How do you change 0.16666 in a fraction

The correct answer is: If 0.16666… (repeating) then $1/6$. If it is exactly 0.16666 (five 6’s) then $\tfrac{8333}{50000}$.

Explanation

Case A — Repeating decimal $0.16666\ldots$

1. Let $x=0.16666\ldots$.
2. Multiply by 10: $10x=1.6666\ldots$.
3. Subtract: $10x-x=9x=1.6666\ldots-0.16666\ldots=1.5$.
4. So $9x=1.5=\tfrac{3}{2}$, hence $x=\tfrac{3}{2}\div9=\tfrac{3}{18}=\tfrac{1}{6}$.

Case B — Finite decimal 0.16666 (exactly five 6’s)

1. Write as a fraction: $0.16666=\dfrac{16666}{100000}$.
2. Reduce by 2: $\dfrac{16666}{100000}=\dfrac{8333}{50000}$.
3. Check gcd: $50000\mod8333=2$, so gcd$(8333,50000)=1$, so $\dfrac{8333}{50000}$ is fully simplified.

If you meant a repeating decimal, the fraction is $1/6$; if you meant the finite decimal with five 6’s, it is $8333/50000$.