Better decisions.
One timeline.
Should you buy the machine, change the loan, or invest in the upgrade? Connect costs, time, and value—and learn to justify the choice.
Explore thirty visual lessons, live mathematical models, and practice that connects each answer to its meaning.
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An investment today. Benefits over time.
A common date makes the comparison possible.
Every design has a cost.
Every choice has a future.
Clean energy equipment, a production line, a student loan, or a replacement pump: the most useful option depends on when money moves and what you give up to commit it.
Move between a physical decision, its cash-flow diagram, and its equation. Learn to calculate the result and defend the assumptions behind it.
Three questions. One useful framework.
What is money worth later?
A future receipt and money available today are different opportunities. Interest and equivalence put both on a common date.
Try this idea ↗When is a project worth its cost?
Change the investment, annual benefit, MARR, and life. Follow the cash flows and watch PW change. PW, FW, and AW are the same project expressed on different clocks.
Explore the full explanation ↗Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The independent project meets the MARR on these assumed cash flows.
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
A little structure.
A better decision.
Start with costs and cash-flow timing. Build the interest toolkit, evaluate an investment, then compare the alternatives fairly.
Each lesson pairs a live model with two numerical problems and a conceptual check. Twenty-one mixed problems help you choose a method independently.
Begin the course ↗01Frame the decision+
Principles, costs, design, estimation
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- Make a defensible engineering decision ↗Visual lesson
- Sort costs before calculating ↗Visual lesson
- Find where costs and revenue meet ↗Visual lesson
- Estimate a cost model from observations ↗Visual lesson
- Scale a plant without scaling every dollar ↗Visual lesson
02Put cash flows on one clock+
Interest, equivalence, loans, factors, annuities, gradients, interpolation
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- Watch interest earn interest ↗Visual lesson
- Move money to a common date ↗Visual lesson
- Follow the loan balance ↗Visual lesson
- Choose the right factor ↗Visual lesson
- Put a deferred annuity on the right date ↗Visual lesson
- Separate a base series from a gradient ↗Visual lesson
- Distinguish growth from interest ↗Visual lesson
- Read between factor-table entries ↗Visual lesson
03Match rates and periods+
Nominal/effective rates, different periods, continuous compounding
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- Convert nominal into effective interest ↗Visual lesson
- Match interest to payment periods ↗Visual lesson
- Separate continuous interest from deposits ↗Visual lesson
04Evaluate a project+
Marr, pw/fw/aw, recovery, perpetual life
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- Set an acceptance threshold ↗Visual lesson
- Make PW, FW, and AW tell the same story ↗Visual lesson
- Recover capital and cover operation ↗Visual lesson
- Value service that continues forever ↗Visual lesson
05Understand returns and recovery+
Irr, interpolation, non-simple flows, err, payback, mcdm
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- Find the zero-worth interest rate ↗Visual lesson
- Refine IRR in two interpolation steps ↗Visual lesson
- Spot an ambiguous rate of return ↗Visual lesson
- Make the external rate explicit ↗Visual lesson
- Track recovery without confusing it with profit ↗Visual lesson
- Compare dollars, returns, and liquidity carefully ↗Visual lesson
06Compare alternatives fairly+
Pw, incremental irr, unequal lives, project relationships
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- Choose among alternatives, not only projects ↗Visual lesson
- Ask whether the extra investment earns its keep ↗Visual lesson
- Give unequal lives a fair comparison ↗Visual lesson
- Choose a feasible portfolio ↗Visual lesson
Make your study time diagnostic.
- Work one topic without opening a solution; note exactly where you get stuck.
- Use its visual model to repair the misconception, then solve the new scenario.
- Complete mixed practice on paper to test method selection, not recognition of a heading.
- Build your own concise reference. Reserve space for timing reminders, rate conversion, IRR caveats, and unequal-life assumptions.
Start with a real decision and follow it through the picture, the mathematics, and the assumptions. Progress records the topics you choose to mark understood.