A maximum of how many pieces of 12.6 cm can be cut from a 857 cm long rod?

## Concept & Formula To find the maximum number of whole pieces that can be obtained, perform integer division (floor division) of the total length by the piece length. $$ \text{Number of pieces} = \lfloor \frac{\text{Total Length}}{\text{Piece Length}} \rfloor $$ ## Step-by-Step Solution * **Given:** Total length of the rod = 857 cm Length of each piece = 12.6 cm * **Calculation:** Set up the division: $\text{Number of pieces} = \frac{857}{12.6}$ Multiply both the numerator and denominator by 10 to clear the decimal: $\text{Number of pieces} = \frac{8570}{126}$ Simplify the fraction by dividing by 2: $\frac{4285}{63}$ Perform long division: $4285 \div 63 = 68.015...$ Since we can only cut whole pieces, truncate the decimal value to the nearest lower integer. $\text{Number of pieces} = 68$ ## Exam Strategy & Shortcut Use fast multiplication and estimation. $12.6 \times 60 = 756$ Remaining length = $857 - 756 = 101$ Now find how many times 12.6 goes into 101: $12.6 \times 8 = 100.8$ Total whole pieces = $60 + 8 = 68$. This eliminates long division entirely. ## Common Pitfall Students sometimes round up to the nearest integer if the decimal fraction is high. In cutting/partitioning problems, you can never round up because you cannot create a piece out of missing material. ## Final Answer **Therefore, the correct answer is 68.**