Let n be the number of bottles she bought for R100.

Then the price she paid for each single bottle was .


She sold (n-10) bottles for R100 + R50 = R150, in total.

The price she sold each bottle was  .


The difference between prices was R10

     -  = 10.


The setup is done.
To solve this equation, multiply both sides by n*(n-10), then simplify

    150n - 100*(n-10) = 10n*(n-10)

    150n - 100n + 1000 = 10n^2 - 100n

    10n^2 - 150n - 1000 = 0

      n^2 - 15n  - 100 = 0

      (n-20)*(n+5) = 0.


Of two roots, 20 and -5, we select positive value n=20 for the number of bottles that she bought.


Then the price at which she bought each bottle was 100/20 = R5.          ANSWER

The price      at which she sold   each bottle was R5 + R10 = R15.       ANSWER

Let C be the cost of each bottle purchased
Then # of bottles purchased = 
Also, # of bottles sold would be: , and sale price of each bottle = C + 10
We then get: 
                            ------ Multiplying by LCD, C
                                 
                                      0 = (C - 5)(C + 20)
                                      0 = C - 5       or      0 = C + 20
 Cost of each bottle purchased or     or      C = - 20 (ignore)

 Sale price of each bottle:  C + 10 = 5 + 10 = R15.