1. **Problem statement:** We need to find a linear function that converts the highest mark 285 to 200 and the lowest mark 75 to 60. 2. **Formula and explanation:** A linear function can be written as $y = mx + c$, where $m$ is the slope and $c$ is the intercept. 3. **Find the slope $m$:** Using the two points $(285, 200)$ and $(75, 60)$, $$m = \frac{200 - 60}{285 - 75} = \frac{140}{210} = \frac{\cancel{140}}{\cancel{210}} = \frac{2}{3}$$ 4. **Find the intercept $c$:** Using $y = mx + c$ and the point $(285, 200)$, $$200 = \frac{2}{3} \times 285 + c$$ $$200 = 190 + c$$ $$c = 200 - 190 = 10$$ 5. **Linear function:** $$y = \frac{2}{3}x + 10$$ 6. **Use the function to find new marks for original marks 95, 175, 215, and 255:** - For $x=95$: $$y = \frac{2}{3} \times 95 + 10 = \frac{190}{3} + 10 = 63.33 + 10 = 73.33$$ - For $x=175$: $$y = \frac{2}{3} \times 175 + 10 = \frac{350}{3} + 10 = 116.67 + 10 = 126.67$$ - For $x=215$: $$y = \frac{2}{3} \times 215 + 10 = \frac{430}{3} + 10 = 143.33 + 10 = 153.33$$ - For $x=255$: $$y = \frac{2}{3} \times 255 + 10 = \frac{510}{3} + 10 = 170 + 10 = 180$$ **Final answers:** The linear function is $y = \frac{2}{3}x + 10$. The new marks corresponding to original marks 95, 175, 215, and 255 are approximately 73.33, 126.67, 153.33, and 180 respectively.