ski Arizona word problem

Question

Last winter we went skiing in Flagstaff, Arizona. In the parking lot I noticed license plates from Arizona as well as from various other states. There were also some license plates from other countries. Some of the vehicles were trucks, and the rest were not trucks. Some of the vehicles had four-wheel drive (FWD) and the rest did not. Here are some other observations: a. There were 56 total vehicles. b. Every truck had FWD c. Thirty-five of the vehicles had FWD and 21 did not. d. There were 20 trucks. e. There were 42 vehicles from the United States, 28 of which were from Arizona. f. There were four times as many Arizona trucks as trucks from other states in the United States. g. There were two more non-FWD vehicles from Arizona than there were FWD vehicles that were not trucks and not from the United States. h. The number of trucks was eight less than the number of vehicles from Arizona. i. There were seven more Arizona trucks than there were trucks from other countries. j. The Arizona vehicles that were not trucks were split evenly between FWD and non-FWD. How many of the U.S. vehicles that weren’t from Arizona were not trucks? How many foreign FWD vehicles were there?

 

I have been trying to solve this problem for hours now using a ven diagram. Someone please help and show their work!

Answer 1

Total Vehicles: 56  (clue a)

 

US Plates: 42 (clue e)                             International Plates: 56 - 42 = 14

Arizona Plates: 28  (clue e)                         (clue a less clue e)

Other US States: 42 - 28 = 14

  (Clue e and doing the math)

 

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                                                                                                            NTN4WD

4WD Trucks (clues b and d)    4WD Non-Truck (clues b, c, d)            Non-Truck (no 4WD) (clue c - 21 did not)

     20                                          56-20-21= 15                                         21

                                                  or  35 - 10 =15      

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Total # Trucks = 20 (clue d)

 

1)  # AZ Trucks/4 = 4 x # US Trucks (clue f)     

2)  # AZ NTN4WD=# INT NTN4WD + 2 (clue g)

3)  # Total Trucks = # AZ Vehicles - 8, so 28 - 8 = 20 (clues h, d, e)

4)  # AZ Trucks - 7 = # INT Trucks  (clues i, d)

 

Total # Trucks = #AZ Trucks + #US Trucks + #INT Trucks

                 20 = #AZ Trucks + #AZ Trucks/4 + #AZ Trucks - 7    (substitute from equations 1 and 4)

                 20 = x + x/4 + x -7                                                  (let x = # AZ trucks and solve)

              20 + 7 = 2x + x/4                                                       (add 7 to both sides)

                 27  = 8x/4 + x/4                                                      (Rewrite with common denominator and add)

                 27 = 9x/4  

             4(27) = 9x                                                                   (Multiply both sides by 4, and divide by 9)

           4(27/9) = x                                                                    (Simplify)

            4(3)  = x        so x = 12   or 12 AZ trucks.

 

#US Trucks = x/4 = 12/4 = 3.  Therefore there are 3 US Trucks

#INT Trucks = x - 7 = 12 - 7 = 5. Therefore there are 5 INT Trucks.      (check: 12 + 3 + 5 = 20. 20 truck total. Check!)    

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Skip down to Clue J.  We know there are 28 AZ vehicles with 12 AZ Trucks. Which leave 28 - 12 = 16 AZ non-Trucks

 

Clue J, states the remaining vehicles were split evenly between 4WD and Non-4WD, therefore 8 of each.  

 

AZ Vehicles: 28 total - 12 Trucks - 8  NT4WD - 8 NTN4WD.     (Non-Truck 4WD, Non-Truck Non-4WD)

 

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Go back to Clue g.  #AZ NTN4WD - 2 =# INT NTN4WD 

                                                         # INT NTN4WD = 8 - 2 = 6   There are 6 INT NTN4WD

 

INT Vehicle:  14 Total - 5 Trucks - 6 NTN4WD - 3 NT4WD  (14 Ttl - 5 Tr - 6 NTN4) 

 

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15 Total NT4WD Vehicles - 8 AZ and 3 INT,  leaving (15 - 8 - 3) with US plates, or 4 NT4WD

 

21 Total NTN4WD Vehicles - 8 AZ and 6 INT, leaving (21 - 8 - 6) with US plates, or 7 NTN4WD

 

US Vehicles: 14 Total - 3 Trucks - 4 NT4WD - 7 NTN4WD

                            

                       

Answer 2

The idea here is to split all vehicles into disjoint sets and express each of the given conditions a.-j. in terms of the number of vehicles in each of those sets. By b. any vehicle is a truck, or FWD non-truck, or a non-FWD non-truck.

Also any vehicle is from Arizona, or another US state, or not from US. Therefore, each vehicle falls in exactly one of the 9 categories below whose sizes I denote by x1, x2, ... x9 as follows:

 

x1 = number of Arizona trucks

x2 = number of Arizona FWD non-trucks

x3 = number of Arizona non-FWD non-trucks

x4 = number of non-Arizona US trucks
x5 = number of non-Arizona US FWD non-trucks
x6 = number of non-Arizona US non-FWD non-trucks

x7 = number of non-US trucks
x8 = number of non-US FWD non-trucks
x9 = number of non-US non-FWD non-trucks

 

Then the given observations yield:

a. x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 = 56

c. x1 + x2 + x4 + x5 + x7 + x8 = 35 and x1 + x4 + x9 = 21

d. x1 + x4 + x7 = 20

e. x1 + x2 + x3 = 28 and x1 + x2 + x3 + x4 + x5 + x6 = 42

f. x1 = 4 * x4

g. x3 = 2 + x9

h. x1 + x4 + x7 = x1 + x2 + x3 - 8 (however, this is nothing new; we already know it from d. and e.)

i. x1 = 7 + x7

j. x2 = x3

 

The two questions asked are to find x5 + x6 and x7 + x8. To this end we will find every single one of the nine unknowns x1, x2, ... x9.

From d., f., and i. we get a system of three equations involving x1, x4, and x7 only. Solving it, we find x1 = 12, x4 = 3, and x7 = 5.

From the first part of e. and j., using the found value of x1, we obtain a system of two equations involving only x2 and x3. Solving it, we find x2 = x3 = 8.

From g., using the found value of x3, we find x9 = 6.

From c., using the found values of x3 and x9, we find x6 = 7.

From the second part of e. and all the values we already have, we find x5 = 4.

From a. and all the values we have already found, we get x8 = 3.

 

Thus, there were x5 + x6 = 11 U.S. vehicles that weren’t from Arizona were not trucks.

And there were x7 + x8 = 8 foreign FWD vehicles.