Discount

Sommaire

In short: A discount is the amount subtracted from an original price during a sale, and it is usually given as a percent of that original price.

7th GradeRatios and Percents7.RP.A.3

What is Discount?

When a store runs a sale, the money it takes off the regular price is the discount. A sign that reads 35% off tells you the reduction is 35 hundredths of the tag price.

With original price $P$ and discount rate $r$ as a decimal, $\text{discount} = P \times r$ and $\text{sale price} = P – P \times r = P(1 – r)$. The percent is measured against the original price, never against the sale price.

How to spot it

A sign or ad says percent off, sale, clearance, coupon, or markdown next to a regular price.
The problem asks what you will pay after a reduction, or how much money you save.
You are given the regular price and the sale price and asked for the percent that was taken off.
Two different offers, such as 30% off versus a flat coupon, must be compared in dollars.

The method

Change the percent to a decimal by dividing by $100$.
Multiply the original price by that decimal to get the savings in dollars.
Subtract the savings from the original price, or compute $P \times (1 – r)$ directly.
Check that the sale price is lower than the original price and that savings plus sale price equals the original.

A 30% discount on $85.00 saves $25.50 and leaves a $59.50 sale price.

Discount: solved example

Problem

A hiking backpack is tagged $\$85.00$ and a weekend sale takes 30% off. How much do you save, and what is the sale price?

Solution

Savings: $85.00 \times 0.30 = 25.50$ dollars.

Sale price: $85.00 – 25.50 = 59.50$. You save $\$25.50$ and pay $\$59.50$.

Try it

Question 1

Headphones normally cost $\$120$ and are on sale for $\$93$. What is the discount rate?

Show the answer

Savings are $120 – 93 = 27$. Then $\frac{27}{120} = 0.225$, so the discount is 22.5%.

Question 2

A $\$64$ jacket has 15% off, and then a $\$5$ coupon is applied. What is the final price?

Show the answer

$64 \times 0.85 = 54.40$, then $54.40 – 5 = 49.40$, so the final price is $\$49.40$.

Discount vs. Markup

A discount shrinks the tag price and is figured on the tag price. A markup grows the seller’s cost and is figured on the cost. Same arithmetic shape, opposite direction and different starting amounts, so you cannot swap one for the other when the base number changes.

Markup

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