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a shopkeeper sold his entire stock of shirts and ties in a sale for $1. the shirt were priced at 3 for $100 and the ties $20 each. if he had sold only half the shirts and two-thirds of the ties hw would have received $6000. how many did he sell in the sale?

He sold his entire stock for  $10000.

Let the number of shirts be x.
Let the number of ties be     y.

The shirt is at 3 for $100. Therefore, a shirt would cost   $(100/3)
The tie cost $20 each.

Therefore, Multiplying by the number of shirts and ties:

100x / 3    +  20y  =  10000 (i).

If he had sold half the shirts and two-thirds the ties, the equation becomes:
100/3 * (x/2)  +  20 * (2y / 3) =  6000
50x / 3          +    40y / 3        = 6000 (ii)

Multiply equation (ii) by 2.

2*(50x / 3          +    40y / 3 )      = 2* 6000
100x  / 3              + 80y / 3          = 12000 (iii).

Equation (i)  Minus (iii) . Inorder to eliminate x.

100x / 3 - 100x / 3  + 20y - 80y / 3 =  10000 - 12000
                                    -20y / 3        =  - 2000.
                                                       y = 3 * 2000 / 20
                                                       y = 300.
Substituting into equation (i).
100x / 3    +  20y  =  10000
100x / 3    +  20*300  =  10000
100x / 3    +  6000  =  10000
100x / 3      =  10000 - 6000
100x / 3      =  4000
x  =  4000 * 3 / 100
x = 120.

So he actually sold 120 shirts and 300 ties. Cheers.

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