Question: A pitcher's earned-run average $E$ varies directly as the number $R$ of earned runs allowed and inversely as the number $I$ of innings pitched. A certain pitcher had an earned-run average of $2.92$, giving up $85$ earned runs in $262$ innings. How many earned runs would the pitcher have given up having pitched $272$ innings with the same average?

1. **State the problem:** We know that the earned-run average $E$ varies directly as the number of earned runs $R$ and inversely as the innings pitched $I$. Given $E=2.92$, $R=85$, and $I=262$, find $R$ when $I=272$ and $E$ remains the same. 2. **Write the formula:** Since $E$ varies directly as $R$ and inversely as $I$, we have $$E = k \frac{R}{I}$$ where $k$ is a constant of proportionality. 3. **Find the constant $k$ using the known values:** $$2.92 = k \frac{85}{262}$$ Multiply both sides by $\frac{262}{85}$: $$k = 2.92 \times \frac{262}{85}$$ Calculate: $$k = 2.92 \times 3.082352941 = 9.000$$ 4. **Use $k$ to find the new number of earned runs $R_2$ when $I=272$ and $E=2.92$:** $$2.92 = 9.000 \times \frac{R_2}{272}$$ Divide both sides by 9.000: $$\frac{2.92}{9.000} = \frac{R_2}{272}$$ Simplify fraction with cancellation: $$\frac{\cancel{2.92}}{\cancel{9.000}} = \frac{R_2}{272}$$ Calculate left side: $$0.3244 = \frac{R_2}{272}$$ Multiply both sides by 272: $$R_2 = 0.3244 \times 272 = 88.25$$ 5. **Round to the nearest whole number:** $$R_2 \approx 88$$ **Final answer:** The pitcher would have given up approximately **88** earned runs.