Question

Tree pruning company A charges a one-time $100 equipment fee and charges $50 for each tree that it prunes. Tree pruning company B charges a one-time $80 equipment fee and charges $60 for each tree that it prunes.

Let the variable t represent the number of trees pruned and let the variable c represent the cost.

For how many pruned trees will the cost be the same for both companies?

Which system of equations can be used to solve this problem?




{c=100−50tc=80−60t

{c=150tc=140t

{c=100+50tc=80+60t

{c=50+100tc=60+80t

Answer 1

• 2 trees ($200 from either service)
• c=100+50t
• c=80+60t

Explanation:

It can work well to solve problems like this by considering how many times the difference in per-tree costs it takes to make up the difference in one-time costs.

Here the difference of one-time costs is $20, and the difference in per-tree costs is $10, so it takes the pruning of 2 trees for the per-tree cost difference to equal the one-time cost difference (2×10 = 20).

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The cost equations are ...

cost = (one-time cost) + (per tree cost) × (number of trees)

c = 100 + 50t . . . . . company A

c = 80 + 60t . . . . . . company B

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If you use these equations to solve the problem, you want to find t such that the two costs are equal:

100 +50t = 80 +60t

(100 -80) = t(60 -50) . . . . . . . subtract 80+50t, factor out t

(100 -80)/(60 -50) = t . . . . . . . looks a lot like the verbal description above

The difference in fixed price divided by the difference in per-tree cost) is the number of trees required to make costs equal.