Please help me solve math problem step by step

Question

The U-Drive Rent-A-Truck company plans to spend $14 million on 260 new vehicles. Each commercial van will cost $45,000, each small truck $70,000, and each large truck $60,000. Past experience shows that they need twice as many sans as small trucks. How many of each type of vehicle can they buy?

Answer 1

The first step is to define the variables that you need to solve the problem.  Usually it's good if these variables are closely related to the question you're trying to answer.  In this case the question is, how many of each type of vehicle, so let 

 

V = number of vans

S = number of small trucks 

L = number of large trucks  

 

As you can see, I prefer to use variable names that are a little more descriptive than X, Y and Z, so that later when I get an answer I don't have to wonder, now what did Y stand for again?  But you can call the variables anything you like, so if you're more comfortable using X,Y,Z or A,B,C that's fine too.  

 

The next step is to translate from the words in the problem into mathematical statements that you can use to solve the problem.  I like to start with the simplest parts 

 

"twice as many vans as small trucks" - this means that V = 2*S 

 

"260 new vehicles" - this means that V + S + L = 260 

 

The last part is the cost equation which is a little more complicated - 

45000*V + 70000*S + 60000*L = 14,000,000 

In fact this is so complicated it's probably a good idea to divide both sides of this equation by 1000 to simplify it a little 

45*V + 70*S + 60*L = 14000

 

Now you have 3 equations in 3 unknowns, so the next step is to go ahead and solve the system of equations.  Again, start with the simplest one V = 2*S.  This means that wherever you see a V in the other 2 equations you can replace it with 2*S, which will leave you with 2 equations in 2 unknowns 

 

(2*S) + S + L = 260  ==>  3*S + L = 260

45*(2*S) + 70*S + 60*L = 14000  ==>  160*S + 60*L = 14000  

 

Now you can solve the top equation for L to get  L = 260 - 3*S 

and substitute this value for L into the bottom equation 

 

160*S + 60*(260 - 3*S) = 14000

160*S + 15600 - 180*S = 14000

-20*S = -1600

S = 80  

 

Once you have 1 of the answers then plug in back in to previous equations to find the others 

 

L = 260 - 3*S = 260 - 3*(80) = 260 - 240 = 20 

 

V = 2*S = 2*(80) = 1560

 

So they can buy 160 commercial vans, 80 small trucks, and 20 large trucks 

 

The last step, which is very important but often overlooked, is to check your answer by A) making sure you are answering the question that is being asked, and B) verifying that the solution you found satisfies the conditions in the problem  

 

Check:  V + S + L = 160 + 80 + 20 = 260 new vehicles

            V/S = 160/80 = 2 = twice as many vans as small trucks

            Total cost = 45000*(160) + 70000*(80) + 60000*20 

                           = 7,200,000 + 5,600,000 + 1,200,000 = $14,000,000 = $14 million