Example 3
Completion requirements
| Example 3 |
Shelly makes and sells jewelery at the local Farmers’ Market. She purchased \(\$75\) worth of supplies to make new bracelets. She plans to keep \(3\) bracelets for her friends and sell the remaining bracelets for a total of \(\$250\). If Shelly wants to make a profit of \(\$12.50\) per bracelet, how many bracelets must she make?
Let \(b\) represent the number of bracelets made.
Note that Shelly gives away three of her bracelets to friends.
The equation for this question will be:
Profit from each bracelet = earnings from each bracelet \(-\) cost of making each bracelet
\(12.50 = \frac{{250}}{{b - 3}} - \frac{{75}}{b}, \thinspace b \ne 0,\thinspace 3\)
LCD: \(b\left(b - 3\right)\)
\[\begin{align}
\left( {12.5} \right)\left( b \right)\left( {b - 3} \right) &= \left( {\frac{{250}}{{{\color{red}\cancel{\color{#444}{b - 3}}}}}} \right)\left( b \right){\color{red}\cancel{\color{#444}{\left(b - 3\right)}}} - \left( {\frac{{75}}{{{\color{red}\cancel{\color{#444}{b}}}}}} \right){\color{red}\cancel{\color{#444}{\left(b\right)}}}\left( {b - 3} \right) \\
12.5b\left( {b - 3} \right) &= 250b - 75\left( {b - 3} \right) \\
12.5b^2 - 37.5b &= 250b - 75b + 225 \\
12.5b^2 - 212.5b - 225 &= 0 \\
12.5\left( {b^2 - 17 - 18} \right) &= 0 \\
\left( {b - 18} \right)\left( {b + 1} \right) &= 0 \\
b &= 18 \thinspace {\rm{and}} \thinspace b = - 1 \\
\end{align}\]
Note that Shelly cannot make a negative number of bracelets; therefore, \(b = -1\) is not a possible solution.
Verify for \(b = 18\).
Shelly will need to make \(18\) bracelets in order to make a profit of \(\$12.50\) per bracelet.
Cost of making each bracelet \(= \frac{\$75.00}{b}\)
Earnings for each bracelet \(=\frac{\$250.00}{b - 3}\)
Earnings for each bracelet \(=\frac{\$250.00}{b - 3}\)
Note that Shelly gives away three of her bracelets to friends.
The equation for this question will be:
Profit from each bracelet = earnings from each bracelet \(-\) cost of making each bracelet
\(12.50 = \frac{{250}}{{b - 3}} - \frac{{75}}{b}, \thinspace b \ne 0,\thinspace 3\)
LCD: \(b\left(b - 3\right)\)
\[\begin{align}
\left( {12.5} \right)\left( b \right)\left( {b - 3} \right) &= \left( {\frac{{250}}{{{\color{red}\cancel{\color{#444}{b - 3}}}}}} \right)\left( b \right){\color{red}\cancel{\color{#444}{\left(b - 3\right)}}} - \left( {\frac{{75}}{{{\color{red}\cancel{\color{#444}{b}}}}}} \right){\color{red}\cancel{\color{#444}{\left(b\right)}}}\left( {b - 3} \right) \\
12.5b\left( {b - 3} \right) &= 250b - 75\left( {b - 3} \right) \\
12.5b^2 - 37.5b &= 250b - 75b + 225 \\
12.5b^2 - 212.5b - 225 &= 0 \\
12.5\left( {b^2 - 17 - 18} \right) &= 0 \\
\left( {b - 18} \right)\left( {b + 1} \right) &= 0 \\
b &= 18 \thinspace {\rm{and}} \thinspace b = - 1 \\
\end{align}\]
Note that Shelly cannot make a negative number of bracelets; therefore, \(b = -1\) is not a possible solution.
Verify for \(b = 18\).
| Left Side | Right Side |
|---|---|
|
\(12.50\) |
\[\begin{array}{l} \frac{{250}}{{b - 3}} - \frac{{75}}{b} \\ \frac{{250}}{{\left( {18} \right) - 3}} - \frac{{75}}{{\left( {18} \right)}} \\ \frac{{250}}{{15}} - \frac{{25}}{6} \\ \frac{{100}}{6} - \frac{{25}}{6} \\ \frac{{75}}{6} \\ 12.50 \\ \end{array}\] |
| \(\hspace{25pt}\)LS = RS | |
Shelly will need to make \(18\) bracelets in order to make a profit of \(\$12.50\) per bracelet.