BTGmoderatorDC wrote:The average weight of a class is x pounds. When a new student weighing 80 pounds joins the class, the average decreases by 1 pound. In a few months the student's weight increases to 110 pounds and the average weight of the class becomes x + 4 pounds. None of the other students' weights changed. What is the value of x?
A. 85
B. 86
C. 88
D. 90
E. 92
Source: Veritas Prep
(All weights are in pounds.)
$$? = x$$
Excellent opportunity to use the homogeneity nature of the average:
$$\sum\nolimits_n { = \,\,nx\,\,\,\,\,\left( {n\,\,{\rm{students}}} \right)} $$
$$\left\{ \matrix{
80 + \sum\nolimits_n {\, = \sum\nolimits_{n + 1} {\, = \,\,\left( {n + 1} \right)\left( {x - 1} \right)} } \hfill \cr
110 + \sum\nolimits_n {\, = \sum\nolimits_{n + 1} {\, = \,\,\left( {n + 1} \right)\left( {x + 4} \right)} } \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,110 - 80 = \left( {n + 1} \right)\left[ {\left( {x + 4} \right) - \left( {x - 1} \right)} \right]$$
$$30 = 5\left( {n + 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n = 5$$
$$80 + 5x = \left( {5 + 1} \right)\left( {x - 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = x = 86$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
