Solve Sales Tax, Commission, and Discount Applications

Solve sales tax applications

Sales tax and commissions are applications of percent in our everyday lives. To solve these applications, we will follow the same strategy we used in the section on decimal operations.

Do you pay a tax when you shop in your city or state? In many parts of the United States, sales tax is added to the purchase price of an item. The sales tax is determined by computing a percent of the purchase price.

To find the sales tax, multiply the purchase price by the sales tax rate. Remember to convert the sales tax rate from a percent to a decimal. Once the sales tax is calculated, it is added to the purchase price. The result is the total cost — this is what the customer pays.

Example. Cathy bought a bicycle in Washington, where the sales tax rate was $6.5\%$ of the purchase price. What was (a) the sales tax and (b) the total cost of a bicycle if the purchase price of the bicycle was $\text{\textdollar}392$?

(a) Let $t =$ sales tax. The sales tax is $6.5\%$ of the purchase price: $t = 0.065 \cdot 392$. Simplify: $t = 25.48$. Check: this is reasonable because the sales tax amount is less than $10\%$ of the purchase price. The sales tax is $\text{\textdollar}25.48$.

(b) Let $c =$ total cost of the bicycle. The total cost is the purchase price plus the sales tax: $c = 392 + 25.48$. Simplify: $c = 417.48$. Check: this is reasonable because the total cost is a little more than the purchase price. The total cost of the bicycle is $\text{\textdollar}417.48$.

Example. Evelyn bought a new smartphone for $\text{\textdollar}499$ plus tax. She was surprised when she got the receipt and saw that the tax was $\text{\textdollar}42.42$. What was the sales tax rate for this purchase?

Let $r =$ sales tax rate. What percent of the price is the sales tax? $r \cdot 499 = 42.42$. Divide: $\tfrac{499r}{499} = \tfrac{42.42}{499}$. Simplify: $r = 0.085$. Check: this is reasonable, because $8.5\%$ is close to $10\%$, and $10\%$ of $\text{\textdollar}500$ is $\text{\textdollar}50$, which is close to $\text{\textdollar}42.42$. The sales tax rate is $8.5\%$.

Solve commission applications

Sales people often receive a commission, or percent of total sales, for their sales. Their income may be just the commission they earn, or it may be their commission added to their hourly wages or salary. The commission they earn is calculated as a certain percent of the price of each item they sell. That percent is called the rate of commission.

To find the commission on a sale, multiply the rate of commission by the total sales. Just as we did for computing sales tax, remember to first convert the rate of commission from a percent to a decimal.

Example. Helene is a realtor. She receives $3\%$ commission when she sells a house. How much commission will she receive for selling a house that costs $\text{\textdollar}260{,}000$?

Let $c =$ the commission. The commission is $3\%$ of the price: $c = 0.03 \cdot 260{,}000$. Simplify: $c = 7{,}800$. Check: $1\%$ of $\text{\textdollar}260{,}000$ is $\text{\textdollar}2{,}600$, and $\text{\textdollar}7{,}800$ is three times $\text{\textdollar}2{,}600$, so this is reasonable. Helene will receive a commission of $\text{\textdollar}7{,}800$.

Example. Rikki earned $\text{\textdollar}87$ commission when she sold a $\text{\textdollar}1{,}450$ stove. What rate of commission did she get?

Let $r =$ the rate of commission. The commission is what percent of the sale? $87 = r \cdot 1{,}450$. Divide: $\tfrac{87}{1{,}450} = \tfrac{1{,}450r}{1{,}450}$. Simplify: $0.06 = r$. Change to percent form: $r = 6\%$. Check: a $10\%$ commission would have been $\text{\textdollar}145$; the $6\%$ commission, $\text{\textdollar}87$, is a little more than half of that, so this is reasonable. The commission was $6\%$ of the price of the stove.

Solve discount applications

Applications of discount are very common in retail settings. When you buy an item on sale, the original price of the item has been reduced by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of discount. To determine the amount of discount, we multiply the discount rate by the original price.

The sale price should always be less than the original price. In some cases, the amount of discount is a fixed dollar amount. Then we just find the sale price by subtracting the amount of discount from the original price.

Example. Jason bought a pair of sunglasses that were on sale for $\text{\textdollar}10$ off. The original price of the sunglasses was $\text{\textdollar}39$. What was the sale price of the sunglasses?

Let $s =$ the sale price. The sale price is the original price minus the discount: $s = 39 - 10$. Simplify: $s = 29$. Check: the sale price, $\text{\textdollar}29$, is less than the original price, $\text{\textdollar}39$, so this is reasonable. The sale price of the sunglasses was $\text{\textdollar}29$.

In the example above, the amount of discount was a set amount, $\text{\textdollar}10$. Often, though, the discount is given as a percent of the original price, and we must find the amount of discount first.

Example. Elise bought a dress that was discounted $35\%$ off of the original price of $\text{\textdollar}140$. What was (a) the amount of discount and (b) the sale price of the dress?

(a) Let $d =$ the amount of discount. The discount is $35\%$ of the original price: $d = 0.35 \cdot 140$. Simplify: $d = 49$. Check: a $\text{\textdollar}49$ discount is reasonable for a $\text{\textdollar}140$ dress. The amount of discount was $\text{\textdollar}49$.

(b) Let $s =$ the sale price. The sale price is the original price minus the discount: $s = 140 - 49$. Simplify: $s = 91$. Check: the sale price, $\text{\textdollar}91$, is less than the original price, $\text{\textdollar}140$, so this is reasonable. The sale price of the dress was $\text{\textdollar}91$.

There may be times when you buy something on sale and want to know the discount rate.

Example. Jeannette bought a swimsuit at a sale price of $\text{\textdollar}13.95$. The original price of the swimsuit was $\text{\textdollar}31$. Find (a) the amount of discount and (b) the discount rate.

(a) Let $d =$ the amount of discount. The discount is the original price minus the sale price: $d = 31 - 13.95$. Simplify: $d = 17.05$. Check: the $\text{\textdollar}17.05$ discount is less than the original price, so this is reasonable. The amount of discount was $\text{\textdollar}17.05$.

(b) Let $r =$ the discount rate. The discount is what percent of the original price? $17.05 = r \cdot 31$. Divide: $\tfrac{17.05}{31} = \tfrac{r(31)}{31}$. Simplify: $0.55 = r$. Check: the rate of discount was a little more than $50\%$, and the amount of discount is a little more than half of $\text{\textdollar}31$, so this is reasonable. The rate of discount was $55\%$.

Solve mark-up applications

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the wholesale price. The retailer then adds a mark-up to the wholesale price to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the wholesale price. The percent is called the mark-up rate. To determine the amount of mark-up, multiply the mark-up rate by the wholesale price.

The list price should always be more than the wholesale price.

Example. Adam’s art gallery bought a photograph at the wholesale price of $\text{\textdollar}250$. Adam marked the price up $40\%$. Find (a) the amount of mark-up and (b) the list price of the photograph.

(a) Let $m =$ the amount of mark-up. The mark-up is $40\%$ of the wholesale price: $m = 0.40 \cdot 250$. Simplify: $m = 100$. Check: the mark-up rate is less than $50\%$ and $\text{\textdollar}100$ is less than half of $\text{\textdollar}250$, so this is reasonable. The mark-up on the photograph was $\text{\textdollar}100$.

(b) Let $p =$ the list price. The list price is the wholesale price plus the mark-up: $p = 250 + 100$. Simplify: $p = 350$. Check: the list price, $\text{\textdollar}350$, is more than the wholesale price, $\text{\textdollar}250$, so this is reasonable. The list price of the photograph was $\text{\textdollar}350$.

Key terms

commission — a percentage of total sales, as determined by the rate of commission; $\text{commission} = \text{rate of commission} \cdot \text{total sales}$. discount rate — the percent used to determine the amount of discount off an original price. mark-up — the amount a retailer adds to the wholesale price of an item to reach the list price; $\text{list price} = \text{wholesale price} + \text{mark-up}$.

This section is adapted from Prealgebra 2e, Section 6.3: Solve Sales Tax, Commission, and Discount Applications by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Self Check checklist, the restaurant-receipt and sale-window photos, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.