Question 631079
I agree with the beginning of your work:
Let the articles bought be x
Then C.P of each article is 4800/x
The Selling price of x articles is 100x.
 
The mistake is in the next step.
Buying x articles the merchant paid 4800/x for each one.
That does not mean that the merchant would pay 4800/15=320 for each one of only 15 of those articles.
It does not mean that the total cost for 15 articles is 320, either.
You cannot say x=15 for this step, when the value of x is what you are trying to find.
 
Instead, you should think that the cost for 15 articles (at a cost per article of 4800/x, as you found before) is {{{15*(4800/x)=72000/x}}}.
The merchant sells all the articles for a total of 100x, which is, of course, more than the 4800 cost of all the articles.
The difference is the profit, that happens to be {{{72000/x}}} , so
{{{highlight(100x-4800=72000/x)}}}
Now let's solve, and then we can make sure that it makes sense.
{{{100x-4800=72000/x}}} --> {{{100x^2-4800x=72000}}} (multiplying both sides times x).
{{{100x^2-4800x=72000}}} --> {{{x^2-48x=720}}} (dividing both sides by 100).
{{{x^2-48x=720}}} --> {{{highlight(x^2-48x-720=0)}}} (subtracting 720 from both sides, to get a nice quadratic equation).
We know that solving by factoring could be an option, because x should be a nice positive integer.
We need two factors that multiplied equal -720, and add up to -48.
The factors -60 and 12 work.
The quadratic polynomial can be factored as:
{{{x^2-48x-720=(x-60)(x+12)}}}
So {{{x^2-48x-720=0}}} --> {{{(x-60)(x+12)=0}}}
with solutions {{{highlight(x=60)}}} and {{{x=-12}}} .
Of course we know that x, the number of articles bought and sold, is not a negative number, so we realize that {{{x=-12}}} must be discarded because it does not make sense.
The merchant bought {{{highlight(60)}}} articles for a total of 4800 (dollars, or whatever the currency is).
 
COMMON SENSE VERIFICATION:
The cost per article was {{{4800/60=80}}}.
On selling the 60
articles at 100 each, the merchant collected {{{100*60=6000}}}
The profit was the 6000 collected minus the 4800 cost, or
{{{6000-4800=1200}}}
The cost of 15 item was {{{15*80=1200}}} , the same amount as the total profit.
If the merchant had borrowed the original 4800 (from a relative, in a no-interest loan), the loan can now be repaid, and the merchant is left with 1200 to use to buy maybe 15 more of those articles (assuming he/she can still get the same good price from his supplier).