AAPL wrote:Manhattan Prep
If the original price of an item in a retail store is marked up by m percent and the resulting price is then discounted by d percent, where m and d are integers between 0 and 100, is the item's final price (after both changes) greater than its original price?
1) m > d
2) m = 1.5d
$$1\,\, \le m,d\,\, \le \,\,99\,\,\,\,\,{\rm{ints}}$$
$$P\,\,\,\, \to \,\,\,\,\left( {1 + {m \over {100}}} \right)P\,\,\,\, \to \,\,\,\,\left( {1 - {d \over {100}}} \right)\left( {1 + {m \over {100}}} \right)P\,\,\,\,\,\,\,\,\,\,\,\,\left[ {P > 0} \right]$$
$$\left( {1 - {d \over {100}}} \right)\left( {1 + {m \over {100}}} \right)\,\,\,\mathop > \limits^? \,\,\,1$$
$$\left( {1 + 2} \right)\,\,\,\,m = {3 \over 2}d\,\,\,\,\,\,\left[ {d > 0\,\,\,\, \Rightarrow \,\,\,m > d} \right]\,\,\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,d} \right) = \left( {3,2} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left[ {{{98} \over {100}} \cdot {{103} \over {100}} > 1} \right]\, \hfill \cr
\,{\rm{Take}}\,\,\left( {m,d} \right) = \left( {90,60} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\left[ {{{40} \over {100}} \cdot {{190} \over {100}} < 1} \right]\,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left( {\rm{E}} \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
