Simplify the fraction 3/9.

Step-by-step solution

1. 1

   Find the greatest common factor of numerator and denominator. The factors of $3$ are $1,3$ and the factors of $9$ are $1,3,9$. The largest shared factor is

   $\gcd(3,9)=3$

2. 2

   Divide both parts by that factor. Dividing top and bottom by the same nonzero number does not change the value of a fraction:

   $\frac{3\div 3}{9\div 3}=\frac{1}{3}$

3. 3

   Confirm the result is fully reduced. The numerator is now $1$, whose only factor is itself, so no further cancellation is possible. A fraction is in lowest terms exactly when $\gcd(\text{numerator},\text{denominator})=1$, and $\gcd(1,3)=1$ $\checkmark$.

4. 4

   Check by cross-multiplication and by decimal. $3\times 3=9\times 1$, so $\tfrac39=\tfrac13$ $\checkmark$. As decimals, $3\div 9=0.\overline{3}$ and $1\div 3=0.\overline{3}$ $\checkmark$. Structurally, $\tfrac39$ means three of nine equal parts, and grouping the nine parts into three groups of three shows this is one group out of three.