1. **Problem statement:** Given $a=3.0$, $b=4.24$, and $c=3.654$, find the percentage error of the expression $$\frac{a+b}{c+a}$$. 2. **Formula and explanation:** Percentage error is calculated as $$\text{Percentage Error} = \left| \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right| \times 100\%$$. Here, we interpret $a$, $b$, and $c$ as measured values with possible errors, and we want to find the percentage error of the ratio $$\frac{a+b}{c+a}$$. 3. **Calculate numerator and denominator:** - Numerator: $a+b = 3.0 + 4.24 = 7.24$ - Denominator: $c+a = 3.654 + 3.0 = 6.654$ 4. **Calculate the ratio:** $$\frac{a+b}{c+a} = \frac{7.24}{6.654} \approx 1.0883$$ 5. **Calculate the exact value if no errors were present:** Assuming $a$, $b$, and $c$ are exact, the ratio is as above. 6. **Calculate the percentage error:** Since the problem does not specify a true value or measured value separately, we interpret the percentage error as the relative difference between numerator and denominator sums. Calculate the percentage error of numerator and denominator separately: - Percentage error in numerator $a+b$ assuming $a$ and $b$ have errors: - Error in $a$ is zero (given as exact 3.0), error in $b$ is unknown, so we cannot calculate exact percentage error. Since the problem is ambiguous about which values have errors, the best interpretation is to calculate the percentage error of the ratio using propagation of errors formula: 7. **Propagation of errors formula for division:** $$\frac{\Delta R}{R} = \sqrt{\left(\frac{\Delta (a+b)}{a+b}\right)^2 + \left(\frac{\Delta (c+a)}{c+a}\right)^2}$$ where $R = \frac{a+b}{c+a}$. 8. **Calculate absolute errors:** - $\Delta a = 0$ (assuming exact) - $\Delta b = 0$ (not given, assume zero) - $\Delta c = 0$ (not given, assume zero) Therefore, $\Delta (a+b) = 0$, $\Delta (c+a) = 0$, so percentage error is zero. **Final answer:** Without additional error data, the percentage error of $$\frac{a+b}{c+a}$$ is 0%. If the problem intended to find the percentage error of the value $$\frac{a+b}{c+a}$$ given errors in $a$, $b$, and $c$, please provide the errors. --- **Summary:** - Calculated $$\frac{a+b}{c+a} \approx 1.0883$$ - Without error data, percentage error cannot be computed.