Tax, Tips, Commission, Discounts and Markups | Grade 7 Math | LevelKite

A price tag says 50 dollars, but at the register you pay 54 dollars. The extra 4 dollars is not random. It is 8% of the price.

A percent is a rate per 100 of a stated positive whole. The whole is the amount the percent refers to. Here the whole is the price, 50 dollars. A rate of 8% means 8 dollars for every 100 dollars of the whole. Because 50 dollars is half of 100 dollars, the extra amount is half of 8 dollars, which is 4 dollars.

So a percent never stands alone. Before you compute, ask: percent of what? In this lesson every amount is in dollars, and each problem tells you which positive amount is the whole.

Key concept

Every percent has a whole

A percent is a rate per 100 of a stated positive whole. Let $W$ be the whole in dollars, a number greater than 0. Let $p$ be the number in the percent rate, so a rate of 8% has $p = 8$ . In this lesson $p \ge 0$ . The percent amount $A$ , in dollars, is $p$ hundredths of $W$ .

Why multiply by $\frac{p}{100}$ ? One hundredth of the whole is $\frac{W}{100}$ , and the rate takes $p$ of those hundredths.

What you do with $A$ next depends on the situation. You may add it to $W$ , subtract it from $W$ (only when $p \le 100$ , so the result is not negative), or report it by itself.

A = \dfrac{p}{100} \times W

Three everyday percent amounts use this idea: sales tax, tips and commission. The table below tells you what each one is. As you read each row, find the whole first.

For a fixed rate, the percent amount is proportional to the whole. That means the amount is always the whole times the same number, $\frac{p}{100}$ . At a rate of 8%, a whole of 50 dollars gives 4 dollars. A whole twice as large, 100 dollars, gives twice as much, 8 dollars.

Three percent amounts and their wholes. Each whole is a positive dollar amount.

Amount	What it is	Its whole (percent of what?)	Example
Sales tax	Money a buyer pays to the government, added on top of a price	The price before tax	8% tax on a price of 50 dollars: $0.08 \times 50 = 4$ , so the tax is 4 dollars
Tip	Extra money a customer adds to a bill to thank someone for a service	The bill the problem names, such as the bill before tax	20% tip on a bill of 30 dollars: $0.20 \times 30 = 6$ , so the tip is 6 dollars
Commission	Pay a salesperson earns for making sales; it is not added to the buyer's bill	The total dollar amount of the sales	5% commission on sales of 800 dollars: $0.05 \times 800 = 40$ , so the commission is 40 dollars

Worked example

Sales tax and a tip on a meal

The menu price of your meal is 40 dollars. Sales tax is 6% of the menu price. You also leave a tip of 15% of the menu price. What is the total amount you pay?

Name the whole for each percent. The problem says the tax and the tip are both percents of the menu price, so the whole for both is 40 dollars, a positive amount.
$W = 40$
Find the sales tax. Write 6% as 6/100 = 0.06, because percent means per 100. Multiply by the whole to take 6 hundredths of 40 dollars: 6 times 40 is 240, and 240 divided by 100 is 2.40. The tax is 2.40 dollars.
$\dfrac{6}{100} \times 40 = 0.06 \times 40 = 2.40$
Find the tip the same way. Write 15% as 15/100 = 0.15 and multiply by the same whole: 15 times 40 is 600, and 600 divided by 100 is 6. The tip is 6.00 dollars.
$\dfrac{15}{100} \times 40 = 0.15 \times 40 = 6.00$
Add both amounts to the menu price. The tax and the tip are extra amounts you pay on top of the price, so the total is the menu price plus the tax plus the tip.
$40 + 2.40 + 6.00 = 48.40$
Check. Both rates have the same whole, 40 dollars, so you may add them: 6% + 15% = 21% of 40 dollars. The distributive property allows this: 0.06 times 40 plus 0.15 times 40 equals (0.06 + 0.15) times 40. Then 21 times 40 is 840, and 840 divided by 100 is 8.40, which matches 2.40 + 6.00. The total is again 48.40 dollars, more than 40 dollars, as it must be when the rates are above 0%.
$0.21 \times 40 = 8.40, \quad 40 + 8.40 = 48.40$

Answer

You pay a total of 48.40 dollars: 40 dollars for the meal, 2.40 dollars of sales tax and a tip of 6.00 dollars.

Now look at prices that change. A discount is an amount taken off a price. Stores often write a discount rate as a percent off. The price before the discount is the original price. The price you pay after the discount is the sale price.

A markup is an amount a store adds to its cost, which is what the store paid for an item. The cost plus the markup is the store's selling price.

Which amount is the whole? A discount lowers a price, so it is a percent decrease. A markup turns a cost into a higher selling price, so it is a percent increase. You already know that a percent of change is measured against the positive original amount, never the new amount. So the whole for a discount is the original price, never the sale price. The whole for a markup is the cost, never the selling price.

The rates have limits. A discount rate is above 0% and at most 100%. That keeps the discount from being more than the original price, so the sale price is less than the original price and never negative. A markup rate is above 0%, so the selling price is more than the cost.

Discount and markup. Each whole is a positive dollar amount.

Change	Its whole (percent of what?)	How to get the new price	Example
Discount	The original price	Subtract the discount from the original price to get the sale price	20% off an original price of 50 dollars: $0.20 \times 50 = 10$ and $50 - 10 = 40$ , so the sale price is 40 dollars
Markup	The store's cost	Add the markup to the cost to get the selling price	50% markup on a cost of 10 dollars: $0.50 \times 10 = 5$ and $10 + 5 = 15$ , so the selling price is 15 dollars

Worked example

From a discount to a sale price

A jacket has an original price of 60 dollars. It is on sale for 35% off the original price. What is the sale price?

Name the whole. The discount is a percent of the original price, so the whole is 60 dollars, a positive amount. The rate, 35%, is above 0% and at most 100%, so the sale price must be less than 60 dollars and not negative.
$W = 60$
Find the discount. Write 35% as 35/100 = 0.35 and multiply by the whole to take 35 hundredths of 60 dollars: 35 times 60 is 2100, and 2100 divided by 100 is 21. The discount is 21 dollars.
$0.35 \times 60 = 21$
Subtract the discount from the original price, because a discount is taken off the price. What is left is the sale price, 39 dollars. It is less than 60 dollars and not negative, as step 1 said it must be.
$60 - 21 = 39$
Check another way. The original price is 100% of itself, and the discount is 35% of that same whole, so you may subtract the rates: 100% minus 35% leaves 65% of the original price. By the distributive property, 60 minus 0.35 times 60 equals (1 - 0.35) times 60. Then 65 times 60 is 3900, and 3900 divided by 100 is 39, the same sale price.
$(1 - 0.35) \times 60 = 0.65 \times 60 = 39$

Answer

The sale price of the jacket is 39 dollars, which is 21 dollars less than the original price of 60 dollars.

Worked example

Markup, then discount, then commission

A skate shop pays 80 dollars for a skateboard. That amount is the shop's cost. The shop marks up the cost by 25% to set its selling price. Later the board goes on sale for 25% off that selling price. The salesperson earns a 4% commission on the sale price. Find the sale price and the commission.

Find the markup. Its whole is the cost, 80 dollars, a positive amount. Write 25% as 0.25: 25 times 80 is 2000, and 2000 divided by 100 is 20. The markup is 20 dollars.
$0.25 \times 80 = 20$
Add the markup to the cost, because a markup raises the price above the cost. The selling price is 100 dollars.
$80 + 20 = 100$
Find the discount. The selling price is the original price for this change, so the whole is now 100 dollars, not the cost. Then 25 times 100 is 2500, and 2500 divided by 100 is 25. The discount is 25 dollars.
$0.25 \times 100 = 25$
Subtract the discount from the selling price to get the sale price. The rate, 25%, is above 0% and at most 100%, so the sale price is less than 100 dollars and not negative.
$100 - 25 = 75$
Find the commission. Its whole is the sale price, 75 dollars. Write 4% as 0.04: 4 times 75 is 300, and 300 divided by 100 is 3. The commission is the salesperson's pay, so you report it by itself and do not add it to the price.
$0.04 \times 75 = 3$

Answer

The sale price of the skateboard is 75 dollars, and the salesperson earns a commission of 3 dollars.

Here is the idea to keep: every percent is a rate per 100 of a stated positive whole, so always ask: percent of what? Multiply the whole by $\frac{p}{100}$ to get the percent amount. Then the situation tells you the last step. Add a sales tax, a tip or a markup to its whole. Subtract a discount, with a rate of at most 100%, from the original price to get the sale price. Report a commission by itself as the seller's pay.

Next, you will work backward and find the original price when you are given a sale price.