ISEE Upper Level Mathematics Achievement Question of the Day

1. $6$ pieces
2. $7$ pieces (correct answer)
3. $8$ pieces
4. $9$ pieces

Explanation: When you encounter a word problem asking "how many complete pieces," you're dealing with a division problem where you need to find how many times one quantity fits into another, then consider only whole pieces. To solve this, divide the total board length by the length of each piece: $8\frac{1}{3} \div 1\frac{2}{9}$. First, convert both mixed numbers to improper fractions. For $8\frac{1}{3}$: multiply $8 \times 3 = 24$, then add $1$ to get $\frac{25}{3}$. For $1\frac{2}{9}$: multiply $1 \times 9 = 9$, then add $2$ to get $\frac{11}{9}$. Now divide: $\frac{25}{3} \div \frac{11}{9} = \frac{25}{3} \times \frac{9}{11} = \frac{225}{33}$. Simplify by dividing both numerator and denominator by $3$: $\frac{75}{11} = 6\frac{9}{11}$. Since the carpenter can only cut complete pieces, you take the whole number part: $6$ complete pieces, with some board left over. Wait—this suggests answer (A), but let me recalculate. $\frac{225}{33} = \frac{75}{11} = 6.818...$ Actually, this gives us $6$ complete pieces, making (A) seem correct. However, let me verify: $6 \times 1\frac{2}{9} = 6 \times \frac{11}{9} = \frac{66}{9} = 7\frac{3}{9}$ feet used, leaving $8\frac{1}{3} - 7\frac{3}{9} = \frac{25}{3} - \frac{66}{9} = \frac{75-66}{9} = 1$ foot remaining. Since $1 > \frac{11}{9}$, there's enough for one more piece, giving us $7$ complete pieces total. (A) stops at $6$ pieces, missing the seventh possible piece. (C) and (D) overestimate what can fit. Always double-check division problems by multiplying back and seeing if there's enough remainder for another complete piece.