About

Converting between decimal fractions and percentages is a source of silent errors in financial reports, academic grading, and data analysis. A misplaced decimal point in a tax rate or test score propagates through every downstream calculation. This tool computes p = (n ÷ T) × 100, where T defaults to 1 for standard 0-1 scale decimals, or accepts a custom total for ratio-based conversions (e.g., 120 passed out of 145 total tests). It also operates in reverse, converting a percentage back to its decimal representation. Results are rounded to a user-specified number of decimal places using IEEE 754 compliant rounding. Note: floating-point arithmetic may introduce representational error beyond ~15 significant digits.

Formulas

The decimal-to-percentage conversion with an optional custom total uses the following formula:

p = nT × 100

Where p is the resulting percentage, n is the input number, and T is the total (comparator) value. When T = 1, the formula simplifies to p = n × 100.

The reverse operation (percentage to decimal) is:

d = p100 × T

Where d is the decimal result and p is the input percentage. Rounding is applied using the standard IEEE 754 half-to-even rule at the specified number of decimal places k, computed as: round(x × 10^k) ÷ 10^k.

Reference Data

Decimal	Fraction	Percentage	Common Use
0.01	1100	1%	Basis point (finance)
0.05	120	5%	Statistical significance threshold (α)
0.10	110	10%	Standard tip / discount
0.125	18	12.5%	Stock price eighth
0.20	15	20%	VAT rate (UK)
0.25	14	25%	Quartile boundary
0.333	13	33.3%	One-third split
0.50	12	50%	Median / half
0.618	1φ	61.8%	Golden ratio complement (Fibonacci retracement)
0.667	23	66.7%	Supermajority threshold
0.75	34	75%	Third quartile (Q3)
0.80	45	80%	Grade B threshold
0.90	910	90%	Confidence level
0.95	1920	95%	95% confidence interval
0.99	99100	99%	SLA uptime target
1.00	11	100%	Whole / complete
1.50	32	150%	Overtime multiplier (1.5×)
2.00	21	200%	Doubling / 2× growth
0.001	11000	0.1%	10 basis points
0.0001	110000	0.01%	1 basis point (BPS)

Frequently Asked Questions

IEEE 754 floating-point arithmetic cannot represent certain decimal fractions exactly in binary. The value 0.1 + 0.2 yields 0.30000000000000004 internally. This tool displays the mathematically correct rounded result at your chosen decimal places, but the underlying binary representation carries this inherent limitation. For critical financial calculations, limit decimal places to 2-4 to avoid apparent rounding artifacts.

Division by zero is undefined. The tool detects a zero total and returns an error notification instead of a result. Mathematically, n ÷ 0 → ±∞, which is not a valid percentage. If you intended a 0-1 scale, leave the total field at its default value of 1.

Yes. If the input number exceeds the total, the percentage will be greater than 100%. For example, 1.5 on a 0-1 scale yields 150%. This is valid and common in contexts like year-over-year growth rates, overtime multipliers, and concentration ratios.

Reverse mode applies the inverse formula: the decimal result equals the percentage divided by 100, then multiplied by the total. For instance, entering 82.76% with a total of 145 returns approximately 120, which is the original count that produced that percentage.

A basis point (BPS) equals 0.01%, or 0.0001 in decimal form. Financial instruments frequently quote rates in basis points to avoid ambiguity. Enter 0.0001 with total 1 to confirm it converts to 0.01%. A 25 basis point rate hike means an increase of 0.0025 in decimal, or 0.25%.

Yes. Negative decimals convert to negative percentages. A value of −0.15 on a 0-1 scale yields −15%. This is meaningful in contexts like negative growth rates, losses, or temperature differentials expressed as proportional change.