A fraction is one number

Open a JavaScript console and type 3/4. It answers 0.75. One number out, not a pair, not a little stacked object.

That’s the whole truth about fractions, and it’s right there in the syntax: $\frac{a}{b}$ is just $a \div b$ with the division not done yet. A fraction is a division you’re choosing to leave un-evaluated, because the un-evaluated form is exact and often more useful.

Because $\frac{3}{4}$ is a number, it has a home on the number line: a single point, three-quarters of the way from $0$ to $1$. Not two points. Not a picture of a pie. One position.

Each bar shows one fixed fraction. The steppers re-slice the bar into more or fewer pieces, but the bar’s length never moves. Slicing is cosmetic. The number is the number.

Multiply the top and the bottom by the same number and you have not changed the value; you’ve only cut the same length into finer pieces:

$\frac{3}{4} = \frac{6}{8} = \frac{9}{12}$: same point on the line, three slicings. Going the other way, dividing top and bottom by a common factor, is reducing.

A fraction is in lowest terms when its top and bottom share no common factor bigger than $1$: you’ve reduced as far as you can go.

Take $\frac{18}{24}$. Both numbers divide by $6$, which gives $\frac{3}{4}$. Nothing divides $3$ and $4$ together, so $\frac{3}{4}$ is the floor.

Here is the one rule people get wrong, and there’s a real reason behind it, not a ritual.

The denominator is the unit. $\frac{3}{4}$ is “3 of a thing called a quarter.” $\frac{5}{6}$ is “5 of a thing called a sixth.” You can’t add 3 quarters to 5 sixths any more than you can add 3 metres to 5 seconds (different units). Re-slice both into a shared unit first, then the counts add.

Step the two bars until their denominators match. Only then does the result bar appear, and it lands exactly on a tick.

With that understood, here is the whole toolkit:

Add: find a common denominator, then add the counts. Multiply: straight across, no common denominator needed. Divide: multiply by the reciprocal (the inverse trick from the last lesson, back on the job).

A fraction, a decimal, and a percent are not three kinds of number. They’re three spellings of one number, one position on the line. $\frac{3}{8} = 0.375 = 37.5\%$. A percent is just a fraction whose denominator is locked to $100$: $37.5\% = \frac{37.5}{100}$.

Drag the dot. All three spellings update together, because there’s only one thing being spelled. Engineers do this every day: the same byte is 255, 0xFF, or 0b11111111. Pick the spelling that makes the next step easy.

A ratio compares two quantities by division; a proportion sets two ratios equal. The one move that matters: scaling both quantities by the same factor leaves the ratio unchanged.

A recipe for 6 servings needs 250 g of flour. For 15 servings you don’t add a fixed amount; you scale. The factor is $\frac{15}{6} = 2.5$, and the flour scales by that identical factor.