Reading Challenge Scoring
The Grade 7 reading challenge runs for 14 school days. Class A begins with 18 points from a head start and earns 7 points each day. Class B begins with 46 points and earns 4 points each day. Points are awarded only at the end of each full day. Class A claims its head start makes it certain to win, while Class B claims its higher daily rate will catch up. The student council must decide whether the scoring system is fair or whether the starting scores should be adjusted so the final result is a tie. The principal will award a pizza party to the class with more points after day 14.
1. Write a rule for each class's total after d days. Make a table for d = 0, 5, 9, 10, and 14. Use it to find when the lead changes.
Read the worked answer
Class A: A = 18 + 7d. Class B: B = 46 + 4d.
| d | A | B | Leader |
|---|---|---|---|
| 0 | 18 | 46 | B by 28 |
| 5 | 53 | 66 | B by 13 |
| 9 | 81 | 82 | B by 1 |
| 10 | 88 | 86 | A by 2 |
| 14 | 116 | 102 | A by 14 |
The lead changes between day 9 and day 10. Solving 18 + 7d = 46 + 4d gives 3d = 28, so d = 9 1/3. Since points are awarded only at the end of a full day, B leads through day 9 and A leads from day 10.
2. Who wins after 14 days, and by how many points? If the challenge were only 7 days, who would win? What does this show about how the length of the challenge affects fairness?
Read the worked answer
After 14 days: A = 18 + 7(14) = 116; B = 46 + 4(14) = 102. Class A wins by 14 points. After 7 days: A = 18 + 49 = 67; B = 46 + 28 = 74. Class B wins by 7 points. The scoring system is not automatically unfair; the longer challenge gives A's higher daily rate time to overcome B's head start, while a short challenge favors B. Fairness depends on the chosen duration, so the council should decide the length before choosing the scoring system.
3. What starting score for Class B would make the day-14 totals equal? Show the equation and check. Would that adjustment make the early part of the challenge more or less balanced?
Read the worked answer
Let S be B's starting score. Set the day-14 totals equal: 18 + 7(14) = S + 4(14). Then 116 = S + 56, so S = 60. Check: B = 60 + 56 = 116, tying A. However, at day 0 B would lead 60 to 18, and at day 9 B would lead 96 to 81. The original system has a lead change and is closer early; raising B's start to 60 makes B lead the whole time and ties only at the end. That is less balanced, even though the final score is a tie.