Question 1000074: When Mr. and Mrs. Smith took the airplane, they had together 94 pounds of baggage. He paid 1.50 and she paid 2.00 for excess weight. If Mr. Smith made the trip by himself with the combined baggage of both of them, he would have to pay $13.50. How many pounds of baggage can one person take along without being charged.
The variable in the problem is not explicitly stated and I have tried multiple ways of starting it. I realized i understand what the question is asking and can figure out how to answer it, but not algebraically. How can I answer it algebraically? Any help would be appreciated. Thanks!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! it took some doing, but after several false starts, i finally got it.
let x = how much his luggage weighed.
let y = how much her luggage weighed.
let a = how much weight is allowed free.
let c = the cost per pound for the excess weight.
presumably the cost per pound of the excess luggage is the same.
you get:
(x - a) * c = 1.5 (first equation)
(y - a) * c = 2.0 (second equation)
(x + y - a) * c = 13.5 (third equation)
add the first 2 equations together to get:
(x-a) * c + (y - a) * c = 1.5 + 2.0
simplify to get:
xc - ac + yc - ac = 3.5
combine like terms to get:
xc + yc - 2ac = 3.5
simplify the third equation to get:
xc + yc - ac = 13.5
you have 2 equations that can be solved simultaneously.
they are:
xc + yc - ac = 13.5
xc + yc - 2ac = 3.5
factor out the c in each equation to get:
(x + y - a) * c = 13.5
(x + y - 2a) * c = 3.5
since x + y = 94, you can replace x + y with 94 in these equations to get:
(94 - a) * c = 13.5
(94 - 2a) * c = 3.5
solve for c in both these equations to get:
c = 13.5 / (94 - a)
c = 3.5 / (94 - 2a)
subtract the second equation from the first to get:
0 = 13.5 / (94 - a) - 3.5 / (94 - 2a)
flip sides in the equation to get:
13.5 / (94 - a) - 3.5 / (94 - 2a) = 0
multiply both sides of this equation by (94 - a) * (94 - 2a) to get:
13.5 * (94 - 2a) - 3.5 * (94 - a) = 0
simplify to get:
13.5 * 94 - 13.5 * 2a - 3.5 * 94 + 3.5 * a = 0
simplify further to get:
1269 - 27a - 329 + 3.5a = 0
combine like terms to get:
940 - 23.5a = 0
add 23.5a to both sides of the equation to get:
940 = 23.5a
divide both sides of this equation by 23.5 and solve for a to get:
a = 40
that's your solution.
the allowed weight is 40 pounds.
now that you know that, you can solve for c.
your original 3 equations are, once again:
(x - a) * c = 1.5 (first equation)
(y - a) * c = 2.0 (second equation)
(x + y - a) * c = 13.5 (third equation)
when a = 40, these equations become:
(x - 40) * c = 1.5 (first equation)
(y - 40) * c = 2.0 (second equation)
(x + y - 40) * c = 13.5 (third equation)
add the first 2 equations together to get:
(x - 40) * c + (x - 40) * c = 3.5
simplify and combine like terms to get:
(x + y - 80) * c = 3.5
you now have two equations that can be solved simulteneously.
they are:
(x + y - 80) * c = 3.5
(x + y - 40) * c = 13.5
since x + y = 94, these equations become:
(94 - 80) * c = 3.5
(94 - 40) * c = 13.5
simplify to get:
14c = 3.5
54c = 13.5
solve for c in both equations to get:
c = 3.5/14 = .25
c = 13.5/54 = .25
c is equal to .25
you can now solve for the weight of his luggage and the weight of her luggage.
(x-a)*.25 = 1.5 becomes (x-40)*.25 = 1.5 which becomes .25x - .25*40 = 1.5 which becomes .25x - 10 = 1.5 which becomes .25x = 11.5 which gets you x = 11.5/.25 which gets you x = 46.
if the weight of his luggage is 46 pounds, then the weight of her luggage has to be 48 pounds because 94 = 46 + 48.
he's 6 pounds over.
she's 8 pounds over.
6*.25 = 1.5
8*.25 = 2.0
everything is confirmed as good.
your answer is that the allowed weight = 40 pounds.
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