1. **State the problem:** A catering company provided quiches, each cut into a certain number of pieces. After the luncheon, some pieces were left over. We need to find how many quiches were eaten, expressed as a mixed number in simplest form. 2. **Identify the variables:** - Let $q$ be the number of quiches provided. - Each quiche is cut into $p$ pieces. - Let $l$ be the number of leftover pieces. - Let $e$ be the number of quiches eaten. 3. **Formulate the equation:** The total pieces provided are $q \times p$. The pieces eaten are total pieces minus leftover pieces: $q \times p - l$. Since each quiche has $p$ pieces, the number of quiches eaten is: $$ e = \frac{q \times p - l}{p} $$ 4. **Simplify the expression:** Rewrite the fraction: $$ e = q - \frac{l}{p} $$ 5. **Convert to a mixed number:** Express $e$ as a mixed number by separating the whole number and fractional parts. 6. **Example:** Suppose each quiche is cut into 8 pieces ($p=8$), there were 5 quiches ($q=5$), and 6 pieces left over ($l=6$). Calculate: $$ e = 5 - \frac{6}{8} = 5 - \frac{3}{4} = 4 \frac{1}{4} $$ So, 4 and one-quarter quiches were eaten. **Final answer:** The number of quiches eaten is $q - \frac{l}{p}$ expressed as a mixed number in simplest form.