Question 1203277
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James used 1/4 of his money to buy 3 pencil cases and 7 key chains. The cost of each pencil case is 3 times the cost of each key chain. He bought some more key chains with 5/6 of his remaining money. He spent $30.40 more on all the key chains than on all the pencil cases. How much was the cost of one key chain?

Let original amount he had, and cost of each key chain, be A and K, respectively
He used {{{1/4}}} of his money {{{matrix(1,3, (1/4)A, or, A/4)}}} to buy 3 pencil cases and 7 key chains, leaving {{{matrix(1,3, (3/4)A, or, 3A/4)}}} 
Therefore, cost of the 7 key chains (K) = 7K, and the cost of the 3 pencil cases = {{{A/4  -  7K}}}
So, cost of 1 pencil case = {{{(A/4  -  7K)/3}}}
Since the cost of each pencil case is 3 times the cost of each key chain, we get: {{{matrix(3,3, (A/4 - 7K)/3, "=", 3K, A/4 - 7K, "=", 9K, A - 28K, "=", 36K)}}}
                                                                                         A = 36K + 28K
                                                                                         A = 64K --- eq (i)

Since he then used {{{5/6}}} of remaining money to buy more key chains, he used {{{matrix(1,3, (5/6) * (3A/4), or, 5A/8)}}} to buy more key chains. 

Amount spent on key chains: {{{7K + 5A/8}}}
Amount spent on pencil cases: {{{A/4  -  7K}}}
He spent $30.40 more on ALL key chains than on pencil cases, and so: {{{matrix(3,3, 7K + 5A/8, "=", A/4  -  7K + 30.4, 7K + 7K + 5A/8  -  A/4, "=", 30.4, 14K + 5A/8  -  A/4, "=", 30.4)}}}
                                                                    112K + 5A - 2A = 243.2 -- Multiplying by LCD, 8 
                                                                         112K + 3A = 243.2 -- eq (ii)
                                                                     112K + 3(64K) = 243.2 -- Substituting 64K for A in eq (ii)
                                                                       112K + 192K = 243.2
                                                                              304K = 243.2
                                                     Cost of each key chain, or {{{highlight_green(matrix(1,5, K, "=", 243.2/304, "=", highlight("$0.80")))}}}</pre>