Sixths and Fifths as Percentages
Sixths (fractions with denominator 6) and fifths (fractions with denominator 5) are common in mathematics, music, cooking, and everyday proportion problems. Fifths convert cleanly to percentages (since 5 is a factor of 100), while sixths — like thirds — produce repeating decimals.
Sixths Reference Table (÷6)
| Fraction | Decimal | Percentage |
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Fifths Reference Table (÷5)
| Fraction | Decimal | Percentage |
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Why Fifths Are "Clean" and Sixths Are Not
A fraction produces a terminating decimal if and only if its denominator (in lowest terms) has no prime factors other than 2 and 5. Since 5's only prime factor is 5, all fifths terminate: 1/5 = 0.2, 2/5 = 0.4, etc. These are particularly easy to work with.
6 = 2 × 3. The factor of 3 in the denominator causes repeating decimals for fractions that don't simplify away the 3. 1/6 = 0.16666..., 1/3 = 0.33333..., 5/6 = 0.83333... However, fractions like 2/6 = 1/3, 3/6 = 1/2 (0.5), and 4/6 = 2/3 follow their simplified forms.
Sixths in Music Theory
In Western music, an octave is divided into 12 semitones. A major sixth interval spans 9 semitones; a minor sixth spans 8. Rhythmically, a sixth note value in compound time signatures creates 6/8, 6/4, and 6/16 time signatures — felt as two groups of three or three groups of two.
Fifths in Everyday Life
Fifths convert perfectly to multiples of 20%: 1/5 = 20%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%, 5/5 = 100%. This makes them unusually practical. A "fourth of a group" might get 25%, but "a fifth" gets exactly 20% — a clean, round number. Liquor bottles are sold in "fifths" (1/5 of a gallon = 750ml). Musical fifths define the harmonic circle of fifths.
Converting Between Fractions and Percentages: A Summary
The key principle: to convert any fraction to a percentage, divide numerator by denominator and multiply by 100. The ease or complexity of the decimal depends entirely on the denominator's prime factors. Denominators made only of 2s and 5s always produce clean terminating decimals. Any other prime factor in the denominator produces a repeating decimal.