How Calculations Are Derived

Last updated: May 2026

This page documents the exact formulas behind Utilia's most-used calculators. Variables, constants, and sources are defined for each. For broader methodology context, see our Calculator Methodology page.

Finance Calculators

Mortgage / Loan Monthly Payment

The standard amortization formula computes a fixed monthly payment M that pays off principal P over n months at monthly interest rate r:

        M = P × [r(1 + r)ⁿ] / [(1 + r)ⁿ − 1]
      

Where: P = loan principal, r = annual rate ÷ 12, n = loan term in months. Source: standard financial mathematics; consistent with CFPB mortgage disclosures.

Compound Interest

The compound interest formula computes the future value A of a principal P compounded n times per year at annual rate r over t years:

        A = P × (1 + r/n)^(n×t)
      

For continuous compounding: A = P × e^(r×t). Source: standard financial mathematics textbooks.

Compound Interest with Regular Contributions

When a regular monthly contribution C is added, the future value is calculated as the sum of the compounded principal and the future value of the annuity:

        A = P×(1+r/n)^(n×t) + C × [(1+r/n)^(n×t) − 1] / (r/n)
      

Debt Payoff (Fixed Payment)

The number of months to pay off balance B at monthly rate r with fixed payment M:

        n = −log(1 − B×r/M) / log(1 + r)
      

Requires M > B×r (payment must exceed the first month's interest).

Debt-to-Income Ratio

        Front-end DTI = Monthly housing costs / Gross monthly income × 100

        Back-end DTI  = All monthly debt payments / Gross monthly income × 100

Qualification thresholds: Conventional loans ≤43–45% back-end; FHA ≤50% with compensating factors. Source: Fannie Mae Selling Guide; CFPB ability-to-repay rules.

Return on Investment (ROI)

        ROI (%) = (Final Value − Initial Value) / Initial Value × 100

        CAGR    = (Final Value / Initial Value)^(1/years) − 1

Health Calculators

Body Mass Index (BMI)

        BMI = weight (kg) / height (m)²

        BMI = 703 × weight (lb) / height (in)²  [imperial]

Classification thresholds: Underweight <18.5, Normal 18.5–24.9, Overweight 25–29.9, Obese ≥30. Source: WHO Global Database on Body Mass Index.

Basal Metabolic Rate (BMR) — Mifflin–St Jeor

Selected over Harris-Benedict (1919/1984) based on Frankenfield et al. (2005) validation study showing Mifflin–St Jeor is more accurate in 82% of non-obese and obese subjects.

        Men:   BMR = 10×weight(kg) + 6.25×height(cm) − 5×age + 5

        Women: BMR = 10×weight(kg) + 6.25×height(cm) − 5×age − 161

Source: Mifflin MD et al. "A new predictive equation for resting energy expenditure in healthy individuals." Am J Clin Nutr. 1990;51(2):241–7.

Total Daily Energy Expenditure (TDEE)

        TDEE = BMR × Activity Multiplier

        Sedentary (desk job, little exercise): × 1.2

        Lightly active (1–3 days/week):        × 1.375

        Moderately active (3–5 days/week):     × 1.55

        Very active (6–7 days/week):           × 1.725

        Extremely active (physical job + exercise): × 1.9

Maximum Heart Rate and Target Zones

        Max HR = 220 − age  [Fox & Haskell formula; most widely used]

        Karvonen target HR = ((Max HR − Resting HR) × intensity%) + Resting HR

Source: Fox SM, Naughton JP, Haskell WL. Physical activity and the prevention of coronary heart disease. Ann Clin Res. 1971.

Math Calculators

Standard Deviation

        Population σ = √[Σ(xᵢ − μ)² / N]

        Sample s      = √[Σ(xᵢ − x̄)² / (N−1)]

Sample standard deviation uses N−1 (Bessel's correction) as an unbiased estimator of population variance.

Compound Annual Growth Rate (CAGR)

        CAGR = (End Value / Start Value)^(1 / Years) − 1
      

Percentage Change

        % Change = (New Value − Old Value) / |Old Value| × 100
      

Additional Math Formulas

Greatest Common Factor (GCF) and Least Common Multiple (LCM)

GCF is found using the Euclidean algorithm: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder, repeating until the remainder is zero. The last non-zero remainder is the GCF. LCM is then derived as:

        LCM(a, b) = |a × b| / GCF(a, b)
      

Percentage Calculations

        What is X% of Y?          → Y × (X / 100)

        X is what % of Y?         → (X / Y) × 100

        % increase from X to Y    → (Y − X) / X × 100

        % decrease from X to Y    → (X − Y) / X × 100

Mean, Median, and Mode

The arithmetic mean (average) of a dataset is the sum of all values divided by the count of values. The median is the middle value when the dataset is sorted; if there is an even number of values, the median is the average of the two middle values. The mode is the value that appears most frequently; a dataset may have multiple modes or none.

        Mean = Σxᵢ / N
      

Quadratic Formula

For a quadratic equation ax² + bx + c = 0, the roots are:

        x = (−b ± √(b² − 4ac)) / 2a
      

The discriminant (b² − 4ac) determines the nature of the roots: positive = two real roots, zero = one real root, negative = two complex roots.

Exponents and Logarithms

        log_b(x) = ln(x) / ln(b)   [change of base]

        ln(x)    = log_e(x)        [natural log, base e ≈ 2.71828]

        log(x)   = log_10(x)       [common log]

Formula Sources and References

Utilia's formulas are derived from the following categories of sources:

Financial formulas — standard financial mathematics references, Consumer Financial Protection Bureau (CFPB) guidelines, Fannie Mae Selling Guide, and IRS publications.
Health and fitness formulas — peer-reviewed journals including the American Journal of Clinical Nutrition, World Health Organization (WHO) Global Database on Body Mass Index, and American College of Sports Medicine guidelines.
Mathematical formulas — standard mathematics textbooks and curricula consistent with high school and undergraduate coursework. No proprietary or novel formulas are used — all are standard public-domain mathematics.

For formulas where multiple variants exist (for example, Harris-Benedict vs. Mifflin–St Jeor for BMR), we document which variant we use and why on this page and in the relevant calculator's methodology section. We default to the variant that is most widely validated in recent literature.

If you need to cite the underlying source for a specific formula — for example, the original research paper behind the Mifflin–St Jeor equation — please refer to the citations shown on this page or contact us for the specific reference.

Questions or Corrections

If you believe a formula shown here or applied in a calculator is incorrect, please contact us with the specific calculator, the formula reference, and your source. Formula corrections are prioritized and applied promptly.