The Snack Cart Cash Box Shortage
The fifth-grade service club runs a snack cart during morning arrival. The cash box starts each week with $150.00. Prices are $0.75 for a granola bar, $1.25 for a juice box, and $0.50 for an apple. On Monday the club sells 36 granola bars, 28 juice boxes, and 45 apples. That afternoon a volunteer buys supplies with money from the cash box and submits a receipt for $62.40. The club advisor says the box should hold exactly $172.10 on Tuesday morning. When the box is counted, it holds $169.85. Nothing was stolen, the advisor says - most likely an item was handed out and never recorded. The club has no itemized log of the day, only the price list and the totals.
1. Verify the advisor's $172.10 figure. Show the revenue by item and the running balance.
Read the worked answer
Revenue by item
- Granola bars: 36 × $0.75 = $27.00
- Juice boxes: 28 × $1.25 = $35.00
- Apples: 45 × $0.50 = $22.50
Total sales: $27.00 + $35.00 + $22.50 = $84.50
Cash box balance: start with $150.00, subtract supplies, add sales. $150.00 − $62.40 = $87.60, then $87.60 + $84.50 = $172.10
(The order of the two operations does not matter: $150.00 + $84.50 = $234.50, and $234.50 − $62.40 = $172.10.) The advisor's figure is correct, so the $2.25 gap is real and needs an explanation rather than a recalculation.
2. The box is short by $2.25. Give at least two different combinations of items from the price list that total exactly $2.25, and explain why the club cannot tell which one happened.
Read the worked answer
Finding combinations that total $2.25 (treat this as making change with items priced $0.75, $1.25, and $0.50):
- 3 granola bars: 3 × $0.75 = $2.25 ✔
- 1 granola bar + 3 apples: $0.75 + $1.50 = $2.25 ✔
- 1 juice box + 2 apples: $1.25 + $1.00 = $2.25 ✔
Why the count cannot settle it: the cash box only records the total amount of money present. Many different item lists produce the same total, so the missing money identifies a sum, not a list. Three granola bars and one juice plus two apples leave the box in exactly the same state.
What would settle it: an itemized log, a count of remaining inventory compared with the morning count, or the volunteer's memory of who was served. This is the key reasoning - a single total is not enough information to reconstruct individual transactions, so the club should compare leftover stock to the expected remainder rather than guess from the dollar amount alone.
3. The treasurer rounds every price to the nearest whole dollar, computes 36 + 28 + 45 = $109 of revenue, and concludes the count is wrong. Evaluate this reasoning.
Read the worked answer
The treasurer's estimate is not reliable enough to accuse the count of being wrong, and the rounding error can be predicted exactly.
Error from each price:
- Granola bars: $0.75 rounds to $1.00, which is $0.25 too high × 36 = $9.00 too high
- Juice boxes: $1.25 rounds to $1.00, which is $0.25 too low × 28 = $7.00 too low
- Apples: $0.50 rounds to $1.00, which is $0.50 too high × 45 = $22.50 too high
Net error: $9.00 − $7.00 + $22.50 = $24.50 too high
Check: $109.00 − $24.50 = $84.50, which matches the exact revenue from the earlier task. So the estimate is off by nearly a third of the true amount.
The rule the treasurer missed: rounding to the nearest dollar is only safe when the prices are near the dollar, and $0.50 is exactly half a dollar, so it inflates every single apple sale. A better check is to round to the nearest $0.25 (giving $0.75, $1.25, $0.50 unchanged) or to use compatible numbers such as 36 × $0.75 ≈ 36 × 3/4 = 27.