Math Core

Almost every price you pay has a percent hiding in it. Stores mark up what they buy, then run sales. The government adds sales tax. At a restaurant, you add a tip. Each one is a percent increase or decrease with its own name.

Four names for percent change

Situation	What happens	Direction
Markup	A store raises the price it paid to get its selling price	increase
Discount	A store lowers a price during a sale	decrease
Sales tax	A percent of the price is added at the register	increase
Tip (gratuity)	You add a percent of a restaurant bill for good service	increase

You already know how to handle all of these. For each one, you can either find the dollar amount and add or subtract it, or use a one-step multiplier.

Two ways to find a final price

Two steps: find the amount (percent $\times$ price), then add it (markup, tax, tip) or subtract it (discount).

One step: multiply the price by $1 + \text{rate}$ for an increase, or by $1 - \text{rate}$ for a discount.

Markup and discount

Worked example: Markup

A shoe store pays $48 for a pair of sneakers and marks them up $75\%$ . What is the selling price?

Markup: $0.75 \times 48 = 36$ dollars.

Selling price: $48 + 36 = 84$ dollars.

One step: $1.75 \times 48 = 84$ . The selling price is $84.

Worked example: Discount

A jacket costs $65. It is on sale for $30\%$ off. What is the sale price?

Discount: $0.30 \times 65 = 19.50$ dollars.

Sale price: $65 - 19.50 = 45.50$ dollars.

One step: you pay $100\% - 30\% = 70\%$ of the price, and $0.70 \times 65 = 45.50$ . The sale price is $45.50.

Tax and tips

Sales tax and tips work just like a markup: they add a percent of the price. Money is rounded to the nearest cent (hundredth of a dollar).

Worked example: Tax

A video game costs $59.99, and the sales tax rate is $7\%$ . What is the total cost?

Tax: $0.07 \times 59.99 = 4.1993$ , which rounds to $4.20.

Total: $59.99 + 4.20 = 64.19$ . The total cost is $64.19.

Tip

To estimate a $15\%$ or $20\%$ tip in your head, find $10\%$ first by moving the decimal point one place left. For a $46 bill, $10\%$ is $4.60. Double it for $20\%$ ($9.20), or add half of it for $15\%$ ($4.60 + $2.30 = $6.90).

Several percents in a row

When there's more than one percent, apply them one at a time, in order. Each percent is taken of the amount at that step.

Worked example: Discount, then tax

A lamp costs $80. It's $25\%$ off, and then $6\%$ sales tax is added to the sale price. What do you pay?

Sale price: $0.75 \times 80 = 60$ dollars.

With tax: $1.06 \times 60 = 63.60$ dollars.

You pay $63.60.

Common mistake

You can't just combine the percents. A $25\%$ discount followed by $6\%$ tax is not the same as a $19\%$ discount. Here, $0.81 \times 80 = 64.80$ , which is not $63.60. The tax is a percent of the sale price, not the original price.

Finding the rate

To find a markup, discount, tax or tip rate, use percent change: divide the dollar amount by the original price.

Worked example: Find the tip rate

The Garcia family's dinner bill was $72. They left a tip of $12.96. What percent tip did they leave?

\frac{12.96}{72} = 0.18 = 18\%

They left an 18% tip.