Business 9...Show work, Explain?

Business: The president of Speedy Airlines has discovered that her competitor, Zip Airlines, has purchased 13 new airlines form Commuter Aviation for a total of $15.9 million. She knows that Commuter Aviation produces three types of planes and that type A sells for $1.1 million, type B sells for $1.2 million, and type C sells for $1.7 million. The president of Speedy Airlines also managed to find out that Zip Airlines purchased 5 more type A plane than type C plane. How many planes of each type did Zip Airlines purchase?

Business 9...Show work, Explain?
A = number of planes of type "A" at 1.1 million each


B = number of planes of type "B" at 1.2 million each


C = number of planes of type "C" at 1.7 million each





Since there were a total of 13 planes, we know:





    A + B + C = 13


    [Equation #1]





Since the total purchase price was 15.9 million, we know:





    A*1.1 + B*1.2 + C*1.7 = 15.9


    [Equation #2]





Since we know that there are 5 more of A than C, we know:





    A = C + 5


    [Equation #3]





Substituting #3 into #1:





    A + B + C = 13


    (C + 5) + B + C = 13


    C + 5 + B + C = 13


    2C + B = 8


    B = 8 - 2C


    [Equation #4]





Substituting Equations #3 and #4 into #2:





    A*1.1 + B*1.2 + C*1.7 = 15.9


    (C + 5)*1.1 + (8 - 2C)*1.2 + C*1.7 = 15.9


    1.1C + 5.5 + 9.6 - 2.4C + 1.7C = 15.9


    1.1C + 5.5 + 9.6 - 2.4C + 1.7C = 15.9


    15.1 + .4C = 15.9


    .4C = .8


    C = 2





By equation #3:





    A = C + 5


    A = 2 + 5


    A = 7





By equation #4:





    B = 8 - 2C


    B = 8 - 4


    B = 4








To check:





    A + B + C = 13


    7 + 4 + 2 = 13


    OK





    A*1.1 + B*1.2 + C*1.7 = 15.9


    7*1.1 + 4*1.2 + 2*1.7 = 15.9


    7.7 + 4.8 + 3.4 = 15.9


    15.9 = 15.9


    OK





Therefore, Zip airlines purchased:





    7 planes of type A


    4 planes of type B


    2 planes of type C








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Reply:Let the quantities purchased of each airplane be a,b,c





Given:


[13 planes purchased] a+b+c=13


[also] a=5c


[total of $15.9m]


1.1a + 1.2b + 1.7c = 15.9


11a + 12b + 17c = 159





This is pretty easy; since a is an integer, it can only be 5 or 10. (it can't really be 0 since 5*0 = 0 is not what they intended)


So your only choices are (a=5,c=1,b=7) or (a=10,c=2,b=1)





(Formally, we substituted a=5c into a+b+c=13 to get


b+6c =13 )





It's easy to test (a=10,b=1,c=2) so let's do that:


11a + 12b + 17c = 110 + 12 + 34 = 156 !=159





So it looks like (a=5,b=7,c=1):


11a + 12b + 17c = 55 + 84 +17 = 156 !=159





(a=0,b=13,c=0) doesn't work either


12b = 156 !=159





[you must have made a minor typo?]

roses