A nursery has two hundred 7-gallon palm trees to sell, consisting of coconut palm trees, saw palmetto palm trees and christmas palm trees. Coconut palm trees sell for $75 each, saw palmetto palm trees sell for $45 each, and christmas palm trees sell for $30 each. The nursery has three times as many saw palmetto palm trees as christmas palm trees. A developer purchases all of the nursery's palm trees for a total price of 12,300. How many of each type of palm tree does the developer purchase?

Total number of trees available to be sold = 200

Selling price of coconut palm trees sell (Pco) = $75 each

Selling price of saw palmetto palm trees sell (Pp) = $45 each

Selling price of christmas palm trees (Pch) = $30 each

Number of saw palmetto palm trees (Np) = 3 * Number of christmas palm trees (Nch)

Let number of coconut palm trees be Nco

Total price paid by developer for all the palm trees = 12,300

So Total number of trees = Nco + Np + Nch

⇒ 200 = Nco + 3*Nch + Nch

⇒ 200 = Nco + 4Nch ... (i)

Total selling price of trees = Sco * Nco + Sp * Np + Sch * Nch

⇒ 12,300 = 75*Nco + 45*Np + 30*Nch

Dividing above equation by 5

⇒ 2,460 = 15Nco + 9Np + 6Nch

Substituting Np = 3Nch in the above equation

⇒ 2,460 = 15Nco + 9 (3Nch) + 6Nch

⇒ 2,460 = 15Nco + 33Nch ... (ii)

Mutiply (i) with 15 and subtract from (ii)

⇒ 2,460 - 3,000 = 15Nco + 33Nch - 15Nco - 60Nch

⇒ - 540 = 33Nch - 60Nch

⇒ - 540 = 27Nch

⇒ Nch=20

If Nch = 20, Np = 3*20 = 60

Since Nco + Np + Nch = 200

⇒ Nco = 200 - (80)

⇒ Nco = 120

Hence, the developer purchased 120 coconut palm trees, 20 christmas palm trees and 60 saw palmetto palm trees.