Cumulative review: Numbers behind a recommendation
Check a promotion’s arithmetic and revise the recommendation when the corrected numbers miss its target.
Solve mixed percentage, weighted-average, and growth problems, then revise a week 3-style decision memo when a calculation changes.
Overview
Rina owns Cedar Trail, a small outdoor-goods shop. Her supervisor recommends a new gift-kit promotion. Check the numbers, then revise the recommendation.
Allow two minutes for recall, thirteen for the questions, and ten for the worked answers. Brief notes and a calculator are enough.
Cedar Trail, its policies, and all figures are fictional teaching assumptions.
Recall from memory
From Lesson 016, explain why an unrecoverable past payment should not determine today's choice. From Lesson 023, identify the denominator used for margin and the different denominator used for markup. Hold your answers for the debrief.
Recent sales
Cedar Trail's existing kits sold as follows in two equal-length months. Prices stayed unchanged. All sales were completed, with no discounts, returns, or taxes in the figures.
| Existing kit | Price | Month A units | Month B units |
|---|---|---|---|
| Standard | $40 | 30 | 20 |
| Deluxe | $80 | 10 | 20 |
The supervisor says the increase in average revenue per kit proves customers will accept higher prices. He also projects two more monthly revenue increases of 20%, adding them to predict 40% total growth from Month B.
The proposed promotion
A separate promotion would run next month:
| Item | Supplied fact or estimate |
|---|---|
| Selling price | $60 per new kit |
| Complete product cost | $42 per kit, also the full variable cost |
| Other additional campaign expense | $300 total, regardless of units sold |
| Estimated sales | 70 kits, untested |
| Available packing time | 40 productive hours after existing work |
| Packing and checking time | 30 minutes per kit |
| Cash available for the campaign | $3,500 after existing obligations |
Product costs and the $300 expense must be paid before customers pay. Assume every kit produced sells and is collected in full, with no leftover stock or other costs. Existing business results stay unchanged. No work is displaced, and no extra packing time is available. Ignore income taxes.
Only Rina can approve the campaign. She requires at least $1,200 of additional profit under the chosen sales estimate, within available cash and capacity. Revise any proposal that misses these limits. She must decide by Friday.
The shop already spent $200 on nonrefundable design work, with no resale or reuse value. It is separate from the future expenses above.
The supervisor's draft memo reads: “Approve 70 kits by Friday. At a 40% margin, each $60 sale leaves $24; after the $300 campaign expense, profit improves by $1,380. The campaign fits our resources, and the money already spent on design is another reason to proceed.”
Review questions
- Calculate revenue and weighted average revenue per kit in Months A and B. Find the percentage increase in each. Did either listed price rise?
- Calculate revenue after two further 20% monthly increases from Month B. Compare it with the supervisor's 40% total-growth projection. Explain what the calculation does not establish.
- For the proposed kit, calculate gross profit per unit and gross margin. State the percentage-point error in the memo. Calculate additional campaign profit at 70 sales and cash remaining before collection.
- Convert packing time into kit capacity. Solve for the minimum whole kits needed to earn $1,200 after the $300 expense. Can more sales alone meet the requirement within existing capacity?
- Rewrite the memo in three or four sentences. Include the decision, corrected numbers, strongest reason to consider a revised proposal, and an assigned next action. Address the $200 design payment.
Use weighted average price = revenue ÷ units, and campaign profit = units × (price − variable cost) − campaign expense.
Try the questions before reading the worked answers.
Answer
Sales mix and growth
Month A revenue is (30 × $40) + (10 × $80) = $2,000. Forty kits were sold, so average revenue per kit is $2,000 ÷ 40 = $50.
Month B revenue is (20 × $40) + (20 × $80) = $2,400. Forty kits again gives $2,400 ÷ 40 = $60 per kit.
Revenue growth is $400 ÷ $2,000 × 100 = 20%. Average revenue per kit rose by $10 ÷ $50 × 100 = 20%. The deluxe share of units rose from 25% to 50%, while both prices stayed unchanged. This mix change does not establish willingness to pay a higher price for the new kit.
Two further 20% increases give $2,400 × 1.20 = $2,880, then $2,880 × 1.20 = $3,456. Total growth is 1.20 × 1.20 − 1 = 0.44, or 44%.
The supervisor's calculation gives $2,400 × 1.40 = $3,360, understating the compounded scenario by $96. Neither figure is a verified forecast; both depend on future demand assumptions.
Margin, profit, and cash
The proposed kit leaves $60 − $42 = $18 gross profit. Gross margin is $18 ÷ $60 × 100 = 30%. The draft overstates it by ten percentage points and $6 per kit.
Margin divides gross profit by sales revenue, while markup divides it by product cost, as shown in Business Victoria's formulas. Source: Business Victoria. Here markup is $18 ÷ $42 × 100 ≈ 42.9%; that is also different from the claimed 40% margin.
At 70 sales, additional profit is (70 × $18) − $300 = $960. The draft's $1,380 is too high by 70 × $6 = $420. The corrected result misses Rina's $1,200 requirement by $240.
Cash required before collection is (70 × $42) + $300 = $3,240. It leaves $3,500 − $3,240 = $260. The plan fits the stated cash limit. Its positive profit does not remove the need to fund these earlier payments.
Capacity and the target
Forty hours × 60 minutes per hour = 2,400 productive minutes. Dividing by 30 minutes per kit gives capacity of 80 kits. Seventy kits need 35 hours and fit.
Let q mean kits sold. The profit target requires $1,200 = ($18 × q) − $300. Add $300, then divide by $18: q = $1,500 ÷ $18 ≈ 83.33. Round upward to 84 whole kits.
That exceeds capacity by four kits. Even 80 sales produce only (80 × $18) − $300 = $1,140, missing the target by $60. At 84 kits, upfront cash would also reach (84 × $42) + $300 = $3,828, exceeding available funds. More sales alone cannot solve the problem.
Revised recommendation
A suitable memo is: “Rina, do not approve the current campaign by Friday: its actual 30% margin produces $960 additional profit at 70 sales, below your $1,200 requirement. It fits cash and capacity and adds profit, which makes a revised proposal worth considering, but the current economics require 84 kits against capacity of 80. The supervisor will check lower costs or a higher price and obtain demand evidence before resubmitting; review actual sales, costs, and packing time after any approved trial. The unrecoverable $200 design payment does not favor proceeding.”
Recall answer: Today's decision cannot recover the design payment. Compare future amounts that change; keep that past expense in historical records. Margin uses revenue as its base, while markup uses product cost.
Applying this review
Keep your corrected memo with the calculation sheet. Mark one mistake to watch for: a wrong denominator, an unweighted average, an added growth rate, or a capacity result rounded upward. Next week turns these checks into a small spreadsheet model with visible assumptions and formulas.
Sources
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