(a) In the diagram, < KLM = x, < LMK = y, < KJH = r and < KGF = 110°. If 2x = r = y, find the value of x. (b) Ten boys and twelve girls collected donations ...

Answer Details

(a) Finding x.

From the diagram, J-H-G-F is a straight line along the bottom. The line J-K-L is straight (K lies on JL), and the line K-M-G is straight (M lies on KG). The marked angles are \(\angle KLM = x\) at L, \(\angle LMK = y\) at M, \(\angle KJH = r\) at J, and \(\angle KGF = 110^\circ\) at G, with the condition \(2x = r = y\).

Step 1: angle the line KG makes with the base at G. Since J-G-F is straight, \(\angle KGF\) and \(\angle KGJ\) are on a straight line:

\[\angle KGJ = 180^\circ - 110^\circ = 70^\circ\]

Step 2: angle at K in triangle KLM. The angles of \(\triangle KLM\) sum to \(180^\circ\):

\[\angle LKM = 180^\circ - x - y\]

Because J-K-L is a straight line, the angle on the other side of K (angle JKG, which is the angle of triangle JKG at K) is supplementary:

\[\angle JKG = 180^\circ - \angle LKM = 180^\circ - (180^\circ - x - y) = x + y\]

Step 3: use triangle JKG. Its vertices are J and G on the base and K above. Its angles are \(\angle KJG = r\), \(\angle JKG = x+y\) and \(\angle KGJ = 70^\circ\), summing to \(180^\circ\):

\[r + (x + y) + 70^\circ = 180^\circ \quad\Rightarrow\quad r + x + y = 110^\circ\]

Step 4: apply \(r = 2x\) and \(y = 2x\):

\[2x + x + 2x = 110^\circ\]\[5x = 110^\circ \quad\Rightarrow\quad x = 22^\circ\]

x = 22\(^\circ\). (Then \(r = 44^\circ\), \(y = 44^\circ\).)

(b) Total amount collected by the two groups.

Let the total collected by the girls be \(G\). Then the boys collected \(G + 600\) (N600 more).

• Average of the 12 girls: \(\dfrac{G}{12}\)
• Average of the 10 boys: \(\dfrac{G + 600}{10}\)

The boys' average is N100 more than the girls' average:

\[\frac{G + 600}{10} = \frac{G}{12} + 100\]

Multiply through by 60 (the LCM of 10 and 12):

\[6(G + 600) = 5G + 6000\]\[6G + 3600 = 5G + 6000\]\[G = 2400\]

So the girls collected \(\text{N}2400\) and the boys collected \(2400 + 600 = \text{N}3000\).

\[\text{Total} = 2400 + 3000 = \text{N}5400.00\]

Total collected = N5,400.00. (Check: girls' average \(=200\), boys' average \(=300\), difference \(=100\).)