--- name: math-thumb-financial-mathematics description: "Apply Chapter 17 (Financial Mathematics: Comparing Money Across Time) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to place hypothetical cash flows on one time and rate basis before comparing them." --- # Financial Mathematics: Comparing Money Across Time Use this chapter to help the reader make a checked mathematical decision. All 12 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Cash-flow amounts, signs, dates and currency; valuation date; effective period rate; nominal versus real basis; payment timing. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule 1. Write the objective: quick scale estimate, valuation, affordability, or contract comparison. 2. Draw a timeline and mark every amount, sign, date, and currency. 3. Put rates on a common effective-period basis and decide whether amounts and rates are nominal or real. 4. Move every cash flow to one valuation date. Use a geometric closed form only after recognizing an exact pattern. 5. Check domain conditions: payment timing, \(r>0\), \(r>g\), fixed count, and compatible periods. 6. Stress-test cash flows and rates. Add taxes, fees, default, liquidity, and constraints when relevant. 7. Treat the number as a model output, not personal financial advice; escalate high-stakes decisions to contract-specific analysis and qualified professionals. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Treat exercise rates as explicit assumptions. Match rate periods and cash-flow timing; do not convert a classroom valuation into personalized investment or borrowing advice. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 17.2.2, 17.2.3, 17.3.1 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C17-D01: Change the rate and recheck present value**: rule 17.2.3. - **C17-D02: Use the correct base for recovering a loss**: rule 17.1.3. - **C17-D03: Compare compounding conventions fairly**: rule 17.2.1. - **C17-D04: Check the rule of 72**: rule 17.1.1. - **C17-D05: Watch a growing perpetuity blow up**: rule 17.3.3. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.