3. Scaling up, simplifying and irreducible fractions — Fractions

Scaling up and simplifying

→ Scaling up: multiplying the numerator and the denominator by the same number.

→ Simplifying: dividing the numerator and the denominator by a common divisor.

Either way the fraction changes its look but not its value: it still stands for the same number.

\dfrac{5}{3} \xrightarrow{\ \cdot\,3\ } \dfrac{5 \cdot 3}{3 \cdot 3} = \dfrac{15}{9}

The $3$ multiplies top and bottom. Multiplying only the numerator would give a different number.

We keep dividing by common divisors until none is left:

\dfrac{150}{40} \xrightarrow{\ :\,2\ } \dfrac{75}{20} \xrightarrow{\ :\,5\ } \dfrac{15}{4}

It can be done in several small steps or in a single one: the end result is the same.

A fraction is irreducible when it cannot be simplified any further.

That happens exactly when $\gcd(N, D) = 1$.

Shortcut. Dividing straight by the greatest common divisor gets us there in one step:

\gcd(150, 40) = 10 \qquad \dfrac{150}{40} \xrightarrow{\ :\,10\ } \dfrac{15}{4}

Check: $\gcd(15, 4) = 1$, so $\dfrac{15}{4}$ is already irreducible.