How Calculations Are Derived

Last updated: September 28, 2026

This page documents the exact formulas behind Utilia's most-used US personal-finance calculators and home-project estimators. 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. The mortgage calculator also links the CFPB's material on the cost components of a US mortgage.

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

Source: CFPB definition of the debt-to-income ratio. Commonly cited lender limits are roughly 43–45% back-end for conventional loans and up to about 50% for FHA loans with compensating factors, but limits vary by lender and loan program, so treat these as typical ranges rather than rules.

Return on Investment (ROI)

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

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

Home Project Calculators

The home project calculators are material estimators: geometry from the dimensions you enter, plus a waste allowance and product coverage or unit sizes that each page states. They are not structural or code-compliance calculations.

Volume (Concrete, Gravel, Mulch)

        Cubic feet  = Length (ft) × Width (ft) × Depth (in) / 12

        Cubic yards = Cubic feet / 27

        Order qty   = Cubic yards × (1 + waste %)

Bag counts divide the volume by the yield of the bag size shown on the calculator page (for example, the concrete calculator uses 0.6 cubic feet per 80-lb bag, or 45 bags per cubic yard).

Area Coverage (Paint, Flooring, Tile, Drywall)

        Area         = Length × Width (minus openings where the page asks for them)

        Units needed = Area × (1 + waste %) / Coverage per unit

Coverage per unit (square feet per gallon, per box, per sheet) is the typical figure stated on each calculator page; the figure printed on the product you buy should take precedence. The paint calculator, for example, defaults to 350 sq ft per gallon per coat.

Roofing

        Roof area = Footprint area × Pitch factor

        Squares   = Roof area / 100

        Bundles   = Squares × Bundles per square

The pitch factor converts a flat footprint into sloped surface area. Bundles per square depends on the shingle product; the roofing calculator page states the typical values it assumes.

Formula Sources and References

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

Financial formulas — standard financial mathematics references, plus the agency each calculator links in its "Sources & method" section: IRS publications, the Social Security Administration, the Consumer Financial Protection Bureau (CFPB), the Bureau of Labor Statistics Consumer Price Index, and Medicare.gov.
Home project formulas — standard geometry (area, volume, slope), with the waste allowance and product coverage or unit sizes stated on each calculator page. No proprietary or novel formulas are used.

Where a formula has more than one common variant (for example, how often interest compounds), the calculator page states which one it uses or lets you choose.

If you need to cite the underlying source for a specific formula — for example, the IRS publication behind a required-minimum-distribution calculation — please refer to the "Sources & method" section of that calculator 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.