Finding the Original Price from a Sale Price | Grade 7 Math | LevelKite

You already know how to start with an original price, take a percent off, and find the sale price. Now you will go the other way. A tag says a jacket costs 48 dollars after 20% off. What did it cost before the sale?

To answer, you need to know what the 20% measures. A percent is a rate per 100 of a stated whole, the amount you take the percent of, and that whole must be a positive amount. The discount is the amount a store takes off a price. For a discount, the whole is the original price (the price before the sale), never the sale price. So 20% off means the store takes off 20 dollars for every 100 dollars of the original price.

Think about what is left after the discount. The original price is 100% of itself. When the store takes off 20% of the original price, the part left is 100% - 20% = 80% of the original price. That 80% is the sale price (the price you pay after the discount).

Now 80% means 80 per 100, which is 0.80 as a decimal. So when the discount is 20% of a positive original price, the sale price is 0.80 times that original price. The table shows this for four original prices.

A 20% discount, where the 20% is a percent of each original price

Original price (dollars)	Discount: 20% of original price (dollars)	Sale price (dollars)	Sale price divided by original price
$10$	$0.20 \times 10 = 2$	$10 - 2 = 8$	$8 \div 10 = 0.80$
$25$	$0.20 \times 25 = 5$	$25 - 5 = 20$	$20 \div 25 = 0.80$
$40$	$0.20 \times 40 = 8$	$40 - 8 = 32$	$32 \div 40 = 0.80$
$100$	$0.20 \times 100 = 20$	$100 - 20 = 80$	$80 \div 100 = 0.80$

In every row, the sale price divided by the original price is 0.80, just as the 80% says it must be. For a 20% discount of a positive original price, the sale price and the original price are in a proportional relationship: one is always the same number, here 0.80, times the other.

That gives you two directions. Going forward, you multiply the original price by 0.80. Going backward, you divide the sale price by 0.80, because dividing by a nonzero number undoes multiplying by that number. In the second row, $20 \div 0.80 = 25$ , which is the original price.

Now write the idea with symbols so that it works for other discounts too.

Let $P$ be the original price in dollars. The original price is a positive amount, so $P > 0$ .
Let $S$ be the sale price in dollars.
Let $r$ be the discount rate: the percent the store takes off, written as a decimal. For 20%, $r = 0.20$ . In this lesson a discount is more than 0% and less than 100% of the original price, so $0 < r < 1$ .

The discount is the rate times the original price: $r \times P$ dollars. Subtract it from the original price to get the sale price:

S = P - r \times P = (1 - r) \times P

The two forms are equal because $P$ is the same as $1 \times P$ . You have $1$ times $P$ and you take away $r$ times $P$ , so $(1 - r)$ times $P$ is left. This is the distributive property: $1 \times P - r \times P = (1 - r) \times P$ . In words, all of the original price minus the part the store takes off is the part you still pay.

With $r = 0.20$ , the part you pay is $1 - 0.20 = 0.80$ , the same 0.80 you found in the table.

Key concept

Divide the sale price by the part you pay

A percent discount is a rate per 100 of the original price, which is the positive whole. It is never a rate of the sale price. Write the discount rate as a decimal $r$ . When the original price $P$ is positive and $0 < r < 1$ , the sale price is $S = (1 - r) \times P$ . To get back to the original price, undo that multiplication: divide the sale price by $1 - r$ . You may divide because $r < 1$ makes $1 - r$ greater than $0$ , so it is not zero. Under these same conditions $1 - r$ is less than $1$ , so the original price is greater than the sale price.

P = S \div (1 - r) \qquad \text{when } P > 0 \text{ and } 0 < r < 1

Worked example

A whole-number sale price

A jacket is on sale for 48 dollars after a 20% discount. The 20% is a percent of the original price. What was the original price?

Name each quantity and its unit. Let P be the original price in dollars. The original price is positive, so P is greater than 0. The sale price S is 48 dollars. The discount rate r is 20% of the original price, which is 0.20 as a decimal.
$S = 48, \quad r = 0.20, \quad P > 0$
Find the part of the original price that you still pay. The original price is 100% of itself, and the store takes off 20% of it, so 80% is left. In decimals, subtract 0.20 from 1.
$1 - r = 1 - 0.20 = 0.80$
Write the relationship as an equation. The sale price is 0.80 times the original price, and the sale price is 48 dollars.
$0.80 \times P = 48$
Undo the multiplication by dividing both sides by 0.80. Dividing both sides of an equation by the same nonzero number keeps the two sides equal, and 0.80 is not zero. On the left, multiplying P by 0.80 and then dividing by 0.80 leaves P.
$(0.80 \times P) \div 0.80 = 48 \div 0.80, \quad \text{so } P = 48 \div 0.80$
Do the division. Multiply both 48 and 0.80 by 10 so that you divide by a whole number. Multiplying both numbers by the same nonzero number does not change the result of a division.
$48 \div 0.80 = 480 \div 8 = 60$
Check by going forward with your answer as the whole. The discount is 20% of the original price, 60 dollars. Subtract it and you get the sale price back. The answer 60 is also greater than 48, which fits a discount of more than 0% and less than 100% of a positive original price.
$0.20 \times 60 = 12, \quad 60 - 12 = 48$

Answer

The original price of the jacket was $60$ dollars.

Worked example

A sale price with cents

A backpack costs 42.50 dollars after a 15% discount. The 15% is a percent of the original price. What was the original price?

Let P be the original price in dollars, with P greater than 0. The store takes off 15% of the original price, so you still pay 85% of it. As a decimal, the part you pay is 0.85.
$1 - 0.15 = 0.85$
The sale price is 0.85 times the original price, and the sale price is 42.50 dollars.
$0.85 \times P = 42.50$
Divide both sides by 0.85. The two sides stay equal because 0.85 is not zero.
$P = 42.50 \div 0.85$
Multiply both numbers by 100, a nonzero number, so that you divide by a whole number without changing the result. Since 85 times 50 is 4250, the result is 50.
$42.50 \div 0.85 = 4250 \div 85 = 50$
Check forward with the original price, 50 dollars, as the whole. The discount is 15% of 50 dollars. Subtracting it gives the sale price.
$0.15 \times 50 = 7.50, \quad 50 - 7.50 = 42.50$

Answer

The original price of the backpack was $50$ dollars.

Worked example

Original price and amount saved

Headphones are on sale for 91 dollars. The sign says 35% off the original price. What was the original price, and how many dollars does the sale save you?

Let P be the original price in dollars, with P greater than 0. Taking off 35% of the original price leaves 65% of it, so the sale price is 0.65 times P.
$1 - 0.35 = 0.65, \quad 0.65 \times P = 91$
Divide both sides by 0.65, which is not zero, so the sides stay equal. Multiply both numbers by 100, a nonzero number, to divide by a whole number. Then 65 times 100 is 6500 and 65 times 40 is 2600, which add to 9100, so the result is 140.
$P = 91 \div 0.65 = 9100 \div 65 = 140$
The amount saved is the discount: the original price minus the sale price.
$140 - 91 = 49$
Check the percent. A percent discount is the amount taken off divided by the positive original price, never by the sale price. Dividing both 49 and 140 by 7, a nonzero number, does not change the result.
$49 \div 140 = 7 \div 20 = 0.35 = 35\%$

Answer

The original price was $140$ dollars, and the sale saves you $49$ dollars.

The big idea: a percent discount is a rate per 100 of the positive original price, never of the sale price. When the discount rate $r$ is greater than 0 and less than 1, you pay $1 - r$ of the original price, so $S = (1 - r) \times P$ . To find the original price, divide the sale price by $1 - r$ , then check by going forward from your answer. Next, you will work with simple interest.