Petra is organising a team building activity day with work colleagues . She is considering two options, which both offer packages for groups of up...

(a) 

Number of people attending	12	15	25
Option A	$17\times 12+150=354$	$17\times 15+150=405$	575
Option B	$22\times 12+60=324$	$22\times 15+60=390$	610

(b) 
(i) We multiply the number of people $x$ by $17$ (the cost per person) and add the booking fee $150$.

(ii) We multiply the number of people $x$ by $22$ (the cost per person) and add the booking fee $60$ to get $$ y=22x+60$$

(c)
(i) The graph is attached. 

(ii) (1) The gradients of the lines represent the cost per person. 
      (2) The $y$-intercepts represent the booking fees. 

(d) From the graph it seems that the intersection point is at $x=18$ and $y=455$. It means that for approximately 18 people the two options will cost the same (around 455 pounds).

(e) We have $$22x+60=y=17x+150 \implies 22x-17x=150-60 \implies 5x=90 $$ $$\implies x=18 \implies y=22x+60=22\times 18 +60 =456 $$ 

(f) For number of people less than 18 Option B gives better deal, and for number of people greater than 18 Option A gives better deal. 

(g) Option A cost is $17\times 21 +150 = 507$ pounds. For option B the discounted cost per person is $22 \times 95 \% =22 \times 0.95 =20.9$. So the total cost is $$20.9 \times 21 +60 =498.9$$ pounds, which is less than option A. So the discount should affect the choice.