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?
Steve answered this for you last night.
http://www.jiskha.com/display.cgi?id=1432686891
steve is wrong... the answer is 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 $5(5+x)$ -- which is positive, so we do not have to worry about the direction of the inequality -- 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 $5(5+x)$ -- which is positive, so we do not have to worry about the direction of the inequality -- 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 $\boxed{30}$.
To determine the week when your team will have won at least 65% of its games, we need to calculate the number of games your team will have played after each week and see when the win percentage reaches or exceeds 65%.
Let's break down the problem step by step:
Step 1: Calculate the total number of games played in the first five weeks.
Total games played in the first five weeks = Number of wins + Number of losses
Total games played = 3 + 17 = 20
Step 2: Calculate the number of wins and losses after each subsequent week.
After the first week after Week 5: Number of wins = 3 + 3 = 6, Number of losses = 17 + 1 = 18
After the second week after Week 5: Number of wins = 6 + 3 = 9, Number of losses = 18 + 1 = 19
After the third week after Week 5: Number of wins = 9 + 3 = 12, Number of losses = 19 + 1 = 20
Step 3: Calculate the win percentage after each week.
After the first week after Week 5: Win percentage = (Number of wins / Total games played) x 100 = (6 / 24) x 100 = 25%
After the second week after Week 5: Win percentage = (9 / 28) x 100 = 32.14%
After the third week after Week 5: Win percentage = (12 / 32) x 100 = 37.5%
As we can see, after the third week after Week 5, the win percentage will be 37.5%, which is less than 65%. Therefore, the team will not have won at least 65% of its games by the end of the third week. To find out when the team will reach or exceed a 65% win percentage, we continue the calculation.
After the fourth week after Week 5: Number of wins = 12 + 3 = 15, Number of losses = 20 + 1 = 21
Win percentage = (15 / 36) x 100 ≈ 41.67%
After the fifth week after Week 5: Number of wins = 15 + 3 = 18, Number of losses = 21 + 1 = 22
Win percentage = (18 / 40) x 100 = 45%
After the sixth week after Week 5: Number of wins = 18 + 3 = 21, Number of losses = 22 + 1 = 23
Win percentage = (21 / 44) x 100 ≈ 47.73%
After the seventh week after Week 5: Number of wins = 21 + 3 = 24, Number of losses = 23 + 1 = 24
Win percentage = (24 / 48) x 100 = 50%
After the eighth week after Week 5: Number of wins = 24 + 3 = 27, Number of losses = 24 + 1 = 25
Win percentage = (27 / 52) x 100 ≈ 51.92%
After the ninth week after Week 5: Number of wins = 27 + 3 = 30, Number of losses = 25 + 1 = 26
Win percentage = (30 / 56) x 100 ≈ 53.57%
After the tenth week after Week 5: Number of wins = 30 + 3 = 33, Number of losses = 26 + 1 = 27
Win percentage = (33 / 60) x 100 = 55%
After the eleventh week after Week 5: Number of wins = 33 + 3 = 36, Number of losses = 27 + 1 = 28
Win percentage = (36 / 64) x 100 = 56.25%
After the twelfth week after Week 5: Number of wins = 36 + 3 = 39, Number of losses = 28 + 1 = 29
Win percentage = (39 / 68) x 100 ≈ 57.35%
After the thirteenth week after Week 5: Number of wins = 39 + 3 = 42, Number of losses = 29 + 1 = 30
Win percentage = (42 / 72) x 100 = 58.33%
After the fourteenth week after Week 5: Number of wins = 42 + 3 = 45, Number of losses = 30 + 1 = 31
Win percentage = (45 / 76) x 100 ≈ 59.21%
After the fifteenth week after Week 5: Number of wins = 45 + 3 = 48, Number of losses = 31 + 1 = 32
Win percentage = (48 / 80) x 100 = 60%
After the sixteenth week after Week 5: Number of wins = 48 + 3 = 51, Number of losses = 32 + 1 = 33
Win percentage = (51 / 84) x 100 ≈ 60.71%
After the seventeenth week after Week 5: Number of wins = 51 + 3 = 54, Number of losses = 33 + 1 = 34
Win percentage = (54 / 88) x 100 ≈ 61.36%
After the eighteenth week after Week 5: Number of wins = 54 + 3 = 57, Number of losses = 34 + 1 = 35
Win percentage = (57 / 92) x 100 ≈ 61.96%
After the nineteenth week after Week 5: Number of wins = 57 + 3 = 60, Number of losses = 35 + 1 = 36
Win percentage = (60 / 96) x 100 = 62.5%
After the twentieth week after Week 5: Number of wins = 60 + 3 = 63, Number of losses = 36 + 1 = 37
Win percentage = (63 / 100) x 100 = 63%
As we can see, after the twentieth week after Week 5, the win percentage will be 63%, which is more than 65%. Therefore, your team will have won at least 65% of its games for the first time in the season by the end of the twentieth week.