pratik2018 wrote:
An alloy of gold,silverand bronze contain 90% bronze, 7% gold and3% silver.a second alloy of bronze and silver only is melted with the first and mixture contain 85% of bronze,5% of gold,10% of silver find the percentage of bronze in second alloy?
a. 75%
b. 72.5%
c. 70%
d. 67.5%
e. 65%
Without loss of generality (to be understood later) we may assume we have 100 grams of the first alloy, therefore \(\,\,100\,\,{\text{g}}\,\,\,\left\{ \begin{gathered}
\,\boxed{90\,\,{\text{g}}\,\,{\text{bronze}}} \hfill \\
\,7\,\,{\text{g}}\,\,{\text{gold}} \hfill \\
\,3\,\,{\text{g}}\,\,{\text{silver}} \hfill \\
\end{gathered} \right.\,\,\)
The second alloy does not have gold, and when the alloys are combined, we have 5% gold, therefore we may conclude (in grams of gold) that:
\(\frac{5}{{100}}\left( {{\text{total}}\,\,{\text{combined}}} \right)\,\, = \,\,\,7\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{total}}\,\,{\text{combined}} = \,\,\,140\,\,{\text{g}}\,\,\left\{ \begin{gathered}
\,\,\left( {\frac{{85}}{{100}}} \right)140 = \boxed{119\,\,{\text{g}}\,\,{\text{bronze}}}\,\, \hfill \\
\,\,\left( {\frac{5}{{100}}} \right)140\,\,{\text{g}}\,\,{\text{gold}} \hfill \\
\,\,\left( {\frac{{10}}{{100}}} \right)140\,\,{\text{g}}\,\,{\text{silver}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,{\text{AND}}\,\,\,\,\,\,\,\)
\({\text{second}}\,\,{\text{alloy}} = \,\,\,40\,\,{\text{g}}\,\,\,\left\{ \begin{gathered}
\boxed{x \cdot 40\,\,{\text{g}}\,\,{\text{bronze}}} \hfill \\
\left( {1 - x} \right) \cdot 40\,\,{\text{g}}\,\,{\text{silver}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,{\text{where}}\,\,\,\,? = x\,\,\,\left( {0 < x < 1} \right)\)
The equation (in grams of bronze) obtained using the "frames" presented above ends our solution:
\(90 + x \cdot 40 = 119\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = x = \left( {\frac{{29}}{{40}}} \right) \cdot 100\% = \,\,\underleftrightarrow {\frac{5}{2}\left( {28 + 1} \right)\% } = \left( {70 + 2.5} \right)\%\)
The above follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
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