4.5: Merchandising

Step 1: List what you know.

[latex]\begin{eqnarray*}C&=&\$45.46\;\\E&=&\$25.17\;\\P&=&\$10.55\;\\D&=&25\%\end{eqnarray*}[/latex]

Write the unknowns: [latex]\$D,\;S,\;S_{red},\;P_{red}[/latex]

Step 2: Where can you find the information in known formulas?

Using Formula 4.3a ([latex]S=C+E+P[/latex]) will give us the selling price and the [latex]\$D[/latex].

[latex]\$D=0.10S[/latex]

Using Formula 4.4c, [latex]S_{red}=S-D[/latex], we can find the reduced sale price.

Using Formula 4.5a, [latex]P_{red}=S_{red}-C-E[/latex], we can find the reduced profit.

Using Formula 4.5b, [latex]P_{red}=P-D[/latex], we can find the reduced profit.

Step 3: Where do you find the unknown variables.

Same as Step 2.

Step 4: Solve the Question: There are two methods to solve the question.

Method 1

[latex]\begin{eqnarray*}S&=&{\color{green}{C}}+{\color{blue}{E}}+{\color{purple}{P}}\\S&=&{\color{green}{\$45.46}}+{\color{blue}{\$25.17}}+{\color{purple}{\$10.55}}\\S&=&\$81.18\\\\D&=&0.10{\color{purple}{S}}\\D&=&0.10\times{\color{purple}{\$81.18}}\\D&=&\$8.12\\\\S_{red}&=&{\color{chocolate}{S}}-{\color{Magenta}{D}}\\S_{red}&=&{\color{chocolate}{\$81.18}}-{\color{Magenta}{\$8.12}}\\S_{red}&=&\$73.06\\\\P_{red}&=&{\color{red}{S_{red}}}-{\color{green}{C}}-{\color{blue}{E}}\\P_{red}&=&{\color{red}{\$73.06}}-{\color{green}{\$45.46}}-{\color{blue}{\$25.17}}\\P_{red}&=&\$2.43\end{eqnarray*}[/latex]

Method 2

Step 5: Write as a statement.

At the sale price of [latex]\$73.06[/latex] the skateboard shop will make a profit of [latex]\$2.43[/latex].

a.

Step 1: List what you know from the question. 

[latex]\begin{eqnarray*}C&=&\$500\\E\;&=&0.20C\\P&=&0.30\;C\\D&=&.25S\end{eqnarray*}[/latex]

Write the unknowns: [latex]S,D,E,P,\text{ Markup on Sale Price}[/latex]

Step 2: Where can your find the information in known formulas?

Using Formula 4.3a, [latex]S=C+E+P[/latex], will give us the selling price.

Using Formula 4.4c, [latex]S_{red}=S-D[/latex], will give us the reduced sale price.

Using Formula 4.5a, [latex]P_{red}=S_{red}-C-E[/latex], will give us the reduced profit.

Using Formula 4.5b, [latex]P_{red}=P-D[/latex], will give us the reduced profit.

Step 3: Where do you find the unknown variables: 

Same as Step 2.

Step 4: Solve the Question: There are two methods to solve the question.

Method 1

[latex]\begin{eqnarray*}{\color{green}{\text{Cost}}}&=&{\color{green}{\$500}}\\\$E&=&0.20(\$500)\\{\color{blue}{\$E}}&=&{\color{blue}{\$100}}\\\\\$P&=&0.30(\$500)\\{\color{red}{\$P}}&=&{\color{red}{\$150.00}}\\\\\text{Using Formula 4.3a: }S&=&{\color{green}{C}}+{\color{blue}{E}}+{\color{red}{P}}\\S&=&{\color{green}{\$500}}+{\color{blue}{\$100}}+{\color{red}{\$150.00}}\\{\color{purple}{S}}&=&{\color{purple}{\$750.00}}\\\\\$D&=&0.25{\color{purple}{S}}\\\$D&=&0.25({\color{purple}{\$750}})\\{\color{Magenta}{\$D}}&=&{\color{Magenta}{\$187.50}}\\\\S_{red}&=&{\color{purple}{S}}-{\color{Magenta}{D}}\\S_{red}&=&{\color{purple}{\$750.00}}-{\color{Magenta}{\$187.50}}\\{\color{Mahogany}{S_{red}}}&=&{\color{Mahogany}{\$562.50}}\\\\P_{red}&=&{\color{Mahogany}{S_{red}}}-{\color{green}{C}}-{\color{blue}{E}}\\P_{red}&=&{\color{Mahogany}{\$562.50}}-{\color{green}{\$500}}-{\color{blue}{\$100}}\\P_{red}&=&-\$37.50\end{eqnarray*}[/latex]

Method 2

b.

[latex]\begin{eqnarray*}\text{Markup on Sale Price}&=&S_{red}-C\\\text{Markup on Sale Price}&=&\$562.50-\$500\\\text{Markup on Sale Price}&=&\$62.50\end{eqnarray*}[/latex]

Step 5: Write as a statement: 

At the sale price of [latex]\$562.50[/latex] the Cycle Corner will have a loss of [latex]\$37.50[/latex] and the Markup amount on the sale price is [latex]\$62.50[/latex].

Step 1: List what you know. 

[latex]\begin{eqnarray*}\text{Cost}\;(C)&=&\$1794\\\text{Profit}\;(P)&=&20\%\;\text{on selling price}\\\text{Markup}\;(M)&=&35\%\;\text{on Selling price}\end{eqnarray*}[/latex]

Write the unknowns: [latex]D,\;S,\;M,\;P,\;S_{red}\;,\;P_{red}[/latex]

Step 2: Where can you find the information in known formulas?

Using Formula 4.3a, [latex]S=C+E+P[/latex], will give us the selling price.

Using Formula 4.3b, [latex]M=E+P[/latex], will give us the Markup.

Using Formula 4.4c, [latex]S_{red}=S-D[/latex], will give us the reduced sale price.

Using Formula 4.5a, [latex]P_{red}=S_{red}-C-E[/latex], will give us the reduced profit.

Using Formula 4.5b, [latex]P_{red}=P-D[/latex], will give us the reduced profit.

Step 3: Where do you find the unknown variables: 

Same as Step 2.

Step 4: Solve the Question: There are two methods to solve the question.

a.

[latex]\begin{eqnarray*}\text{Selling Price}\\S&=&{\color{green}{C}}+{\color{red}{M}}\\S&=&{\color{green}{\$1794}}+{\color{red}{0.35S}}\\1S-0.35S&=&\$1794\\0.65S&=&\$1794\\{\color{blue}{S}}&=&{\color{blue}{\$2\text{,}760.00}}\end{eqnarray*}[/latex]

b. Profit or Loss

[latex]\begin{eqnarray*}\text{Markup}\\M&=&0.35S\\M&=&0.35(\$2760.00)\\{\color{red}{M}}&=&{\color{red}{\$966}}\\\\\text{Profit}\\P&=&0.20S\\P&=&0.20(\$2760)\\{\color{purple}{P}}&=&{\color{purple}{\$552}}\\\\\text{Expenses }\\E&=&M-P\\E&=&\$966-\$552\\{\color{blue}{E}}&=&{\color{blue}{\$414}}\\\\\text{Discount}\\D&=&0.175(\$2760)\\{\color{Magenta}{D}}&=&{\color{Magenta}{\$483}}\\\\\text{Selling Price Reduced}\\S_{red}&=&S-D\\S_{red}&=&\$2760-\$483\\{\color{Mahogany}{S_{red}}}&=&{\color{Mahogany}{\$2277}}\\\\\text{Reduced Profit}\\P_{red}&=&{\color{Mahogany}{S_{red}}}-{\color{green}{C}}-{\color{blue}{E}}\\P_{red}&=&{\color{Mahogany}{\$2277}}-{\color{green}{\$1794}}-{\color{blue}{\$414}}\\P_{red}&=&+\$69.00\\\end{eqnarray*}[/latex]

c. Rate of Markdown at Break-Even Price

Method 1

[latex]\begin{eqnarray*}\text{Break even is when C + E }\;\\BE&=&C+E\;\\BE&=&\$1794\;+\;\$414\;\\BE&=&\$2208.00\;\;\\\\\text{Markdown}\;(D)&=&\text{Selling Price}-BE\;\;\\\text{Markdown}\;(D)&=&\$2760\;-\;\$2208\\\text{Markdown}\;(D)&=&\$552\;\;\\\\\text{Markdown Rate}&=&\frac{\$552}{\$2760}\times100\%\;\\[1ex]\text{Markdown Rate}&=&20\%\end{eqnarray*}[/latex]

Method 2

Step 5: Write a Statement

The selling price of the computer is [latex]\$2760[/latex] and with a [latex]17.5\%[/latex] discount at the trade show, there is a profit of [latex]\$69.00[/latex].  The rate of markdown if the computer is sold at the break-even price is [latex]20\%[/latex].

Step 1: List what you know.

[latex]\begin{eqnarray*}S&=&\$108.50\\P&=&0.30C\\M&=&0.75C\end{eqnarray*}[/latex]

Step 2: Where can you find the information in known formulas?

Using Formula 4.3a, [latex]S=C+E+P[/latex], will give us the selling price.

Using Formula 4.3b, [latex]M=E+P[/latex], will give us the Markup.

Using Formula 4.4c, [latex]S_{red}=S-D[/latex], will give us the reduced sale price.

Using Formula 4.5a, [latex]P_{red}=S_{red}-C-E[/latex], will give us the reduced profit.

Using Formula 4.5b, [latex]P_{red}=P-D[/latex], will give us the reduced profit.

Step 3: Where do you find the unknown variables: 

Same as Step 2.

Step 4: Solve the questions.

a. Cost of the Printer

[latex]\begin{eqnarray*}S&=&\$108.50\\P&=&0.30C\\M&=&0.75C\\\\S&=&C+M\\\$108.50&=&C+0.75C\\\$108.50&=&1.75C\\C&=&\$62.00\;\text{Each printer cost is \$62.00}\end{eqnarray*}[/latex]

b. Rate of Markdown with [latex]\$5.58[/latex] in profit after discount.

[latex]\begin{eqnarray*}\;M&=&E+P\;\;\\0.75C&=&E\;+\;.30C\;\\E&=&0.45C\;\\E&=&0.45(\$62.00)\;\;\\E&=&\$27.90\\\\P_{red}&=&S_{red}-C-E\;\\\$5.58&=&S_{red}-\$62.00-\$27.90\;\\S_{red}&=&95.48\\\\S_{red}&=&S-D\;\;\\95.48&=&\$108.50-D\;\;\\D&=&\$13.02\\\\\text{Rate of Markdown}&=&\frac{\$13.02}{\$108.50}\times100\%\\[1ex]\text{Rate of Markdown}&=&12\%\end{eqnarray*}[/latex]

c. Rate of Markdown with a price at cost.

Method 1

[latex]\begin{eqnarray*}S_{red}&=&C=\$62.00\\S_{red}&=&S-D\\D&=&S-S_{red}\\D&=&\$108.50-\$62.00\\D&=&\$46.50\\\\\text{Rate of Markdown}&=&\frac{\$46.50}{\$108.50}\times100\%\\[1ex]\text{Rate of Markdown}&=&42.86\%\end{eqnarray*}[/latex]

Method 2

Step 5: Write a Statement

The cost for each printer for Carlton Printers is [latex]\$62.00[/latex]. With a profit of [latex]\$5.58[/latex] per printer the rate of markdown would be [latex]12\%[/latex].

Step 1: List what you know.

[latex]\begin{eqnarray*}L&=&\$600\\M&=&0.55S\\D&=&0.10S\\E&=&0.20S\end{eqnarray*}[/latex]

Step 2:  Where can you find the information in known formulas?

Using Formula 4.1d, [latex]$D=L \times (1-d)(1-d)...[/latex], will give us the discount amount.

Using Formula 4.3c, [latex]S=C+M[/latex], will give us the selling price, [latex]M[/latex] and [latex]P[/latex].

Using Formula 4.4b, [latex]$M=S\times d[/latex], will give us the Markdown amount.

Using Formula 4.4c, [latex]S_{red}=S-D[/latex], will give us the reduced sale price.

Using Formula 4.5a, [latex]P_{red}=S_{red}-C-E[/latex], will give us the reduced profit.

Using Formula 4.5b, [latex]P_{red}=P-D[/latex], will give us the reduced profit.

Step 3: Where do you find the unknown variables?

Same as Step 2.

Step 4: Solve the questions.

Method 1

a.

[latex]\begin{eqnarray*}\text{Discount Amount}&=&L\;x\;(1-d)(1-d)\dots\\\text{Discount Amount}&=&\$600(1-0.10)(1-0.08)\\\text{Discount Amount}&=&\$600(0.9)(0.92)\\\text{Discount Amount}&=&\$496.80\;\\\\S&=&C+M\;\\S&=&\$496.80+0.55S\;\\0.45S&=&\$496.80\;\\S&=&\$1104.00\;\\\text{660 units sold at \$1104.00}\end{eqnarray*}[/latex]

b.

[latex]\begin{eqnarray*}\text{Markdown Amount}&=&S\times d\;\\\text{Markdown Amount}&=&\$1104.00\times(0.10))\\\text{Markdown Amount}&=&\$110.40\\\\S_{red}&=&S-D\;\\S&=&\$1104.00-\$110.40\;\\S&=&\$993.60\\\text{300 Units sold at \$993.00}\end{eqnarray*}[/latex]

c.

[latex]\begin{eqnarray*}BE\;\text{price}&=&C+E\;\;\\BE\;\text{price}&=&\$496.80+(0.20\times\$1104.00)\\BE\;\text{price}&=&\$496.80+\$220.80\\BE\;\text{price}&=&\$717.60\;\end{eqnarray*}[/latex]

The [latex]BE[/latex] price at which the remaining TVs were sold is [latex]\$717.60[/latex].

d. Profits from all TVs.

[latex]\text{Total cost of 1000 Units}=1000\times 496.80 = \$496,800.00[/latex]

[latex]\text{Total operating expenses}=1000\times 0.20 \times \$1104.00 = \$220,800.00[/latex]

[latex]\text{Total Selling price}=(660 \times \$1104.00) + (300 \times \$993.00) + (40 \times \$717.60)\\\text{Total Selling price}= \$1,055,424.00[/latex]

[latex]\begin{eqnarray*}\text{Total Profit}&=&{\color{red}{S}}-{\color{green}{C}}-{\color{blue}{E}}\\\text{Total Profit}&=&{\color{red}{\$1\text{,}055\text{,}424.00}}-{\color{green}{\$496\text{,}800.00}}-{\color{blue}{\$220\text{,}800.00}}\\\text{Total Profit}&=&\$337,824.00\end{eqnarray*}[/latex]

Method 2

[latex]\begin{eqnarray*}\text{Total Profit}&=&(600\times\$386.40)\;+\;(300\times\$276.00)\;+\;(40\times00.00)\\\text{Total Profit}&=&\$337,824.00\end{eqnarray*}[/latex]

Step 5: Write a Statement

The selling price of the [latex]660[/latex] units sold at regular price was [latex]\$1104.00[/latex]. The sale price of the [latex]300[/latex] units sold was [latex]\$993.60[/latex]. The break-even price of the remaining [latex]40[/latex] units sold was [latex]\$717.60[/latex].  Finally, the total profit made from the sale of all [latex]1000[/latex] TVs was [latex]\$337,824.00[/latex].