Calculation Methodology & Audited Formulas

1. Fixed-Rate Mortgage Amortization

Standard fixed-rate home loans follow the classic annuity formula defined by the Consumer Financial Protection Bureau (CFPB) and federal banking regulations.

Monthly Payment (Principal & Interest) Equation

M = P · [ r(1 + r)^n ] / [ (1 + r)^n - 1 ]

P = Principal loan amount (Home Purchase Price minus Down Payment)
r = Monthly periodic interest rate (Annual Interest Rate / 12 / 100)
n = Total number of monthly amortization cycles (Loan Term in Years × 12)

In every payment cycle $t$:

Interest Charge for Month $t$: $I_t = B_{t-1} \times r$ (where $B_{t-1}$ is the remaining principal balance from the prior month).
Principal Repaid for Month $t$: $P_t = M - I_t$.
New Ending Balance: $B_t = B_{t-1} - P_t$.

2. Extra Principal Prepayment Acceleration

When a borrower pays an extra amount $E$ each month, 100% of $E$ goes directly toward reducing the principal balance $B_t$:

Prepayment Balance Reduction

B_t = B_{t-1} - (P_t + E)

Because subsequent interest charges $I_{t+1}$ depend on $B_t$, every dollar of prepayment creates an exponential compound saving on all future months, shaving years off the repayment schedule.

3. Compound Interest with Periodic Deposits & Rule of 72

The future value of an initial investment $P$ compounding at annual interest rate $r$ with regular periodic additions $PMT$ is given by:

Future Value of Investment

FV = P · (1 + r/n)^(n·t) + PMT · [ (1 + r/n)^(n·t) - 1 ] / (r/n)

P = Initial starting principal
PMT = Monthly recurring contribution deposit
r = Nominal annual return rate (as a decimal)
n = Compounding frequency per year (12 for monthly, 365 for daily, 1 for annual)
t = Investment horizon in years

The Rule of 72 Mental Shortcut

The Rule of 72 is derived from the natural logarithm approximation:

Doubling Time Derivation

2 = (1 + r)^t ⇒ ln(2) ≈ 0.693 ⇒ t ≈ 69.3 / (100r) ≈ 72 / R

For interest rates between 5% and 12%, 72 is chosen because it has many factors (2, 3, 4, 6, 8, 9, 12), allowing effortless mental math without touching a calculator.

4. 2026 Progressive Federal Tax & FICA Payroll Math

Income tax in the United States does not apply a single flat rate to all income. It is structured into progressive statutory marginal brackets:

Progressive Tax Computation

Taxable Income = Gross Income - Standard Deduction - 401(k) Pre-tax Contributions

2026 Standard Deduction: $15,000 for Single filers, $30,000 for Married Filing Jointly.
Marginal Brackets: 10%, 12%, 22%, 24%, 32%, 35%, 37%.

Mandatory FICA Payroll Taxes

Social Security (OASDI): 6.2% on wages up to the statutory annual wage base cap ($176,100 for 2025/2026). Income earned above this threshold is taxed at 0% for Social Security.
Medicare (HI): 1.45% on all wages with no wage cap. High earners (>$200,000 single / >$250,000 married) incur an Additional Medicare Tax of 0.9%.

5. Clinical Basal Metabolic Rate (Mifflin-St Jeor Equation)

The Mifflin-St Jeor formula was published in the American Journal of Clinical Nutrition (1990) and validated by the Academy of Nutrition and Dietetics as the most accurate estimation of human Basal Metabolic Rate (BMR).

Mifflin-St Jeor Clinical BMR Equations

Men: BMR = (10 · W) + (6.25 · H) - (5 · A) + 5

Women: BMR = (10 · W) + (6.25 · H) - (5 · A) - 161

W = Body weight in kilograms (kg)
H = Height in centimeters (cm)
A = Age in completed years

Total Daily Energy Expenditure (TDEE) Multipliers

To estimate total daily maintenance calories, BMR is multiplied by the validated physical activity coefficient:

Sedentary (desk job, little exercise): $BMR \times 1.2$
Lightly Active (light workouts 1–3 days/week): $BMR \times 1.375$
Moderately Active (moderate exercise 3–5 days/week): $BMR \times 1.55$
Very Active (hard exercise 6–7 days/week): $BMR \times 1.725$
Extremely Active (athlete / physical labor job): $BMR \times 1.9$

6. Dining Gratuity & Bill Splitting Rounding Precision

Standard tipping calculation follows post-tax or pre-tax subtotal conventions:

Tip & Share per Guest

Tip Amount = Bill Subtotal · (Tip % / 100)

Share per Person = (Bill Total + Tip Amount) / Number of Guests

Our tool provides integer rounding rules ($/€/£ whole unit ceiling) so friends can avoid splitting pennies when settling bills with cash, Venmo, or Revolut.

Transparent Formulas.Zero Black Boxes.