SOLUTION: A shop owner paid a total of $2385 for some shirts and ties. Each shirt cost $14 less than each tie. The number of ties bought was 2/3 of the number of shirts bought. He paid $127.
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-> SOLUTION: A shop owner paid a total of $2385 for some shirts and ties. Each shirt cost $14 less than each tie. The number of ties bought was 2/3 of the number of shirts bought. He paid $127.
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Question 1206418: A shop owner paid a total of $2385 for some shirts and ties. Each shirt cost $14 less than each tie. The number of ties bought was 2/3 of the number of shirts bought. He paid $127.80 more for the ties than the shirts.
(a) How much did he pay for all the shirts?
(b) How many ties did he buy? Found 3 solutions by ikleyn, Theo, greenestamps:Answer by ikleyn(52794) (Show Source):
You can put this solution on YOUR website! .
A shop owner paid a total of $2385 for some shirts and ties.
Each shirt cost $14 less than each tie.
The number of ties bought was 2/3 of the number of shirts bought.
He paid $127.80 more for the ties than the shirts.
(a) How much did he pay for all the shirts?
(b) How many ties did he buy?
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(a) Let x be the amount he paid for all the shirts.
Then the amount which he paid for all the ties was by (x+127.80) dollars.
An equation for the total is
x + (x+127.80) = 2385 dollars.
Simplify and find x
2x = 2385 - 127.80
2x = 2257.20
x = 2257.20/2 = 1128.60.
Thus part (a) is solved.
(b) Let x be the number of shirts.
Then the number of ties is .
We just found in (a) the cost of all shirts. It is 1128.60 dollars.
Hence, the cost of all ties is 2385 - 1128.60 = 1256.40 dollars.
The price of one shirt is then .
The price for one tie is = .
The difference of prices is 14 dollars (given).
So, we write this "price equation"
- = 14.
Simplify and find x
= 14
= 14
x = = 54.
Thus the number of shirts was 54.
Then the number of ties was = 36.
At this point, part (b) is solved, too, and the ANSWER is 36 ties.
x * T - (x-14) * S = 127.80
this means that the cost of the ties minus the cost of the shirts = 127.8.
x * T + (x-14) * S = 2385
this means that the cost of the ties plus the cost of the shirts = 2385.
add these two equations together to get 2 * x * T = 2512.8
solve for x * T to get x * T = 2512.8 / 2 = 1256.4
since T = 2/3 * S, replace T with 2/3 * S to get x * 2/3 * S = 1256.4
multiply both sides of this equation by 3/2 to get x * S = 3/2 * 1256.4 = 1884.6
you now have x * T = 1256.4 and x * S = 1884.6
since x * T + (x-14) * S = 2385, and x * T = 1256.4, then (x-14) * S = 2385 - 1256.4 = 1128.6
you now have:
x * T = 1256.4
x * S = 1884.6
(x-14) * S = 1128.6
x * S - (x-14) * S = 1884.6 - 1128.6 = 756
combine like terms to get 14 * S = 756
solve for S to get S = 756 / 14 = 54
since T = 2/3 * S, then T = 2/3 * 54 = 36.
you now have S = 54 and T = 36.
in the equation x * T = 1256.4, solve for x to get x = 1256.4 / 36 = 34.9.
x - 14 must be equal to 34.9 minus 14 = 20.9.
x is the price of a tie = 34.9 and (x-14) is the price of a shirt = 20.9.
your original equation of x * T + (x - 14) * S = 2385 becomes:
34.9 * 36 + 20.9 * 54 = 2385 which becomes 2385 = 2385, confirming the values of x and S and T are good.
your solution is that you bought 36 ties at 34.9 each and 54 shirts at 20.9 each for a total price of 2385.
the total cost for the ties was 36 * 34.9 = 1256.4.
the total cost for the shirts was 54 * 20.9 = 1128.6.
you paid 1256.4 - 1128.6 = 127.8 more for the ties than you paid for the shirts.
