Question
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?
Answers
Ms. Sue
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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
(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
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 $\boxed{30}$.
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 $\boxed{30}$.