Finding Total Annual Profit with Piecewise Cost Relationships

Number Puzzles 9th-10th Grade
Problem
The profit P, in dollars, for any given month at a certain company is defined by P = I - C, where I represents total income, in dollars, and C represents total costs, in dollars, for the month. For each of the first 4 months of the year, C = I + 32,000; and for each of the next 3 months, I = C + 36,000. If I = C + 10,000 for each of the 5 remaining months of the year, what was the company's total profit for the 12 month year?

Skills This Problem Builds

  • Working with piecewise functions across different time periods
  • Substituting algebraic relationships into formulas
  • Calculating cumulative totals from monthly data
  • Understanding profit as the difference between income and costs
  • Interpreting negative profit as business losses

Let's Draw It

The profit P , in dollars, for any given month at a certain company is defined by P = I C , where I represents total...

Solution: Method 1 — Direct Profit Substitution

We'll use the fundamental profit formula P = I - C and handle each time period separately, substituting the given relationships between income and costs.

Step 1 — Analyze the first 4 months (Loss Period)

For months 1-4, we're told that C = I + 32,000. This means costs exceed income by $32,000 each month.

P = I - C
P = I - (I + 32,000)
P = I - I - 32,000
P = -32,000

Monthly profit: -$32,000 (a loss)
Total for 4 months: 4 × (-32,000) = -128,000

Step 2 — Analyze months 5-7 (Recovery Period)

For months 5-7, we have I = C + 36,000. Now income exceeds costs by $36,000 each month.

P = I - C
P = (C + 36,000) - C
P = C + 36,000 - C
P = 36,000

Monthly profit: +$36,000
Total for 3 months: 3 × 36,000 = 108,000

Step 3 — Analyze months 8-12 (Steady Profit Period)

For the final 5 months, I = C + 10,000. Income exceeds costs by $10,000 each month.

P = I - C
P = (C + 10,000) - C
P = C + 10,000 - C
P = 10,000

Monthly profit: +$10,000
Total for 5 months: 5 × 10,000 = 50,000

Step 4 — Calculate total annual profit

Sum the profits from all three periods:

Total Annual Profit = (-128,000) + 108,000 + 50,000
= -128,000 + 158,000
= 30,000
The company's total profit for the 12-month year was $30,000.

Solution: Method 2 — Net Cash Flow Analysis

We can think of this as a cash flow problem where we track the net monthly change in company wealth.

Step 1 — Set up the cash flow periods

Instead of thinking about abstract income and costs, let's focus on net cash flow per month in each period.

Step 2 — Calculate monthly cash flows

Period 1 (Months 1-4):C = I + 32,000 means the company spends $32,000 more than it earns each month. Net cash flow: -$32,000/month

Period 2 (Months 5-7):I = C + 36,000 means the company earns $36,000 more than it spends each month. Net cash flow: +$36,000/month

Period 3 (Months 8-12):I = C + 10,000 means the company earns $10,000 more than it spends each month. Net cash flow: +$10,000/month

Step 3 — Calculate cumulative impact

Period 1 impact: 4 months × (-$32,000) = -$128,000
Period 2 impact: 3 months × (+$36,000) = +$108,000
Period 3 impact: 5 months × (+$10,000) = +$50,000

Net annual change: -$128,000 + $108,000 + $50,000 = $30,000
The company's total profit for the 12-month year was $30,000.

Verification

Let's verify our answer by checking that our period calculations are consistent and sum correctly.

Period breakdown check:

  • Months counted: 4 + 3 + 5 = 12 months ✓
  • First period: 4 × (-$32,000) = -$128,000 ✓
  • Second period: 3 × $36,000 = $108,000 ✓
  • Third period: 5 × $10,000 = $50,000 ✓
Total verification: -128,000 + 108,000 + 50,000 = 30,000 ✓

Algebraic verification: Let's verify that P = I - C works for each period:

  • Period 1: If C = I + 32,000, then P = I - (I + 32,000) = -32,000 ✓
  • Period 2: If I = C + 36,000, then P = (C + 36,000) - C = 36,000 ✓
  • Period 3: If I = C + 10,000, then P = (C + 10,000) - C = 10,000 ✓

Does This Seem Reasonable?

The answer of $30,000 annual profit makes perfect sense when we consider the business cycle shown in our timeline.

The company started the year losing $32,000 per month for 4 months—perhaps during a startup phase or economic downturn. That's a substantial loss of $128,000 that needed to be recovered.

During months 5-7, the company turned around dramatically, earning $36,000 per month. This strong performance period brought in $108,000, nearly offsetting the initial losses.

The final 5 months show steady, moderate profitability at $10,000 per month, contributing another $50,000.

The net result: -128,000 + 108,000 + 50,000 = 30,000. The company recovered from early losses and ended the year with a modest but positive profit—exactly what you'd expect from a business that struggled initially but found its footing.

Watch Out For These

✗Confusing the relationship direction

Students often mix up C = I + 32,000 with I = C + 32,000. Remember: C = I + 32,000 means costs are HIGHER than income (a loss), while I = C + 32,000 means income is HIGHER than costs (a profit).

✗Forgetting to multiply by the number of months

Computing P = -32,000 but then using -32,000 as the total for all 4 months, rather than 4 × (-32,000) = -128,000. Each period spans multiple months!

✗Sign errors when adding negative profits

Writing -128,000 + 108,000 + 50,000 as 128,000 + 108,000 + 50,000 and getting 286,000 instead of 30,000. Negative profits (losses) must stay negative in your calculation.

The General Formula

This problem illustrates the broader principle of piecewise profit analysis. When a business has different income-cost relationships during different periods, the annual profit is:

Annual Profit = Σ(months_i × profit_per_month_i)

Where each profit_per_month_i is calculated by substituting the period's specific income-cost relationship into P = I - C.

The key insight: profit per month stays constant within each period because the relationship between I and C is linear. This allows us to multiply monthly profit by the number of months in each period, then sum across all periods.

This same structure appears in:

  • Seasonal businesses with different profit margins per quarter
  • Subscription services with promotional pricing periods
  • Manufacturing with varying material costs over time

What You Need to Know First

Before tackling this problem, make sure you're comfortable with:

  • Substituting expressions into algebraic formulas
  • Working with negative numbers in addition and subtraction
  • Understanding that profit = income - costs
  • Multiplying to find totals over multiple time periods

What If?

1
Extended Loss Period
What if the first period lasted 6 months (instead of 4) with C = I + 32,000, the next 3 months had I = C + 36,000, and the final 3 months had I = C + 10,000? Find the annual profit.
Step 1 — First 6 months

With C = I + 32,000: P = I - (I + 32,000) = -32,000 per month
Total: 6 × (-32,000) = -192,000

Step 2 — Next 3 months

With I = C + 36,000: P = (C + 36,000) - C = 36,000 per month
Total: 3 × 36,000 = 108,000

Step 3 — Final 3 months

With I = C + 10,000: P = (C + 10,000) - C = 10,000 per month
Total: 3 × 10,000 = 30,000

Step 4 — Annual total

Annual profit = -192,000 + 108,000 + 30,000 = -54,000

Verification

Months: 6 + 3 + 3 = 12 ✓
Check: -192,000 + 108,000 + 30,000 = -54,000 ✓

2
Break-Even Challenge
The first 4 months have C = I + 32,000, the next 3 months have I = C + 36,000. For the company to break even (annual profit = 0), what should be the relationship between I and C for the final 5 months? Express as I = C + k and find k.
Step 1 — Calculate first two periods

First 4 months: 4 × (-32,000) = -128,000
Next 3 months: 3 × 36,000 = 108,000
Combined: -128,000 + 108,000 = -20,000

Step 2 — Set up break-even condition

For annual profit = 0: -20,000 + (final 5 months profit) = 0
So final 5 months must contribute: +20,000 total

Step 3 — Find required monthly profit

Monthly profit needed: 20,000 ÷ 5 = 4,000 per month

Step 4 — Solve for k

If I = C + k, then P = (C + k) - C = k
We need k = 4,000, so I = C + 4,000

Verification

Check: -128,000 + 108,000 + (5 × 4,000) = -20,000 + 20,000 = 0 ✓

3
Quarterly Reporting
A company reports quarterly instead of monthly. Q1 has C = I + 96,000, Q2 has I = C + 108,000, Q3 has I = C + 45,000, and Q4 has I = C + 15,000. What's the annual profit?
Step 1 — Calculate Q1 profit

With C = I + 96,000: P = I - (I + 96,000) = -96,000

Step 2 — Calculate Q2 profit

With I = C + 108,000: P = (C + 108,000) - C = 108,000

Step 3 — Calculate Q3 profit

With I = C + 45,000: P = (C + 45,000) - C = 45,000

Step 4 — Calculate Q4 profit

With I = C + 15,000: P = (C + 15,000) - C = 15,000

Step 5 — Sum annual profit

Annual profit = -96,000 + 108,000 + 45,000 + 15,000 = 72,000

Verification

Check: -96,000 + 108,000 + 45,000 + 15,000 = 72,000 ✓

4
Reverse Engineering
A company's annual profit was $48,000. The first 4 months had C = I + k, the next 3 months had I = C + 36,000, and the final 5 months had I = C + 10,000. Find the value of k.
Step 1 — Calculate known periods

Months 5-7: 3 × 36,000 = 108,000
Months 8-12: 5 × 10,000 = 50,000
Known total: 108,000 + 50,000 = 158,000

Step 2 — Find first period contribution

Annual profit = (first 4 months) + 158,000 = 48,000
So first 4 months contributed: 48,000 - 158,000 = -110,000

Step 3 — Find monthly loss in first period

Monthly profit for first 4 months: -110,000 ÷ 4 = -27,500

Step 4 — Solve for k

If C = I + k, then P = I - (I + k) = -k
We need -k = -27,500, so k = 27,500

Verification

Check: 4 × (-27,500) + 108,000 + 50,000 = -110,000 + 158,000 = 48,000 ✓

Frequently Asked Questions

How do you solve piecewise profit problems with different conditions per period?+
Analyze each time period separately using the profit formula P = I - C. For each period, substitute the given relationship between income and costs, then calculate the total profit for that period by multiplying monthly profit by the number of months. In this problem, we had three distinct periods with different I-C relationships, calculated each separately, then summed them for the annual total.
What does it mean when costs exceed income in a business profit calculation?+
When C = I + amount, costs exceed income by that amount, resulting in negative profit (a loss). This means the business is spending more money than it's bringing in. In this problem, C = I + 32,000 means the company loses $32,000 per month during the first four months—perhaps during a startup phase or economic downturn.
How do you calculate annual profit when monthly conditions change?+
Calculate the profit for each distinct period separately, then sum all periods. Here we have 4 months at -$32,000 each (total: -$128,000), 3 months at +$36,000 each (total: +$108,000), and 5 months at +$10,000 each (total: +$50,000), giving a total annual profit of $30,000.
NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-09-12

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