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?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

(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.

The problem is solved in full  ( = completely).



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
S = number of shirts
T = number of ties.
x = cost of a tie.
x-14 = cost of a shirt.

you are given that:

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.

answers to your questions are:

(a) How much did he pay for all the shirts?

he paid 1128.6 for all the shirts.

(b) How many ties did he buy?

he bought 36 ties.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The number of ties he bought is 2/3 the number of shirts, so...

Let 3x = # of shirts
then 2x = # of ties

Let y = price of each shirt
then y+14 = price of each tie

The total cost was $2385:

[1]

The ties cost $127.80 more than the shirts:

[2]

Multiply [2] by 5 and add to [1] to eliminate the xy terms to solve for x





The number of shirts was 3x = 54
The number of ties was 2x = 36

Substitute x=18 in [1] to solve for y





The price of each shirt was y = $20.90
The price of each tie was y+14 = $34.90

ANSWERS:
(a) total cost of the shirts: 54($20.90) = $1128.60
(b) number of ties: 36
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