Inputs
Advanced options includedExample Data
Click "Recompute from current settings" to apply your denominator and mode.
| # | Input | Denominator | Mode | Rounded (mixed) | As decimal |
|---|---|---|---|---|---|
| 1 | 1 3/8 | 8 | half_even | ||
| 2 | -5/4 | 8 | half_even | ||
| 3 | 2.75 | 8 | half_even | ||
| 4 | 0.33 | 3 | half_up | ||
| 5 | -1.5 | 2 | half_down |
Formula Used
Goal: round a real number x to the nearest fraction with denominator n.
- Convert any input to a real number: for a fraction a/b, use x = a ÷ b. For a mixed number w a/b, use x = sign(w) × (|w| + a/b).
- Compute k = round_mode(x × n) where round_mode applies your selected rule (half up, half down, half even, toward/away from zero, ceiling, floor).
- The rounded fraction is k/n. Reduce by gcd(|k|, n) and optionally format as a mixed number: k/n = s × (q + r/n) where q = ⌊|k|/n⌋ and r = |k| mod n, sign s ∈ {−1, +1}.
For decimals, the same formula applies because decimals are real numbers.
How to Use
- Enter a value as a fraction (e.g., 5/4), mixed number (e.g., 1 3/8), or decimal (e.g., 2.75).
- Set your target denominator or pick a quick preset.
- Choose a rounding mode: nearest with tie rules, directed, or truncation.
- Select the output format. Use decimal places if decimal output is chosen.
- Click Calculate. Inspect the result chips and the optional step-by-step explanation.
- Use the example table to try common cases and export CSV/PDF.
FAQs
It finds the closest multiple of the step 1/n. For example, with n=8, steps are eighths: …, −2/8, −1/8, 0, 1/8, 2/8, ….
Choose among tie-breaking rules: half up, half down, or half to even (bankers). Each rule decides which neighbor is picked when the distance is identical.
Yes. Choose “mixed number” to display as whole remainder/denominator. Improper and decimal formats are also available.
Yes. Rounding modes are implemented symmetrically, respecting signs for toward/away, ceiling, and floor behaviors.
Use the quick preset buttons or set any integer denominator. The algorithm works identically for all denominators.
Yes. Results are simplified using the greatest common divisor so outputs like 4/8 become 1/2.
Rounding Modes Cheat Sheet
| Mode | Definition | Example (+1.25 to halves) | Example (−1.25 to halves) |
|---|---|---|---|
| Nearest, ties to even | Pick nearest; if tie, choose even neighbor. | 1.25 → 1.0 | −1.25 → −1.0 |
| Nearest, ties go up | Ties move away from zero on positives. | 1.25 → 1.5 | −1.25 → −1.5 |
| Nearest, ties go down | Ties move toward zero on positives. | 1.25 → 1.0 | −1.25 → −1.0 |
| Toward zero (truncate) | Drop fractional step toward zero. | 1.25 → 1.0 | −1.25 → −1.0 |
| Away from zero | Push fractional step away from zero. | 1.25 → 1.5 | −1.25 → −1.5 |
| Ceiling | Always round up (toward +∞). | 1.25 → 1.5 | −1.25 → −1.0 |
| Floor | Always round down (toward −∞). | 1.25 → 1.0 | −1.25 → −1.5 |
Tip: For measurements, “ties to even” reduces systematic bias across many readings.
Denominator Quick Reference & Typical Uses
Examples use nearest with ties to even and input x = 2.73.
| Denominator n | Step 1/n | Rounded value | As mixed | Typical uses |
|---|---|---|---|---|
| 2 | 0.5 | 2.5 | 2 1/2 | Carpentry rough cuts, quick estimates |
| 3 | 0.333… | 2.666… | 2 2/3 | Cooking thirds, tiling layouts |
| 4 | 0.25 | 2.75 | 2 3/4 | Woodworking quarters, fasteners |
| 5 | 0.2 | 2.8 | 2 4/5 | Finance tenths/fifths, machining |
| 8 | 0.125 | 2.75 | 2 3/4 | Cabinetry, pipe fittings |
| 10 | 0.1 | 2.7 | 2 7/10 | Metric approximations, decimals |
| 16 | 0.0625 | 2.75 | 2 3/4 | Fine woodworking, metalwork |
| 32 | 0.03125 | 2.71875 | 2 23/32 | Precision jigs, calibration |
- Maximum absolute error for nearest rounding ≤ 1/(2n).
- Reducing the fraction preserves the same numeric value, just simplified.
- Choose n based on your domain’s common increments and tolerances.