Problem Solving

BTGmoderatorDC wrote:Quentin's income is 60% less than Rex's income, and Sam's income is 25% less than Quentin's income. If Rex gave 60% of his income to Sam and 40% of his income to Quentin, Quentin's new income would be what fraction of Sam's new income?

A. 8/9
B. 11/12
C. 8/13
D. 11/13
E. 12/13

Let Rex's income = 100.
Since Quentin's income is 60% less than Rex's, Quentin's income = 100 - (60% of 100) = 100-60 = 40.
Since Sam's income is 25% less than Quentin's, Sam's income = 40 - (25% of 40) = 40-10 = 30.
After Rex gives 60% of his income to Sam, Sam's new income = 30 + (60% of 100) = 30+60 = 90.
After Rex gives 40% of his income to Quentin, Quentin's new income = 40 + (40% of 100) = 40+40 = 80.
Resulting fraction:
(Quentin's new income)/(Sam's new income) =80/90 = 8/9.

The correct answer is A.

BTGmoderatorDC wrote:Quentin's income is 60% less than Rex's income, and Sam's income is 25% less than Quentin's income. If Rex gave 60% of his income to Sam and 40% of his income to Quentin, Quentin's new income would be what fraction of Sam's new income?

A. 8/9
B. 11/12
C. 8/13
D. 11/13
E. 12/13
Source: Manhattan Prep

$$? = {Q \over S}$$
$$\left. \matrix{
Q = {2 \over 5}R\,\,\,\,\left( {40\% R} \right)\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,{2 \over 5}R\,\,\,\,\left( {40\% R} \right)\,\,\,\,\, = \,\,\,{4 \over 5}R \hfill \cr
S = {3 \over 4}Q\,\,\,\left( {75\% Q} \right) = \,\,{3 \over 4}\left( {{2 \over 5}R} \right) = {3 \over {10}}R\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,{3 \over 5}R\,\,\,\,\left( {60\% R} \right)\,\,\,\,\, = \,\,\,{9 \over {10}}R\,\,\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,?\,\, = \,\,{4 \over 5} \cdot {{10} \over 9} = {8 \over 9}$$

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

Regards,
Fabio.