Question

In planning for a school dance, you find that one band will play for $250 plus 50% of the total ticket sales. Another band will play for a flat fee of $550. IN order for the first band to produce more profit for the school than the other band, what is the highest price you can charge per ticket, assuming 300 people attend?

Answer 1

if the ticket charge is 't'
since 300 people are attending
profit with hiring the 1st band = income - expenditure = 300t - (250 + 150t)
= 150t - 250
profit with hiring the 2nd band = income - expenditure
= 300t - 550
in order for the 1st band to produce more profit for the school than the 2nd band
=> profit with hiring the 1st band > profit with hiring the 2nd band
=> (150t - 250) > (300t - 550)
=> 550 - 250 > 300t - 150t
=> 300 > 150t
=> 150t < 300
=> t < 2
therefore the price of the ticket cannot be more than $2 for the first band to produce more profit for the school than the other band

Answer 2

Profit = Revenue - Costs...

So Revenue = 300 x Ticket
Costs(1) = 250 + 150 x Ticket
Costs(2) = 550

As Revenue is always the same, you are looking for when Costs(1) is less than Costs(2)
250+150T < 550... T<2

It is however, from a commercial point of view, a strangely worded question. :-)