The average (arithmetic mean) weight of 5 crates is 250 pounds. The 2

\(?\,\,\,:\,\,\,x \in \left[ {{x_{\min }}\,\,,\,\,\,{x_{\max }}} \right]\)


By the homogeneity nature of the average, we know that:


\(\sum\nolimits_5 = \,\,\,5 \cdot 250 = 1250\)

\(\sum\nolimits_{2\,{\text{light}}}^{\min } = \,\,\,2 \cdot 200 = 400\,\,\,\,\,\,;\,\,\,\,\,\,\sum\nolimits_{2\,{\text{light}}}^{\max } = \,\,\,2 \cdot 205 = 410\)

\(\sum\nolimits_{2\,{\text{heavy}}}^{\min } = \,\,\,2 \cdot 300 = 600\,\,\,\,\,\,;\,\,\,\,\,\,\sum\nolimits_{2\,{\text{heavy}}}^{\max } = \,\,\,2 \cdot 310 = 620\)


\({x_{\min }} = \sum\nolimits_5 { - \left( {\,\sum\nolimits_{2\,{\text{light}}}^{\max } { + \sum\nolimits_{2\,{\text{light}}}^{\max } {} } } \right)} = 1250 - \left( {410 + 620} \right) = 220\)

\({x_{\max }} = \sum\nolimits_5 { - \left( {\,\sum\nolimits_{2\,{\text{heavy}}}^{\min } { + \sum\nolimits_{2\,{\text{heavy}}}^{\min } {} } } \right)} = 1250 - \left( {400 + 600} \right) = 250\)


\(?\,\,\,:\,\,\,x \in \left[ {220,250} \right]\,\,\,\,\, \Leftrightarrow \,\,\,\,\,220 \leqslant x \leqslant 250\)


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

Regards,
Fabio.

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