Question

In a bicycle shop, the number of bicycles purchased and choice of the colors was as follows. Complete the table to show the information by the pie diagram. In all 36 bicycles were purchased.

COLOR	NUMBER OF BICYCLES	CENTRAL ANGLE OF THE SECTOR
Black	9	\[{90^ \circ }\]
Grey	?	?
Blue	?	\[{70^ \circ }\]
Red	8	?

Answer 1

Hint: Here we assume that in a pi diagram \[{360^ \circ }\]is the total angle representing the total number of cycles. We use a unitary method to find the value of the central angle of the sector of the number of cycles from each color when we are given the total angle and number of cycles of each color. We first find the number of cycles from the angle given in case of blue cycles and then using the total number of cycles purchased we find the number of cycles in grey color, then find angle of sector for grey cycles.
* Unitary method helps us to find the value of multiple units if we are given the value of a single unit by multiplying the value of a single unit to the number of multiple units.
* A pie-diagram is a kind of chart in the shape of a circle, where the sectors depict the amount or quantity of each kind of object. Each sector represents a proportion of the whole.

Complete step-by-step answer:
In a pi diagram the total angle is \[{360^ \circ }\].
Let us assume that \[{360^ \circ }\]represents the total number of cycles purchased i.e. 36.
We find the central angle for 1 cycle using unitary method:
Since, 36 cycles have central angle \[ = {360^ \circ }\]
\[ \Rightarrow \]1 cycle will have central angle \[ = \dfrac{{{{360}^ \circ }}}{{36}}\]
So for any number of cycles, say x the central angle will be \[ = \dfrac{{{{360}^ \circ }}}{{36}} \times x\]
On solving the fraction we get;
\[ \Rightarrow \]Central angle for x cycles \[ = {(10 \times x)^ \circ }\] … (1)
Now we are given in the table, number of red cycles is 8
We use the formula from equation (1) to find the central angle in pi diagram
\[ \Rightarrow \]Central angle for 8 cycles \[ = {(10 \times 8)^ \circ } = {80^ \circ }\]
Now we have the central angle for cycles in blue color, we find the number of blue cycles using equation (1)
Central angle for x cycles \[ = {(10 \times x)^ \circ }\]
\[ \Rightarrow \]the number of cycles \[ = \]central angle of the sector divided by 10
\[ \Rightarrow \]number of blue cycles \[ = \dfrac{{70}}{7}\]
Solving the fraction we see the number of blue color cycles \[ = 7\].
Now we know total number of cycles purchased \[ = 36\]
Sum of cycles of all colors will be equal to 36.
\[ \Rightarrow \]Number of black cycles \[ + \]number of grey cycles \[ + \]number of blue cycles \[ + \]number of red cycles \[ = 36\]
Substituting the values from the table
\[ \Rightarrow 9 + \]number of grey cycles \[ + 7 + 8 = 36\]
Add the values on LHS
\[ \Rightarrow 24 + \]number of grey cycles \[ = 36\]
Shift all constants to RHS
\[ \Rightarrow \]number of grey cycles \[ = 36 - 24\]
\[ \Rightarrow \]number of grey cycles \[ = 12\]
Now using equation (1);
\[ \Rightarrow \]Central angle for 12 cycles \[ = {(10 \times 12)^ \circ }\]
\[ \Rightarrow \]Central angle for 12 cycles \[ = {120^ \circ }\]
So, now the table becomes

COLOR	NUMBER OF BICYCLES	CENTRAL ANGLE OF THE SECTOR
Black	9	\[{90^ \circ }\]
Grey	12	\[{120^ \circ }\]
Blue	7	\[{70^ \circ }\]
Red	8	\[{80^ \circ }\]

Now we draw the pi-diagram.
Using the angles from the table we draw each sector according to the central angle.

Note: Alternate method:
Students can also use hit and trial methods to find angles and number of bicycles.
We have using the unitary method:
Since, 36 cycles have central angle \[ = {360^ \circ }\]
\[ \Rightarrow \]1 cycle will have central angle \[ = \dfrac{{{{360}^ \circ }}}{{36}} = {10^ \circ }\]
So, 8 cycles will have angle \[{80^ \circ }\].
When we are given an angle \[{70^ \circ }\], the number of cycles for this angle will be 7 (as \[{70^ \circ }\] divided by 10 is 7).
Using the sum of cycles as 36, let the number of grey cycles be x.
\[ \Rightarrow 9 + 7 + 8 + x = 36\]
Add values in LHS
\[ \Rightarrow x + 24 = 36\]
Shift constants to one side
\[
   \Rightarrow x = 36 - 24 \\
   \Rightarrow x = 12 \\
 \]
So, the angle for 12 cycles will be \[{120^ \circ }\].