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A certain business produced \(\mathrm{x}\) rakes each month form November through February and shipped \(\frac{\mathrm{x}}{2}\) rakes at the beginning of each month from March through October. The business paid no storage costs for the rakes from November through February, but it paid storage costs of \($0.10\) per rake each month from March through October for the rakes that had not been shipped. In terms of \(\mathrm{x}\), what was the total storage cost, in dollars, that the business paid for the rakes for the 12 months form November through October?
Let's break this problem down into two clear phases to understand what's happening:
Production Phase (November through February - 4 months):
Shipping Phase (March through October - 8 months):
Process Skill: TRANSLATE - Converting the timeline and costs into clear mathematical understanding
Let's think through this step by step, month by month during the shipping phase:
At the start of March:
At the start of April:
Following this pattern each month:
Now we multiply the number of rakes in storage each month by \($0.10\):
Let's add up all the monthly storage costs:
Total storage cost = \(\mathrm{\$0.35x + \$0.30x + \$0.25x + \$0.20x + \$0.15x + \$0.10x + \$0.05x + \$0.00x}\)
Total storage cost = \(\mathrm{\$(0.35 + 0.30 + 0.25 + 0.20 + 0.15 + 0.10 + 0.05 + 0.00)x}\)
Total storage cost = \(\mathrm{\$1.40x}\)
The total storage cost that the business paid for the rakes over the 12 months from November through October is \(\mathrm{\$1.40x}\).
This matches answer choice C: \(\mathrm{1.40x}\)
1. Misunderstanding the timeline and when costs apply
Students often confuse which months have storage costs versus which don't. The problem clearly states "no storage costs from November through February" but "storage costs from March through October." A common error is applying storage costs to all 12 months or incorrectly identifying which 8 months have costs.
2. Confusing when shipments occur versus when storage costs are calculated
The problem states shipments happen "at the beginning of each month" from March through October. Students may incorrectly assume storage costs apply to inventory levels before shipment rather than after shipment. The key insight is that storage costs apply to remaining inventory after each month's shipment.
3. Misinterpreting the shipping rate
The problem says the business ships "\(\mathrm{x/2}\) rakes at the beginning of each month" during the shipping phase. Students might misread this as shipping \(\mathrm{x/2}\) rakes total over the entire 8-month period, rather than \(\mathrm{x/2}\) rakes every single month.
1. Arithmetic errors when tracking monthly inventory levels
The month-by-month inventory calculation involves repeated subtraction of fractions (subtracting \(\mathrm{x/2}\) each month). Students commonly make errors like: \(\mathrm{4x - x/2 = 3.5x}\) instead of \(\mathrm{7x/2}\), or \(\mathrm{7x/2 - x/2 = 3.5x}\) instead of \(\mathrm{6x/2 = 3x}\). These fraction arithmetic mistakes compound across the 8-month tracking period.
2. Incorrectly calculating storage costs for fractional inventory
When inventory levels like \(\mathrm{7x/2}\) or \(\mathrm{5x/2}\) arise, students often struggle with multiplying by \($0.10\). For example, \(\mathrm{(7x/2) \times \$0.10}\) should equal \(\mathrm{\$0.35x}\), but students might incorrectly calculate this as \(\mathrm{\$0.70x/2}\) or make other fraction multiplication errors.
3. Losing track during the sequential monthly calculations
The 8-month tracking process requires careful attention to carry forward the correct inventory from each month. Students frequently lose track mid-calculation, using wrong starting inventory for subsequent months, which throws off all remaining calculations.
No likely faltering points
Step 1: Choose a convenient value for x
Let's set \(\mathrm{x = 8}\) rakes per month. This number is chosen because:
Step 2: Calculate production phase (November through February)
• Production: \(\mathrm{8}\) rakes per month × \(\mathrm{4}\) months = \(\mathrm{32}\) rakes total
• Storage cost: \($0\) (no storage costs during production phase)
• Rakes in inventory at end of February: \(\mathrm{32}\) rakes
Step 3: Track shipping phase month by month (March through October)
Monthly shipments: \(\mathrm{x/2 = 8/2 = 4}\) rakes per month
March:
• Rakes in storage: \(\mathrm{32}\)
• Storage cost: \(\mathrm{32 \times \$0.10 = \$3.20}\)
• Rakes shipped: \(\mathrm{4}\)
• Remaining: \(\mathrm{32 - 4 = 28}\)
April:
• Rakes in storage: \(\mathrm{28}\)
• Storage cost: \(\mathrm{28 \times \$0.10 = \$2.80}\)
• Remaining: \(\mathrm{28 - 4 = 24}\)
May:
• Rakes in storage: \(\mathrm{24}\)
• Storage cost: \(\mathrm{24 \times \$0.10 = \$2.40}\)
• Remaining: \(\mathrm{24 - 4 = 20}\)
June:
• Rakes in storage: \(\mathrm{20}\)
• Storage cost: \(\mathrm{20 \times \$0.10 = \$2.00}\)
• Remaining: \(\mathrm{20 - 4 = 16}\)
July:
• Rakes in storage: \(\mathrm{16}\)
• Storage cost: \(\mathrm{16 \times \$0.10 = \$1.60}\)
• Remaining: \(\mathrm{16 - 4 = 12}\)
August:
• Rakes in storage: \(\mathrm{12}\)
• Storage cost: \(\mathrm{12 \times \$0.10 = \$1.20}\)
• Remaining: \(\mathrm{12 - 4 = 8}\)
September:
• Rakes in storage: \(\mathrm{8}\)
• Storage cost: \(\mathrm{8 \times \$0.10 = \$0.80}\)
• Remaining: \(\mathrm{8 - 4 = 4}\)
October:
• Rakes in storage: \(\mathrm{4}\)
• Storage cost: \(\mathrm{4 \times \$0.10 = \$0.40}\)
• Remaining: \(\mathrm{4 - 4 = 0}\)
Step 4: Calculate total storage cost
Total = \(\mathrm{\$3.20 + \$2.80 + \$2.40 + \$2.00 + \$1.60 + \$1.20 + \$0.80 + \$0.40 = \$14.40}\)
Step 5: Express in terms of x
Since \(\mathrm{x = 8}\), we have: \(\mathrm{\$14.40 = \$1.40 \times 8 = \$1.40x}\)
This confirms answer choice C: \(\mathrm{1.40x}\)