Solve 2560 = x times 1.5.

Step-by-step solution

1. 1

   Identify the single operation to undo. The unknown is multiplied by $1.5$ and nothing else, so one division isolates it: $x = \dfrac{2560}{1.5}$.

2. 2

   Convert the decimal divisor to a fraction. $1.5 = \tfrac32$, and dividing by a fraction means multiplying by its reciprocal: $x = 2560 \div \frac32 = 2560 \cdot \frac23.$ This replaces a decimal division with an exact integer computation.

3. 3

   Multiply out. $x = \frac{2560 \cdot 2}{3} = \frac{5120}{3}.$ Since $5{,}120 = 3\cdot1706 + 2$, the fraction does not reduce and is already in lowest terms.

4. 4

   Express as a decimal. $\frac{5120}{3} = 1706.\overline{6} \approx 1706.67.$ The decimal repeats forever, so $\tfrac{5120}{3}$ is the only exact way to write the answer.

5. 5

   Verify by multiplying back. $\tfrac{5120}{3} \cdot \tfrac32 = \tfrac{5120}{2} = 2560$ exactly - the fractional form makes the check trivial, whereas $1706.67 \times 1.5 = 2560.005$ shows the rounding error.

6. 6

   Estimate for reassurance. Multiplying by $1.5$ adds half, so dividing by $1.5$ removes a third: $2560 - \tfrac{2560}{3} \approx 2560 - 853 = 1707$, matching the answer.