Equations, ratios, and sensitivity
Solve for a required sales quantity and test how changing an input affects the result.
Rearrange a simple business equation to solve for a missing variable and show how one input changes the result.
Overview
A manager at established training company Linden Learning wants $2,400 of profit next month. She knows the price and costs but needs to find how many workshop places to sell. An equation can turn that target into a quantity she can check against demand and capacity.
Both businesses and all figures in this lesson are fictional teaching assumptions.
Recall Lesson 024: Why could ClearPath not describe its entire increase in average revenue per license as a price increase?
Express the relationship
An equation states that two quantities are equal. A variable is a symbol standing for a quantity, such as the number of places sold. Solving an equation means finding the value that makes both sides equal.
Start with the business relationship in words. A variable cost changes with the number sold. A fixed cost stays unchanged within the stated period and operating range.
For today's examples:
Profit = (price per unit − variable cost per unit) × units sold − fixed costs.
The amount left from each sale after its variable cost is the contribution per unit. It covers fixed costs first; anything remaining is profit. This model assumes one product, a constant price and variable cost per unit, and no other expenses. All amounts must cover the same period.
Worked example
Linden charges $200 per workshop place. Materials and other costs that change with attendance total $80 per place. All remaining monthly costs are fixed at $3,600. Monthly capacity is 60 places; the stated costs apply throughout that range. Assume completed, fully paid attendance, no other expenses, and no income taxes.
Contribution is $200 − $80 = $120 per place. Let q mean places sold next month. To earn $2,400:
$2,400 = ($120 × q) − $3,600.
To solve, perform the same operation on both sides so equality remains intact. Adding the same amount, or dividing both sides by the same nonzero number, preserves equality, as explained in OpenStax's algebra text.
- Add $3,600 to both sides: $6,000 = $120 × q.
- Divide both sides by $120 per place: q = 50 places.
- Check the original relationship: (50 × $200) − (50 × $80) − $3,600 = $2,400.
In words, required units = (fixed costs + target profit) ÷ contribution per unit. Dollars divided by dollars per place gives places. If the result is fractional and only whole units can be sold, round upward to meet the target.
The 50 places fit capacity. The calculation does not establish that 50 customers will buy. A zero contribution would make this division impossible; a negative contribution means each extra sale increases the loss under this model.
Interpret the ratio
A ratio compares quantities by division. Linden's contribution ratio is $120 ÷ $200 = 0.60, or 60%. For each dollar of sales, 60 cents remains after variable costs to cover fixed costs and profit.
At 50 places, revenue is $10,000. Contribution is 60% × $10,000 = $6,000. Subtracting $3,600 leaves $2,400 profit. The profit margin is $2,400 ÷ $10,000 = 24%, so the 60% contribution ratio is not the final profit margin. Name the costs deducted before interpreting any ratio.
Change one input
Sensitivity analysis asks how a result changes when an input changes. OpenStax illustrates this with changes in selling price, volume, variable costs, and fixed costs in its sensitivity analysis guidance.
Keep attendance at 50 and change one assumption at a time. Each row below starts from Linden's original figures; the changes do not accumulate.
| Separate scenario | Monthly profit calculation | Profit |
|---|---|---|
| Original figures | 50 × ($200 − $80) − $3,600 | $2,400 |
| Price falls to $180 | 50 × ($180 − $80) − $3,600 | $1,400 |
| Variable cost rises to $100 | 50 × ($200 − $100) − $3,600 | $1,400 |
| Fixed costs rise to $4,200 | 50 × ($200 − $80) − $4,200 | $1,800 |
A $20 price reduction costs $20 × 50 = $1,000 of profit if attendance stays unchanged. It might also change demand, which this row does not predict. Sensitivity calculations show consequences under assumptions, not the likelihood of those assumptions. Check cash timing separately before committing money.
Exercise
Allow 8–10 minutes. You own Maple Cleaning. Next month's price is $150 per job, variable cost is $90 per job, and all remaining costs are fixed at $2,400. You want $1,200 profit. Capacity is 65 jobs. The cost assumptions hold within that capacity.
Assume one standard job type, no other expenses or income taxes, and full payment. A supplier then confirms that variable cost will rise to $100 per job. Fixed costs and capacity stay unchanged.
- Write and solve the equation for jobs needed at the original $90 variable cost. Check the result.
- At that same job count, calculate profit with the new $100 cost. Explain the change.
- Calculate the job count the revised equation requires for $1,200 profit. Compare it with capacity, then calculate profit at 65 jobs.
- At 60 jobs and the new cost, what price would deliver $1,200 profit? Name an assumption to verify before adopting it.
Try the calculations before reading the answer.
Answer
Original contribution is $150 − $90 = $60 per job. The target equation is $1,200 = ($60 × q) − $2,400. Add $2,400 to both sides, then divide by $60: q = $3,600 ÷ $60 = 60 jobs.
Check: (60 × $150) − (60 × $90) − $2,400 = $9,000 − $5,400 − $2,400 = $1,200.
At the new variable cost, contribution falls to $50 per job. With 60 jobs, profit becomes (60 × $50) − $2,400 = $600. The $10 extra cost on each of 60 jobs reduces profit by $600.
The revised equation requires $3,600 ÷ $50 = 72 jobs, seven more than capacity. That mathematical requirement is infeasible under the supplied operating limits. At 65 jobs, profit is (65 × $50) − $2,400 = $850, still $350 below target.
For 60 jobs to earn $1,200, each must contribute ($2,400 + $1,200) ÷ 60 = $60. Add the new $100 variable cost: required price is $160. Check: 60 × ($160 − $100) − $2,400 = $1,200.
Investigate whether 60 customers will buy at $160. Raising the price is a possible response, not an established solution. If demand falls, revise the calculation; if considering more capacity, obtain its additional costs before extending the model.
Recall answer: ClearPath's sales shifted toward the higher-priced Pro plan as well as experiencing price increases within each plan. The overall average combined both effects.
Applying this lesson
Write the relationship, identify the unknown, solve while preserving equality, and put the answer back into the original equation. Then change one important input and check capacity, demand, and cash. Add the equation and one sensitivity row to your checked calculation sheet.
Sources
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