Evaluate 0.05 x 0.8 divided by 1.666... (1.6 repeating).

Step-by-step solution

1. 1

   Convert the repeating decimal to a fraction. Let $x=1.\overline{6}$. Then $10x=16.\overline{6}$, so subtracting gives $9x=15$ and

   $x=\frac{15}{9}=\frac{5}{3}$

   Working with $\tfrac53$ instead of a truncated $1.667$ is what keeps the final answer exact rather than approximate.

2. 2

   Convert the two terminating decimals as well.

   $0.05=\frac{1}{20},\qquad 0.8=\frac{4}{5}$

3. 3

   Multiply the first two factors.

   $\frac{1}{20}\times\frac{4}{5}=\frac{4}{100}=\frac{1}{25}=0.04$

4. 4

   Divide by turning the division into multiplication by the reciprocal.

   $\frac{1}{25}\div\frac{5}{3}=\frac{1}{25}\times\frac{3}{5}=\frac{3}{125}$

   Dividing by a number larger than $1$ must shrink the result, and $\tfrac{3}{125}<\tfrac{1}{25}$ $\checkmark$.

5. 5

   Convert back to a decimal and check.

   $\frac{3}{125}=\frac{24}{1000}=0.024$

   Multiplying back, $0.024\times\tfrac53=0.04\;\checkmark$. Had you rounded $1.\overline{6}$ to $1.67$ first, the result would have been $0.023952$ — wrong in the fourth decimal.