Spirit Shirt Order: Grouping, Budget, and a Color-Pattern Claim
The fourth-grade team at Maple Ridge Elementary is ordering spirit shirts. The four classes have 26, 24, 27, and 23 students, and every student orders exactly one shirt. Nine extra shirts are ordered for staff. The supplier sells shirts in boxes of 8 and charges $7 for each shirt; boxes cannot be split, but the school only pays for the shirts it keeps. The PTO approved $750.
The supplier also offers a bulk plan: for orders of at least 100 shirts, each shirt costs $6, but shirts must be bought in full boxes of 12, and every shirt in a box is paid for.
Separately, the art teacher will print one number on each shirt. Her printer repeats a four-color cycle - red, red, blue, gold - starting again with red for each new shirt, and she claims that exactly one-quarter of the shirts will end up gold because one of every four steps is gold.
1. How many shirts are needed, how many boxes of 8 must be opened, and what is the total cost under the standard plan? Decide whether the PTO budget covers it, and support the answer with an estimate.
Read the worked answer
Shirts: 26 + 24 + 27 + 23 = 100 students, plus 9 staff shirts = 109 shirts.
Boxes: 109 ÷ 8 = 13 remainder 5. Thirteen boxes hold 104 shirts, so a 14th box must be opened for the remaining 5 shirts - 14 boxes. Note that the box count affects packaging, not price, because the school pays per shirt.
Cost: 109 × $7 = $763.
Estimate check: 109 is about 110, and 110 × 7 = 770, so $763 is sensible (the estimate is slightly high because 110 > 109).
Judgment: $763 is $13 more than the $750 budget, so the standard plan cannot be approved as written.
2. Should the team choose the bulk plan instead? Compare the two plans with numbers, and say what happens to any extra shirts.
Read the worked answer
The order is 109 shirts, which meets the 100-shirt minimum, so the $6 price applies. Bulk shirts come in full boxes of 12: 109 ÷ 12 = 9 remainder 1. Nine boxes hold 108 shirts, so a 10th box is required - 10 × 12 = 120 shirts, and all 120 must be paid for.
Cost: 120 × $6 = $720.
Compare: $720 is under the $750 budget, and it is $763 − $720 = $43 cheaper than the standard plan, even though the school receives 11 more shirts than it needs.
Recommendation: take the bulk plan. The 11 extra shirts become spares for new students, field-day staff, or replacement for a misprint. The important reasoning step is that the bulk plan's lower unit price must be tested against the true number of shirts paid for (120, not 109).
3. Evaluate the art teacher's claim that exactly one-quarter of the 109 shirts will get a gold number. If the claim is false, find the actual gold count.
Read the worked answer
The claim is false for this order. One full cycle is 4 shirts (red, red, blue, gold) and contains exactly 1 gold shirt, so a cycle gives a fraction of 1/4 - but only when the total is a multiple of 4.
Divide the order into whole cycles: 109 = 27 × 4 + 1. That is 27 complete cycles (27 gold shirts and 81 non-gold shirts) plus 1 extra shirt. The 28th cycle starts over with red, so the final shirt is red.
Actual result: 27 of 109 shirts are gold. For comparison, 1/4 of 109 is 27.25, so 27 is a little low because the leftover shirt fell on red rather than gold.
The reasoning error: the fraction gold equals 1/4 only when the shirts are counted in complete groups of 4. With a partial cycle, you must count the leftover shirts one at a time.