(a)
In the diagram above, \(\theta = y\) (vertically opposite angles)
\(\alpha = 180° - 110° = 70°\) (angles on a straight line)
Thus, \(\beta = \theta + \alpha\) (sum of interior opp. angles)
\(\beta = y + 70° ..... (1)\)
\(x + r + \beta = 180° ..... (2)\)
But 2x = r = y
Hence, \(x + 2x + (2x + 70) = 5x + 70 = 180\)
\(5x = 180 - 70 = 110\)
\(x = 22°\)
(b) Let Nx represent the donation collected by the girls. Then, the donation collected by the boys = N(x + 600).
Also, let y represent the average collection, \(x_{g}\), of the girls. Then the average collection, \(x_{b}\), of the boys = N(y + 100).
Thus, \(x_{g} = \frac{x}{12} = y\)
\(x = 12y ... (1)\)
\(x_{b} = \frac{x + 600}{10} = y + 100\)
\(x + 600 = 10(y + 100) .... (2)\)
Substitute 12y for x in (2) :
\(12y + 600 = 10y + 1000\)
\(12y - 10y = 1000 - 600 \)
\(2y = 400 \implies y = 200\)
From (1), \(x = 12y\)
\(\therefore x = 12 \times 200 = N2,400\).
Thus, the girls collected N2,400
The boys collected N2,400 + N600 = N3000
Total = N(2,400 + 3,000) = N5,400