1. **Stating the problem:** We have a mapping of percentages to a new scale: - 85.5% maps to 0% - 90% maps to 100% - 94.5% maps to 200% We want to find the value that corresponds to 93.1% on this new scale. 2. **Understanding the problem:** This is a piecewise linear mapping with two segments: - From 85.5% to 90% corresponds to 0 to 100 - From 90% to 94.5% corresponds to 100 to 200 We need to find where 93.1% fits and then interpolate accordingly. 3. **Check where 93.1% lies:** Since 90% < 93.1% < 94.5%, it lies in the second segment. 4. **Formula for linear interpolation:** For a segment from $(x_1,y_1)$ to $(x_2,y_2)$, the value $y$ at $x$ is: $$ y = y_1 + \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) $$ 5. **Apply to second segment:** - $x_1 = 90$, $y_1 = 100$ - $x_2 = 94.5$, $y_2 = 200$ - $x = 93.1$ Calculate slope: $$ \frac{200 - 100}{94.5 - 90} = \frac{100}{4.5} = \frac{100}{4.5} $$ Calculate $y$: $$ y = 100 + \frac{100}{4.5} (93.1 - 90) = 100 + \frac{100}{4.5} \times 3.1 $$ 6. **Simplify:** $$ y = 100 + \frac{100 \times 3.1}{4.5} = 100 + \frac{310}{4.5} $$ 7. **Calculate final value:** $$ y = 100 + 68.888\ldots = 168.888\ldots \approx 168.89 $$ **Final answer:** The value corresponding to 93.1% is approximately **168.89** on the new scale.