A shop window shouts "⅓ OFF!" Next door another shouts "33% OFF!" A website says the price is "0.67 of the original." Are these three different deals? No — they are the very same discount, just wearing three different costumes. A half is \tfrac{1}{2}, or 0.5, or 50\%, and being able to slide fluently between the three costumes is one of the most useful number skills you will ever own.

You'll use it every time you work out a sale price, read a statistic, compare two "best deals", split a bill, or check a test score. Let's make the three outfits feel like one number.

Here is a quiet but powerful idea: the same amount can be written in three different outfits. A fraction, a decimal, and a percentage can all name one and the same quantity — they just dress it up differently.

Take a half. We can write it three ways, and they are exactly equal:

A percentage is simply "out of a hundred" — the sign \% means "per cent", per hundred. So 50\% means \tfrac{50}{100}, which is the same as a half.

A few more turn up so often it is worth knowing them by heart. Split a whole into four equal parts and the quarters line up neatly:

And one tenth, the first step into decimals:

A couple more are worth tucking away — a fifth and a third:

Notice the third never settles down — 1 \div 3 gives 0.3333\ldots forever, so we round it to about 33\%.

See it on a hundred grid

A square of one hundred little cells is the perfect picture. Shade some cells and read off all three names at once: how many cells out of 100 (the percentage), the same as a decimal, and the same as a fraction. Step through it, and press Refresh for a new amount.

How to convert

You do not have to memorise every case — a handful of small moves gets you anywhere. Here is the whole toolkit:

Fraction → decimal: divide the top by the bottom. A fraction is a division, so \tfrac{3}{4} is just 3 \div 4 = 0.75.

Decimal → percentage: multiply by 100 (which slides the point two places right) and add the \% sign. So 0.75 \times 100 = 75\%.

Percentage → decimal: divide by 100 (slide the point two places left). So 75\% \div 100 = 0.75.

Percentage → fraction: write it over 100, then cancel to lowest terms. So 75\% = \tfrac{75}{100} = \tfrac{3}{4}.

The two directions simply undo each other: \times 100 going towards a percentage, \div 100 coming back.

Worked example 1 — a fraction all the way to a percentage

Write \tfrac{3}{8} as a decimal and as a percentage.

To a decimal — divide top by bottom:

To a percentage — multiply the decimal by 100:

So \tfrac{3}{8} = 0.375 = 37.5\% — one amount, three outfits.

Worked example 2 — a percentage back to a tidy fraction

Write 35\% as a fraction in its lowest terms.

A percentage is already "out of a hundred", so start there:

Now cancel. Both 35 and 100 divide by 5:

There's no whole number that divides both 7 and 20, so \tfrac{7}{20} is as tidy as it gets.

Worked example 3 — put them in order

Put these three in order, smallest first: \tfrac{3}{5}, 0.7, and 55\%.

They're in three different costumes, so comparing by eye is guesswork. The fix: turn them all into the same form — decimals are usually easiest:

Now they're easy to line up:

So in order: 55\%, then \tfrac{3}{5}, then 0.7. Convert first, compare second.

See it explained

Sal Khan walks through turning a percentage into a decimal and a fraction, step by step.