Question 15

Since we are talking here of areas of sectors taken from the same circle, we consider only the central angles in computing for the ratio of areas. The central angle of the sector taken out is $135^\circ$ that means that remaining angle is $360-135=225^\circ$ .

Therefore, the ratio is $\dfrac{225}{135}=\dfrac{5}{3}$ . So $A+B=5+3=$ $\boxed{8}$

• Let $\theta$ be the angle of the sector.

$\begin{aligned} \frac{A}{B} &= \frac{\left(\frac{360° - \theta}{360°}\right) \pi r^2}{\left(\frac{\theta}{360°}\right) \pi r^2} \\ &=\frac{360° - \theta}{\theta} \\ &=\frac{360° - 135°}{135°} \\ &=\frac{5}{3} \\ \end{aligned}$

Then, $A+B = 5+3 = \boxed{8}$ .