6.2 Discounts, Markups, and Sales Tax

1. Substituting the values into the formula $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}$, we find that the discount is $\mathrm{discount}=\mathrm{0}\mathrm{.25}\times 75.80=18.95$. The discount is $18.95.

   The sale price of the item is then $\mathrm{sale\; price}=\mathrm{original\; price}-\mathrm{discount}=75.80-18.95=56.85$, or $56.85.

2. Substituting the values into the formula $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}$, we find that the discount is $\mathrm{discount}=\mathrm{0}\mathrm{.30}\times 168.90=50.67$. The discount is $50.67.

   The sale price of the item is then $\mathrm{sale\; price}=\mathrm{original\; price}-\mathrm{discount}=168.90-50.67=118.23$, or $118.23.

1. Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $\mathrm{sale\; price}=\mathrm{original\; price}-\mathrm{discount}$. Substituting and calculating, we find the discount to be $900.00=1,200.00-\mathrm{discount}$. Solving for the discount gives $300.00.

   Step 2. Find the percent discount. Substituting the discount of $300.00 and the original price of $1,200.00, into the formula $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}$, we can find the percent discount.

   $$\begin{array}{ccc}\hfill \mathrm{300}\mathrm{.00}& \hfill =\hfill & \mathrm{percent\; discount}\times \mathrm{1,200}\mathrm{.00}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\mathrm{300}\mathrm{.00}}{1,200.00}}}}& \hfill =\hfill & \mathrm{percent\; discount}\hfill \\ \hfill 0.25& \hfill =\hfill & \mathrm{percent\; discount}\hfill \end{array}$$

   Converting to percent form, the percent discount is 25%.

2. Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $\mathrm{sale\; price}=\mathrm{original\; price}-\mathrm{discount}$. Substituting and calculating, we find the discount to be $29.52=36.70-\mathrm{discount}$. Solving for the discount gives $7.38.

   Step 2. Find the percent discount. Substituting the discount of $7.38 and the original price of $36.70, into the formula $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}$, we can find the percent discount.

   $$\begin{array}{ccc}\hfill \mathrm{29}\mathrm{.52}& \hfill =\hfill & \mathrm{percent\; discount}\times \mathrm{36}\mathrm{.70}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\mathrm{29}\mathrm{.52}}{36.70}}}}& \hfill =\hfill & \mathrm{percent\; discount}\hfill \\ \hfill 0.2& \hfill =\hfill & \mathrm{percent\; discount}\hfill \end{array}$$

   Converting to percent form, the percent discount is 20%.

1. Using the percent discount and the sale price, we can find the original price with the formula $\mathrm{sale\; price}=\mathrm{original\; price}\times \mathrm{(1}-\mathrm{percent\; discount)}$. Substituting and solving for the original price, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{sale\; price}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(1}-\mathrm{percent\; discount)}\hfill \\ \hfill \mathrm{450}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(1}-\mathrm{0}\mathrm{.10)}\hfill \\ \hfill \mathrm{450}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(0}\mathrm{.90)}\hfill \\ \hfill \mathrm{500}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\hfill \end{array}$$
   
   The original price of the item was $500.00.
   

2. Using the percent discount and the sale price, we can find the original price with the formula$\mathrm{sale\; price}=\mathrm{original\; price}\times \mathrm{(1}-\mathrm{percent\; discount)}$. Substituting and solving for the original price, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{sale\; price}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(1}-\mathrm{percent\; discount)}\hfill \\ \hfill \mathrm{90}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(1}-\mathrm{0}\mathrm{.75)}\hfill \\ \hfill \mathrm{90}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\times \mathrm{(0}\mathrm{.25)}\hfill \\ \hfill \mathrm{360}\mathrm{.00}& \hfill =\hfill & \mathrm{original\; price}\hfill \end{array}$$
   
   The original price of the item was $360.00.

We are asked to find the discount, and the sale price. We know the percent discount is 30%, or 0.30 in decimal form. The original price was $80.

Substituting into the percent discount formula, we find that the discount is $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}=\mathrm{0}\mathrm{.30}\times \mathrm{80}=\mathrm{24}$.

The discount is $24 on that shirt. The sale price is the original price minus the discount, so the sale price is $80 – $24 = $56.

We know the original price of the item, $240. We also know the sale price of the item, $168. From this we know the discount is $\text{\$}240-\text{\$}168=\text{\$}72$. Substituting these values into the formula $\mathrm{discount}=\mathrm{percent\; discount}\times \mathrm{original\; price}$, we can find the percent discount.

The student percent discount on the bus pass is 30%.

Using the percent discount and the sale price, we can find the original price with the formula$\mathrm{sale\; price}=\mathrm{original\; price}\times \mathrm{(1}-\mathrm{percent\; discount)}$. Substituting and solving for the original price, we find

The original price of the two tires Kendra bought was $210.00.

1. Substituting the values into the formula $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}$, we find that the markup is $\mathrm{markup}=0.15\times 62.00=9.30$. The markup is $9.30.
   The retail price of the item is then $\mathrm{retail\; price}=\mathrm{cost}+\mathrm{markup}$, or $62.00 + $9.30 = $71.30.
2. Substituting the values into the formula $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}$, we find that the markup is $\mathrm{markup}=0.45\times 750.00=337.50$. The markup is $337.50.
   The retail price of the item is then $\mathrm{retail\; price}=\mathrm{cost}+\mathrm{markup}$, or $750.00 + $337.50 = $1,087.50.

1. Step 1: Using the cost and the retail price, we can find the markup with the formula $\mathrm{retail\; price}=\mathrm{cost}+\mathrm{markup}$. Substituting and calculating, we find the markup to be $103.50=90.00+\mathrm{markup}$. Solving for the markup gives $13.50.
   Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}$, we can find the percent markup.
   
   $$\begin{array}{ccc}\hfill \mathrm{13}\mathrm{.50}& \hfill =\hfill & \mathrm{percent\; markup}\times 90.00\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\mathrm{13}\mathrm{.50}}{90.00}}}}& \hfill =\hfill & \mathrm{percent\; markup}\hfill \\ \hfill 0.15& \hfill =\hfill & \mathrm{percent\; markup}\hfill \end{array}$$
   
   Converting to percent form, the percent markup is 15%.
   

2. Step 1: Using the cost and the retail price, we can find the markup with the formula $\mathrm{retail\; price}=\mathrm{cost}+\mathrm{markup}$. Substituting and calculating, we find the markup to be $9.90=5.20+\mathrm{markup}$. Solving for the markup gives $4.70.
   Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}$, we can find the percent markup.
   
   $$\begin{array}{ccc}\hfill \mathrm{4}\mathrm{.70}& \hfill =\hfill & \mathrm{percent\; markup}\times 5.20\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{4.70}{5.20}}}}& \hfill =\hfill & \mathrm{percent\; markup}\hfill \\ \hfill 0.9038& \hfill =\hfill & \mathrm{percent\; markup}\hfill \end{array}$$
   
   Converting to percent form, the percent markup is 90.38%.

1. Using the percent markup and the retail price, we can find the cost with the formula $\mathrm{retail\; price}=\mathrm{cost}\times \mathrm{(1}+\mathrm{percent\; markup)}$. Substituting and solving for the cost, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{retail\; price}& \hfill =\hfill & \mathrm{cost}\times \mathrm{(1}+\mathrm{percent\; markup)}\hfill \\ \hfill \mathrm{10}\mathrm{.62}& \hfill =\hfill & \mathrm{cost}\times (1+0.2)\hfill \\ \hfill \mathrm{10}\mathrm{.62}& \hfill =\hfill & \mathrm{cost}\times \mathrm{(1}\mathrm{.2)}\hfill \\ \hfill \mathrm{8}\mathrm{.85}& \hfill =\hfill & \mathrm{cost}\hfill \end{array}$$
   
   The cost of the item was $8.85.
   

2. Using the percent markup and the retail price, we can find the cost with the formula$\mathrm{retail\; price}=\mathrm{cost}\times \mathrm{(1}+\mathrm{percent\; markup)}$. Substituting and solving for the original price, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{retail\; price}& \hfill =\hfill & \mathrm{cost}\times \mathrm{(1}+\mathrm{percent\; markup)}\hfill \\ \hfill \mathrm{26}\mathrm{.55}& \hfill =\hfill & \mathrm{cost}\times \mathrm{(1+2}\mathrm{.25)}\hfill \\ \hfill \mathrm{26}\mathrm{.55}& \hfill =\hfill & \mathrm{cost}\times \mathrm{(3}\mathrm{.25)}\hfill \\ \hfill \mathrm{11}\mathrm{.80}& \hfill =\hfill & \mathrm{cost}\hfill \end{array}$$
   
   The cost of the item was $11.80.

We are asked to find the retail price. We know the percent markup is 60%. The cost of the bar was $1.30. Substituting into the percent markup formula, we find that the markup is $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}=0.60\times 1.30=0.78$ . The markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is $\mathrm{retail\; price}=\mathrm{cost}+\mathrm{markup}=1.30+0.78=2.08$. The retail price is $2.08.

We know the cost of the phone, $480. We also know the retail price of the phone, $840.00. From this we know the markup is $\text{\$}840.00-\text{\$}480.00=\text{\$}360.00$. Substituting these values into the formula $\mathrm{markup}=\mathrm{percent\; markup}\times \mathrm{cost}$, we can find the percent markup.

The markup on the phone is 75%.

Using the percent markup and the retail price, $29.50, we can find the cost with the formula $\mathrm{retail\; price}=\mathrm{original\; price}\times \mathrm{(1}+\mathrm{percent\; markup)}$. Substituting and solving for cost we find

The cost of the t-shirt was $25.00.

1. The sales tax is found using $\mathrm{sales\; tax}=\mathrm{purchase\; price}\times \mathrm{tax\; rate}$. The purchase price is $428.99 and the tax rate is 8.25%. Substituting and calculating, the sales tax is $\mathrm{sales\; tax}=\text{\$}428.99\times 0.0825=\text{\$}35.391675$. The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 1, we round down and the sales tax is $35.39. The total price is the sales tax plus the purchase price, so is $\text{\$}428.99+\text{\$}35.88=\text{\$}464.87$.
2. The sales tax on the item is found using $\mathrm{sales\; tax}=\mathrm{purchase\; price}\times \mathrm{tax\; rate}$. The purchase price is $34.88 and the tax rate is 8.25%. Substituting and calculating, the sales tax is $\mathrm{sales\; tax}=\text{\$}34.88\times 0.0825=\text{\$}2.8776$. The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 7, we round up and the sales tax is $2.88. The total price of the item is the sales tax plus the purchase price, so is $\text{\$}34.88+\text{\$}2.88=\text{\$}37.76$.

1. Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $24.71.
   Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula $\mathrm{sales\; tax}=\mathrm{purchase\; price}\times \mathrm{tax\; rate}$, the sales tax rate can be found. Substituting and solving yields
   
   $$\begin{array}{ccc}\hfill \mathrm{Sales\; Tax}& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{tax\; rate}\hfill \\ \hfill \text{\$}24.71& \hfill =\hfill & \text{\$}329.50\times \mathrm{tax\; rate}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\text{\$}24.71}{\text{\$}329.50}}}}& \hfill =\hfill & \mathrm{tax\; rate}\hfill \\ \hfill \mathrm{0}\mathrm{.07499}& \hfill =\hfill & \mathrm{tax\; rate}\hfill \end{array}$$
   
   Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%.
   

2. Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $0.62.
   Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula $\mathrm{sales\; tax}=\mathrm{purchase\; price}\times \mathrm{tax\; rate}$, the sales tax rate can be found. Substituting and solving yields
   
   $$\begin{array}{ccc}\hfill \mathrm{sales\; tax}& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{tax\; rate}\hfill \\ \hfill \text{\$}0.62& \hfill =\hfill & \text{\$}13.77\times \mathrm{tax\; rate}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\text{\$}0.62}{\text{\$}13.77}}}}& \hfill =\hfill & \mathrm{tax\; rate}\hfill \\ \hfill \mathrm{0}\mathrm{.04503}& \hfill =\hfill & \mathrm{tax\; rate}\hfill \end{array}$$
   
   Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%.

1. When the sales tax rate and the total price are known, the formula $\mathrm{total\; price}=\mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{tax\; rate)}$ can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{Total\; price}& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{tax\; rate)}\hfill \\ \hfill \text{\$}36.56& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{0}\mathrm{.0575)}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\text{\$}36.56}{1.0575}}}}& \hfill =\hfill & \mathrm{purchase\; price}\hfill \\ \hfill \text{\$}34.57& \hfill =\hfill & \mathrm{purchase\; price}\hfill \end{array}$$
   
   The purchase price, the price before tax, was $34.57.
   

2. When the sales tax rate and the total price are known, the formula $\mathrm{total\; price}=\mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{tax\; rate)}$ can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find
   
   $$\begin{array}{ccc}\hfill \mathrm{Total\; price}& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{tax\; rate)}\hfill \\ \hfill \text{\$}97.17& \hfill =\hfill & \mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{0}\mathrm{.0425)}\hfill \\ \hfill {\displaystyle {\displaystyle {\displaystyle \frac{\text{\$}97.17}{1.0425}}}}& \hfill =\hfill & \mathrm{purchase\; price}\hfill \\ \hfill \text{\$}93.21& \hfill =\hfill & \mathrm{purchase\; price}\hfill \end{array}$$
   
   The purchase price, the price before tax, was $93.21.

The sales tax rate in Denver is 8.81%. To find the sales tax Keven will pay, find 8.81% of the purchase price. In decimal form, that sales tax rate is 0.0881. Using the formula and substituting 499.00 for purchase price, we find that Keven will pay $\mathrm{purchase\; price}\times \mathrm{tax\; rate}=\mathrm{\$499}\times 0.0881=\text{\$}43.96$ in sales tax for the TV.

The total price that Keven will pay is the purchase price plus the sales tax, or $\text{\$}499.00+\text{\$}43.96=\text{\$}542.96$.

The sales tax paid for this purchase is the difference in the total price and the purchase price. We know the total price is $467.64. We also know the sales tax rate, which is 8.25%. In decimal form, this is 0.0825. Using these values and the formula $\mathrm{total\; price}=\mathrm{purchase\; price}\times \mathrm{(1}+\mathrm{tax\; rate)}$ to find the purchase price.

Knowing both the total price and the now the purchase price, we can find the difference, which is the sales tax.

The total price was $467.64. The purchase price was $432. The difference of the total price and the purchase price, or the sales tax, is then $467.64 − $432.00, which is $35.64. Jillian pays $35.64 in sales tax.