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Question 553995: " An alloy containing copper and silver in the ratio 1 : 2 is mixed with another alloy containing copper and silver in the ration 2 : 3. In what ratio must the two alloys be mixed to produce a new alloy containing copper and silver in the ratio 13 : 22? "
* I have tried it many times and I still got the answer 5:4 because
Let x = the # of first alloy units
Let y = the # of second alloy units
You need 13 copper, so since the first gives you 1 unit while the second gives you 2, you have
1x + 2y = 13
Similarly, you need 22 silver and the first gives you 2 while the second gives you 3, so you have
2x + 3y = 22
Solve this system of equations so you get
x = 13 - 2y
2 (13 - 2y) + 3y = 22
26 - 4y + 3y = 22
- y = -4
y = 4
x + 2(4) = 13
x = 5
therefore, the ratio will be x:y which is alloy 1: alloy 2, or 5:4
But still I still I'm wrong , Please help me about this.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! it took a while but i think i've got it.
i needed to put it into numbers before i could solve it with letters.
here's how i solved it with letters.
when the copper to silver ratio is 1/2, this means that the alloy is 1/3 copper and 2/3 silver.
when the copper to silver ratio is 2/3, this means that the alloy is 2/5 copper and 3/5 silver.
when the copper to silver ratio is 13/22, this means that the alloy is 13/35 copper and 22/35 silver.
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let x equal the amount of the first alloy and lety = the amount of the second alloy and let z equal the amount of the final alloy.
you get 2 equations that need to be solved simultaneously.
the first equation is:
x + y = z
the second equation is:
(1/3)x + (2/5)y = (13/35)z
from the first equation, solve for y to get:
y = z - x)
substitute for y in the second equation to get:
(1/3)x + (2/5)(z-x) = (13/35)z
simplify to get:
(1/3)x + (2/5)z - (2/5)x = (13/35)z
multiply both sides of this equation by (3*5*35) to get rid of the denominators to get:
175x + 210z - 210x = 195z
subtract 210z from both sides of this equation to get:
175x - 210x = 195z - 210z
combine like terms to get:
-35x = -15z
substitute (x+y) for z to get:
-35x = -15(x+y)
simplify to get:
-35x = -15x -15y
add 35x and 15y to both sides of this equation to get:
15y = 35x - 15x
simplify to get:
15y = 20x
divide both sides of this equation by 15 to get:
y = 20x/15
simplify to get:
y = 4x/3
this is the ratio you are looking for.
let's see if it works.
if x = 3, then y = 4
you have 3 units of the first solution and 4 unis of the second solution to yield 7 units of the combined solution.
the first solution yields 1 unit of copper and 2 units of silver.
the second solution yields 1.6 units of copper and 2.4 units of silver.
the ratio of copper to silver in the first solution is 1/2.
the ratio of copper in the second solution is 1.6/2.4 = 2/3.
the combined solution yields 1 + 1.6 = 2.6 units of copper and 2 + 2.4 = 4.4 units of silver.
the ratio of copper to silver in the combined solution is 2.6/4.4 = 13/22.
everything checks out so the answer appears to be good.
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