A certain business produced x rakes each month form November through February and shipped x/2 rakes at the beginning of...

1. Translate the problem requirements: Break down the timeline into two phases - production phase (Nov-Feb) where rakes are made but no storage costs incurred, and shipping phase (Mar-Oct) where rakes are shipped out while remaining inventory incurs \($0.10\) monthly storage cost per rake.
2. Track inventory levels month by month: Calculate how many rakes remain in storage each month during the shipping phase, starting with total production and subtracting monthly shipments.
3. Calculate monthly storage costs: For each month from March through October, multiply the number of rakes in storage by the \($0.10\) monthly storage cost.
4. Sum total storage costs: Add up all monthly storage costs to get the total storage cost over the 12-month period.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break this problem down into two clear phases to understand what's happening:

Production Phase (November through February - 4 months):

• The business produces \(\mathrm{x}\) rakes each month
• Total rakes produced = \(\mathrm{4x}\) rakes
• No storage costs during this phase

Shipping Phase (March through October - 8 months):

• The business ships \(\mathrm{x/2}\) rakes at the beginning of each month
• Any rakes remaining in storage cost \($0.10\) per rake per month
• We need to track how many rakes are left each month to calculate storage costs

Process Skill: TRANSLATE - Converting the timeline and costs into clear mathematical understanding

2. Track inventory levels month by month

Let's think through this step by step, month by month during the shipping phase:

At the start of March:

• We have \(\mathrm{4x}\) rakes in inventory (from 4 months of production)
• We ship \(\mathrm{x/2}\) rakes at the beginning of March
• Rakes remaining for the month = \(\mathrm{4x - x/2 = 8x/2 - x/2 = 7x/2}\)

At the start of April:

• We start with \(\mathrm{7x/2}\) rakes
• We ship \(\mathrm{x/2}\) rakes at the beginning of April
• Rakes remaining for the month = \(\mathrm{7x/2 - x/2 = 6x/2 = 3x}\)

Following this pattern each month:

• May: \(\mathrm{3x - x/2 = 5x/2}\) rakes remaining
• June: \(\mathrm{5x/2 - x/2 = 4x/2 = 2x}\) rakes remaining
• July: \(\mathrm{2x - x/2 = 3x/2}\) rakes remaining
• August: \(\mathrm{3x/2 - x/2 = 2x/2 = x}\) rakes remaining
• September: \(\mathrm{x - x/2 = x/2}\) rakes remaining
• October: \(\mathrm{x/2 - x/2 = 0}\) rakes remaining

3. Calculate monthly storage costs

Now we multiply the number of rakes in storage each month by \($0.10\):

• March storage cost: \(\mathrm{(7x/2) \times \$0.10 = \$0.35x}\)
• April storage cost: \(\mathrm{(3x) \times \$0.10 = \$0.30x}\)
• May storage cost: \(\mathrm{(5x/2) \times \$0.10 = \$0.25x}\)
• June storage cost: \(\mathrm{(2x) \times \$0.10 = \$0.20x}\)
• July storage cost: \(\mathrm{(3x/2) \times \$0.10 = \$0.15x}\)
• August storage cost: \(\mathrm{(x) \times \$0.10 = \$0.10x}\)
• September storage cost: \(\mathrm{(x/2) \times \$0.10 = \$0.05x}\)
• October storage cost: \(\mathrm{(0) \times \$0.10 = \$0.00x}\)

4. Sum total storage costs

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 = \(\mathrm{\$1.40x}\)

Final Answer

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: \(\mathrm{1.40x}\)

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the approach

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.

Errors while selecting the answer

No likely faltering points

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for x

Let's set \(\mathrm{x = 8}\) rakes per month. This number is chosen because:

• It's even, so \(\mathrm{x/2 = 4}\) gives us a clean whole number for monthly shipments
• It's small enough to make calculations manageable
• It allows us to easily track inventory changes

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: \(\mathrm{1.40x}\)