A fraction, a decimal and a percentage walk onto a number line. The fraction says \tfrac{1}{2}. The decimal says 0.5. The percentage says 50\%. Then they all stop at the same spot. If this confuses you, you are not alone — most students have been told the three notations are "equivalent" without being shown why they land in the same place. This satellite puts all three on the same number line and lets you drag them. They never separate.
Three names for one point
Each of the three notations is a different way of writing the same thing: "how much of a unit." A fraction writes it as "this many pieces out of that many." A decimal writes it in powers of ten. A percentage writes it as "out of a hundred." But all three are pointing at the same location on the number line.
The translation rules:
- Fraction → decimal. Divide the numerator by the denominator: 1 \div 2 = 0.5.
- Decimal → percentage. Multiply by 100: 0.5 \times 100 = 50, and write "%".
- Percentage → fraction. Put the number over 100: 50\% = \tfrac{50}{100} = \tfrac{1}{2} after simplifying.
The rules look like three independent tricks. They are not. They are the same quantity expressed in three unit systems. Writing 0.5 is writing the quantity in units of 1. Writing 50\% is writing it in units of \tfrac{1}{100}. Writing \tfrac{1}{2} is writing it in units of "pieces of a whole." The quantity has not moved — only the scale on the ruler changed.
Slide the point and watch all three names update
Why the translations work
The decimal and the percentage are built on the same idea — powers of ten. A decimal 0.5 means "five tenths" — you have 5 in the tenths place. A percentage is "out of a hundred" — 50\% means "fifty hundredths" — which is \tfrac{50}{100}. And \tfrac{5}{10} = \tfrac{50}{100} because both top and bottom multiplied by 10.
So decimal-to-percent is just a shift of the decimal point two places. 0.5 becomes 50.0, which is 50\%. 0.07 becomes 7.0, which is 7\%. 1.25 becomes 125.0, which is 125\% — yes, percentages over 100 are fine, they just mean "more than one whole."
Fractions are the oldest notation, and they connect to decimals through division. \tfrac{1}{2} means "one divided into two parts, take one." Long-divide: 1 \div 2 = 0.5. \tfrac{3}{4} becomes 3 \div 4 = 0.75. The decimal is the numerical answer to the division the fraction represents.
Why all three describe the same quantity: they all answer the same question — "what fraction of the unit is this?" Fractions answer in pieces-of-a-whole. Decimals answer in tenths, hundredths, thousandths. Percentages answer in hundredths, with the denominator baked in. The question is one; the grammar is three.
A table of the most common translations
These are worth memorising so the translations happen without computation.
| Fraction | Decimal | Percentage |
|---|---|---|
| \tfrac{1}{2} | 0.5 | 50\% |
| \tfrac{1}{4} | 0.25 | 25\% |
| \tfrac{3}{4} | 0.75 | 75\% |
| \tfrac{1}{5} | 0.2 | 20\% |
| \tfrac{1}{10} | 0.1 | 10\% |
| \tfrac{1}{8} | 0.125 | 12.5\% |
| \tfrac{1}{3} | 0.333\ldots | 33.33\ldots\% |
| \tfrac{2}{3} | 0.666\ldots | 66.66\ldots\% |
Once these are instant, most shopkeeper problems, tip calculations and quick estimations stop requiring computation. You just see that a 25% discount on ₹ 800 is "₹ 800 minus a quarter" = ₹ 600.
When the notations disagree — a common trap
They never disagree about the quantity. But they often disagree about readability.
- \tfrac{1}{3} is a clean fraction. As a decimal it becomes 0.333\ldots — a non-terminating tail — and as a percentage it is 33.33\ldots\%. The fraction notation is exact; the decimal and percentage are approximations unless you write the bar-over notation 0.\overline{3}.
- \tfrac{22}{7} is a clean fraction and a famous approximation to \pi. As a decimal: 3.142857\ldots — the same kind of repeating issue.
- 0.1 is a clean decimal. As a fraction, \tfrac{1}{10} is clean. But in binary (which is what computers use), 0.1 becomes 0.0\overline{0011} — an infinite repeating binary. Decimal-clean does not mean base-clean.
So the quantity is always one, but the notation that displays it cleanly depends on the denominator. Denominators whose prime factors are only 2 and 5 give terminating decimals. All others give repeating decimals. The fraction notation sidesteps this entirely.
The practical reason for three
Why keep all three around if they are the same?
- Fractions are best for exact arithmetic and for ratios. \tfrac{2}{3} of a recipe is a fraction problem, and converting to 0.666\ldots loses precision.
- Decimals are best for measurement and computation. A ruler reads 12.7 cm, not \tfrac{127}{10} cm. Calculators speak decimal natively.
- Percentages are best for comparison and intuitive human scale. "Test score: 72%" is easier to parse than "0.72" or "\tfrac{18}{25}," even though all three say the same thing.
Choose the notation that fits the situation. Switching between them is a small translation — the quantity stays put.
Summary
- Fractions, decimals and percentages are three notations for the same quantity.
- \tfrac{1}{2} = 0.5 = 50\% is one point on the number line, written three ways.
- Translations: fraction-to-decimal by division; decimal-to-percent by multiplying by 100; percent-to-fraction by putting over 100 and simplifying.
- Memorise the first eight or so conversions. They make mental arithmetic ten times faster.
- Choose notation by situation, not by habit — use whichever displays your quantity most cleanly.
Related: Fractions and Decimals · Number Systems · Slice a Chocolate Bar: Drag the Denominator, Watch the Pieces Shrink