Problem Solving

Magoosh

In the month of August, Pentheus Corporation made $200,000 in profit. Pentheus made 6% of that profit on the second Wednesday of August. If the profits that day were approximately 14.5% of the revenue for that day, then what was Pentheus's revenue on the second Wednesday of August?

A. $65,536
B. $75,025
C. $77,922
D. $80,000
E. $82,756

OA E

AAPL wrote:Magoosh

In the month of August, Pentheus Corporation made $200,000 in profit. Pentheus made 6% of that profit on the second Wednesday of August. If the profits that day were approximately 14.5% of the revenue for that day, then what was Pentheus's revenue on the second Wednesday of August?

A. $65,536
B. $75,025
C. $77,922
D. $80,000
E. $82,756

All numbers presented below are in dollars.
$$? = {\rm{rev}}\,\,\left( {{\rm{certain}}\,\,{\rm{day}}} \right)$$
$${\rm{Aug}}\,{\rm{profit}}\,\,:\,\,\,2 \cdot {10^5}$$
$${\text{certain}}\,{\text{ day}}\,{\text{profit}}\,\,:\,\,\,\frac{6}{{100}}\left( {\,2 \cdot {{10}^5}} \right) = 12 \cdot {10^3}\,\,\,$$
$$12 \cdot {10^3}\,\,\, \cong \,\,\,{{29} \over 2}\% \,\,{\rm{rev}}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {\rm{rev}} \cong \,\,{{2 \cdot {{10}^2}} \over {29}}\left( {12 \cdot {{10}^3}} \right) = {{24} \over {29}}\left( {{{10}^5}} \right)$$
Now let´s operate the "encadrement" a beautiful french(!) technique to do approximations!

We start with 29/24, the reciprocal of 24/29, for the reasons that will become evident right-from-the-start:
$$1 + \frac{5}{{25}} = \boxed{\frac{6}{5}\,}\,\, < \,\,\,\,\frac{{29}}{{24}} = 1 + \frac{5}{{24}}\,\,\,\, < \,\,\,\,1 + \frac{5}{{20}}\,\, = \,\,\,\boxed{\frac{5}{4}}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\frac{4}{5} < \frac{{24}}{{29}} < \frac{5}{6}$$
$$\frac{4}{5}\left( {{{10}^5}} \right) < \frac{{24}}{{29}}\left( {{{10}^5}} \right) < \frac{5}{6}\left( {{{10}^5}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,80,000 < ? < 83,333\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( E \right)$$

The ONLY real calculation we did is 5/6 but, if you look carefully at the alternative choices, even this calculation was NOT needed!


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.