Solve for the Unknown
Turn a business target into a simple equation and solve it.
In your head, jot if needed · no calculator why?
Opens at level 8. You're level 1. You can read and practice here now.
the idea
Turn the story into an equation with a named unknown. Revenue = price × count means a count is revenue ÷ price. For a profit target, each customer's contribution (what each one leaves after its own variable costs) has to pay the fixed costs and the target.
Many costs are a fixed amount plus a cost per job. With $2,000 fixed and $50 a job, 100 jobs cost $7,000 in all, an average of $70 a job. But one more job adds only $50, because the $2,000 is paid either way.
techniques
Write it, then undo it
- Write it out: price × count = revenue, or contribution × customers = fixed costs + target.
- Divide both sides by the amount per customer or per unit.
- People and jobs come in whole numbers: round up, then check.
worked example
Each customer leaves $300 a month after variable costs. Fixed monthly costs are $4,500. How many whole customers are needed for at least $2,000 of monthly operating profit?
- $300 × customers must cover $4,500 + $2,000 = $6,500.
- $6,500 ÷ $300 ≈ 21.67, so round up to 22.
- Check: 21 customers bring $6,300, short of $6,500.
Answer: 22
Extra cost: the per-job part
- Count the extra jobs: new number minus old.
- Multiply by the cost per job.
- Leave the fixed amount out. It is paid either way.
worked example
Total cost is $1,500 plus $25 per job. How much does total cost rise when jobs go from 80 to 100?
- 100 − 80 = 20 extra jobs.
- 20 × $25 = $500.
Answer: $500
Average cost: total ÷ jobs
- Total cost: the fixed amount plus the cost per job × the jobs.
- Divide the total by the number of jobs.
worked example
Total cost is $3,000 plus $40 per job. What is the average cost per job at 150 jobs?
- Total: $3,000 + $40 × 150 = $9,000.
- Average: $9,000 ÷ 150 = $60.
Answer: $60
watch out for
- Covering the fixed costs but forgetting the profit target. The customers have to pay for both.
- Rounding part of a customer down. 21.67 customers means 22, because 21 falls short.
- Using the average cost for each extra job. At 150 jobs the average is $60, but one more job adds only $40.
- Answering in dollars when the question asks for a count.
practice
How many to hit a revenue target?
worked example
Each class seat brings in $1,350. How many class seats produce $68,850 of revenue?
Answer: 51
- Revenue = price × count, so count = $68,850 ÷ $1,350 = 51.
- Check: 51 × $1,350 = $68,850.
Customers for a profit target
worked example
Each customer leaves $380 a month after variable costs. Fixed monthly costs are $5,400. How many whole customers are needed for at least $8,100 of monthly operating profit?
Answer: 36
- Cover fixed costs plus the target: ($5,400 + $8,100) ÷ $380 = $13,500 ÷ $380 ≈ 35.53; round up to 36.
- 35 customers would bring only $13,300, short of the $13,500 needed.
How much does cost rise?
worked example
Total cost is $4,000 plus $100 per job. How much does total cost rise when jobs go from 270 to 360?
Answer: $9,000.00
- Only the per-job part changes: 90 extra jobs × $100 = $9,000.
- The $4,000 is paid once either way.
Average cost per job
worked example
Total cost is $3,000 plus $110 per job. What is the average cost per job at 100 jobs?
Answer: $140.00
- Average = total ÷ jobs = ($3,000 + $110 × 100) ÷ 100 = $14,000 ÷ 100 = $140.
- The fixed part gets shared across more jobs as volume grows.