Garden Bed, Path, and a Soil Budget That Does Not Quite Fit
The fifth-grade team plans a rectangular garden bed 12 meters long and 8 meters wide. Along two adjacent sides, a wood-chip path 1.5 meters wide runs inside the bed's outer edges, and vegetables are planted only in the rectangle that remains. Soil comes in bags costing $6 each, and one bag covers 2.5 square meters at the depth the team wants; the team plans to apply that depth evenly across the entire planted area. The team has $150 to spend on soil.
Marcus computes the planted area as 12 x 8 minus (1.5 x 8) minus (1.5 x 12), which gives him 66 square meters. He then divides by 2.5, gets 26.4, and says the team should round down to 26 bags because "rounding up wastes soil." A classmate points out that a narrower path would free up room and therefore save money.
1. Find Marcus's error in the area calculation, give the correct planted area, and explain the geometric reason his method falls short.
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Marcus's error: subtracting both path strips at full length double-counts the 1.5-by-1.5 square where the two strips overlap at the corner. He subtracted that corner twice.
Correct area, method 1 (subtract): 96 − 12 − 18 + 2.25 = 68.25 m². The 2.25 is added back once.
Correct area, method 2 (multiply the remaining rectangle): the planted rectangle measures (12 − 1.5) by (8 − 1.5), or 10.5 by 6.5, which equals 68.25 m².
The two methods agreeing is the check that should have caught the error. Students who subtract full-length strips often forget the overlap; drawing the corner region once, and marking it as counted, prevents it. Marcus's 66 is 2.25 m² low - small, but it matters in the next step.
2. Using the correct area, determine how many bags the team needs and whether $150 is enough. Then explain why Marcus's rounding-down rule is wrong for this particular quantity.
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Bags: 68.25 ÷ 2.5 = 27.3, so the team must round up to 28 bags.
Cost: 28 x $6 = $168, which is $18 over the $150 budget.
Why rounding down fails: 26 bags cover 65 m², and even 27 bags cover only 67.5 m² - both are less than 68.25 m². The soil would be spread thinner than the planned depth over part of the bed, so the team would not actually get the product it budgeted for. Rounding rules follow the meaning of the remainder: bags are whole units and you cannot buy a partial bag, so the remainder forces a purchase. Marcus's rule would be right for a quantity like "how many full bags can we fill," but wrong for "how much material must cover this area."
3. Two proposals are on the table: (i) narrow the path to 1 meter to save money, or (ii) keep the layout and apply the soil only 2 centimeters deep instead of 3. Evaluate both against the $150 limit, and state the trade-off each one makes.
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Proposal (i) is based on backwards reasoning. Narrowing the path to 1 m leaves a planted rectangle of 11 by 7, or 77 m². That is more area, not less: 77 ÷ 2.5 = 30.8, so 31 bags, or $186 - further over budget. The classmate assumed "less path" automatically means "less cost," but path and planting area trade off in the opposite direction.
Proposal (ii) works arithmetically. At 2 cm instead of 3, each bag covers about 3.75 m². Then 68.25 ÷ 3.75 = 18.2, so 19 bags, or $114 - comfortably under $150.
Trade-off: (ii) keeps the planting area and the budget but applies a thinner layer, which dries out faster and supports shallower rooting. A third path - keep 3 cm and raise the extra $18 - preserves the growing plan at the cost of fundraising time. A complete answer names a trade-off rather than declaring one proposal simply "best."