Question

The table shows the number of three different brands of laptops sold by a certain store. The data are represented by the pie chart which shows a semicircle divided into six equal sectors. The angle representing the number of the Bell laptop is;

Brand	Number of laptops
Ace	60
Bell	40
Compact	20

(a)\[\angle POR\]
(b)\[\angle ROV\]
(c)\[\angle QOT\]
(d)\[\angle POU\]

Answer 1

Hint: We must use a unitary method approach to solve this question. At first, we have to consider the whole circle to the total number of laptops. Then we have to estimate the angle of the sector defined by one laptop. We will find out the angle of the sector in the pie chart for the 40 laptops. In the next step we will find out the angles \[\angle PQR\], \[\angle ROV\], \[\angle QOT\] and \[\angle POU\]. Finally, we compare the obtained angle with the angle of the sector in the pie chart for the 40 laptops and find the correct one.

Complete step-by-step solution
The unitary method aims at determining values in relation to a single unit. In other words, we must find out the value of one unit and further use it to find the value of multiple units.
Now the total number of laptops is given by \[60+20+40=120\]
We know that the angle made by a complete rotation along the circumference is \[{{360}^{\circ }}\]. Consider the whole circle to the total number of laptops.
Then 120 laptops subtend \[{{360}^{\circ }}\]of the pie chart.
The angle of the sector subtended by one laptop is \[\dfrac{{{360}^{\circ }}}{120}={{3}^{\circ }}\]
Hence for the 40 number of Bell laptop, the angle of the sector is given by \[40\times {{3}^{\circ }}={{120}^{\circ }}\]
In the question, it is given that the semicircle is divided into 6 equal parts. We know that the angle subtended by a semicircle is\[{{180}^{\circ }}\]. Then the angle of each part is \[\dfrac{{{180}^{\circ }}}{6}={{30}^{\circ }}\] that means \[\angle POQ=\angle QOR=\angle ROS=\angle SOT=\angle TOU=\angle UOV={{30}^{\circ }}\].
Then,
\[\begin{align}
  & \angle POR=\angle POQ+\angle QOR={{30}^{\circ }}+{{30}^{\circ }}={{60}^{\circ }} \\
 & \angle ROV=\angle ROS+\angle SOT+\angle TOU+\angle UOV \\
 & ={{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }} \\
 & ={{120}^{\circ }} \\
 & \angle QOT=\angle QOR+\angle ROS+\angle SOT \\
 & ={{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }} \\
 & ={{90}^{\circ }} \\
 & \angle POU=\angle POQ+\angle QOR+\angle ROS+\angle SOT+\angle TOU \\
 & ={{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }}+{{30}^{\circ }} \\
 & ={{150}^{\circ }}
\end{align}\]
Therefore option (B) is correct.

Note: The angle subtended by a complete rotation along the circumference of a circle is \[{{360}^{\circ }}\]and along the circumference of a semicircle, the angle subtended is\[{{180}^{\circ }}\]. The semicircle is divided into 6 equal parts thus they have equal angles subtended at the center.