Question 1203277
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x = cost of 1 key chain
3x = cost of 1 pencil case
costs are in dollars
x > 0


3(3x) = 9x = cost of 3 pencil cases
7x = cost of 7 key chains
9x+7x = 16x = cost of 3 pencil cases and 7 key chains


James used 1/4 of his money to buy 3 pencil cases and 7 key chains. 
He used 1/4 of his money to spend 16x dollars.


(1/4)*(his starting amount of money) = amount spent
(1/4)*(his starting amount of money) = 16x dollars
his starting amount of money = 4*16x dollars
his starting amount of money = 64x dollars


James started with 64x dollars.
3/4 of 64x = 48x dollars left over


Or we could say: 64x - 16x = 48x


Then,
5/6 of 48x = 40x dollars spent on more key chains


James spent 7x dollars on 7 key chains, and then an additional 40x dollars on some more key chains. 
That's a total of 7x+40x = 47x dollars spent on all of the key chains.


Now we'll use this fact: "He spent $30.40 more on all the key chains than on all the pencil cases."


Amount spent on key chains = Amount spent on pencil cases + 30.40
47x = 9x + 30.40
47x-9x = 30.40
38x = 30.40
x = 30.40/38
x = <font color=red>0.80</font>
<font color=red>One key chain costs $0.80, aka 80 cents</font>


3x = 3*$0.80 = $2.40 is the cost of 1 pencil case.


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Check:


1 pencil case = $2.40
3 pencil cases = 3*$2.40 = $7.20
1 key chain = $0.80
7 key chains = 7*$0.80 = $5.60


3 pencil cases and 7 key chains = $7.20 + $5.60 = $12.80


He spent 1/4 of his money and spent $12.80
Therefore he started with 4*12.80 = $51.20
3/4 of which is what he has left: (3/4)*$51.20 = $38.40
Or we could say 51.20 - 12.80 = 38.40


After buying that first batch of pencil cases and key chains, he has $38.40 left over.
5/6 of $38.40 = $32 is spent on additional key chains
Each key chain is $0.80, so he must have bought 32/0.80 = 40 additional key chains.


He bought a total of 7+40 = 47 key chains
47*0.80 = $37.60 was spent on key chains
$7.20 was spent on pencil cases
37.60-7.20 = 30.40 is the difference in what was spent on each item type.
We have confirmed the answer.


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Answer: One key chain costs <font color=red>$0.80</font> (aka <font color=red>80 cents</font>)
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