwards). So, we must have
\[\frac{3+3x}{20+4x} \ge \frac{65}{100}.\]
Simplifying the right-hand side gives
\[\frac{3+3x}{20+4x} \ge \frac{13}{20}.\]
Multiply both sides by 4 gives
\[\frac{3+3x}{5+x} \ge \frac{13}{5}.\]
Multiplying both sides by $5(5+x)$ gives
$5(3+3x) \ge 13(5+x)$, so $15+15x \ge 65+13x$, or $2x \ge 50$. Therefore, $x \ge 25$.
Checking our work, we see that after 25 additional weeks, we have $3 + 25\cdot 3 = 78$ wins out of $20 + 4\cdot 25=120$ games, and $\frac{78}{120} = \frac{13}{20} = 65\%$. Since we play for 25 weeks after Week 5, we hit the $65\%$ mark at the end of Week $\boxed{30}$.
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To determine at what point your basketball team will have won at least 65% of its games, we need to track the number of wins and losses each week. Let's break it down step-by-step:
1. Find the total number of games played in Week 5:
Total games played = Wins + Losses = 3 + 17 = 20
2. Calculate the minimum number of games needed to achieve a 65% win rate:
Total games needed = (Total games played) + (Total games played * 0.65)
= 20 + (20 * 0.65)
= 20 + 13
= 33
3. Determine how many games your team needs to win in future weeks:
Remaining games needed = Total games needed - Total games played
= 33 - 20
= 13
4. Evaluate how many weeks it will take to win the remaining games:
Number of weeks needed = Remaining games needed / (Wins per week - Losses per week)
= 13 / (3 - 1)
= 13 / 2
= 6.5
Since we cannot have half a week, we need to round up to the next whole number. Therefore, it will take your team 7 more weeks to win at least 65% of its games total for the first time in the season.
after w weeks more, we have
(3+3w)/(17+w) = .65
w = 3.4
check:
after 3 more weeks, the win/loss ratio will be (3+9)/(17+3) = 0.60
after 4 more weeks, it will be (3+12)/(17+4) = 0.71