Question 935828: The land of Jabral has two coins, the Jabr and the Al.
Kirsten buys a typewriter with two Jabrs and gets an Al
in change. Meanwhile, with one Jabr and two Als, her
friend Sophia buys two of the same typewriters Kirsten
purchased. How many of these typewriters could be
purchased with ten Jabrs?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x represent the value of 1 jabr.
let y represent the value of 1 al.
cost of 1 typewriter = 2x - y
cost of 2 typewriter = 1x + 2y
you get 2 equations.
they are:
2x - y = 1
1x + 2y = 2
these equations mean:
2 * the value of one jabr minus 1 * the value of one al = 1 * the cost of one typewriter
1 * the value of one jabr plus 2 * the value of 1 al = 2 * the cost of one typewriter.
in the first equation, solve for y to get:
2x - y = 1 becomes y = 2x - 1
in the second equation, replace y with 2x - 1 to get:
1x + 2y = 2 becomes 1x + 2(2x - 1) = 2 which becomes 1x + 4x - 2 = 2 which becomes 5x - 2 = 2.
add 2 to both sides of this equation to get:
5x = 4
divide both sides of this equation by 5 to get:
x = 4/5
this means that the value of 1 jabr is equal to 4/5 * the cost of 1 typewriter.
in the first original equation, replace x with 4/5 to get:
2x - y = 1 becomes 8/5 - y = 1
add y to both sides of this equation and subtract 1 from both sides of this equation to get:
8/5 - 1 = y which becomes 8/5 - 5/5 = y which becomes 3/5 = y
this means that the value of 1 al is equal to 3/5 * the cost of 1 typewriter.
you now know that:
x = 4/5 * the cost of 1 typewriter.
y = 3/5 * the cost of 1 typewriter.
go back to your original equations to see if this is true.
first equation is 2x - y = 1 which becomes 2 * 4/5 - 3/5 = 1 which becomes 8/5 - 3/5 = 1 which becomes 5/5 = 1 which becomes 1 = 1 which his true.
second equation is 1x + 2y = 2 which becomes 4/5 + 2 * 3/5 = 2 which becomes 4/5 + 6/5 = 2 which becomes 10/5 = 2 which becomes 2 = 2 which is true.
the values of x and y are confirmed to be true.
you want to know how many typewriters you can buy with 10 jabr.
since x represents the value of one jabr and x is equal to 4/5, this means that 1 jabr is equivalent to 4/5 of the cost of 1 typewriter.
10 jabr = 10 * the value of 1 jabr = 10 * 4/5 = 40/5 = 8.
this means that 10 * the cost of 1 jabr = 8 times the cost of 1 typewriter.
this means you can buy 8 typewriters with 10 jabr.
that's your solution.
read on for a further explanation if you like, but you don't need to if you're happy with the solution as it stands.
if we put this in terms of dollars, it might make more sense.
let's assume that 1 jabr is equal to 100 dollars.
if we can buy 8 typewriters with 10 jabr, and 1 jabr = 100 dollars, then the cost of 8 typewriters is equal to 10 * 100 = 1000 dollars.
the cost per typewriter is therefore equal to 1000 / 8 = 125 dollars.
we know that 2x - y = 1 which means that 2 times the value of one jabr - 1 times the value of one al is equal to the cost of one typewriter.
by substitution, this equation becomes:
2*100 - y = 125
simplify to get 200 - y = 125
solve for y to get y = 75 dollars.
we know that x = 100 and y = 75
x is the value of one jabr
y is the value of one al
1 jabr is worth 100 dollars because we said it was.
1 al is worth 75 dollars because we determined that from the equations.
the cost of 1 typewriter = 125 dollars because we determined that from the equations.
100/125 = 4/5 which means that one jabr is worth 4/5 * the cost of 1 typewriter.
75/125 = 3/5 which means that one al is worth 3/5 * the cost of 1 typewriter.
100/75 = 4/3 which means that the value of one jabr is worth 4/3 * the value of one al.
everything checks out so the solution is good.
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