Finding Original Price from Sale Price
Finding the original price from a sale price is a Grade 7 percent application skill in Big Ideas Math, Course 2. When an item is discounted by a percent, the sale price equals the original price multiplied by (1 − discount rate). To work backwards: divide the sale price by (1 − discount rate). For a 25% discount where the sale price is $60, the original price is 60 ÷ 0.75 = $80. This reverses the percent-decrease formula and requires careful interpretation: the sale price is already a fraction of the original, not the discount amount alone. The skill applies to tax, tip, and markup problems using the same inverse approach.
Key Concepts
When given a sale price and discount percentage, the original price can be found using: $$\text{original price} = \frac{\text{sale price}}{\text{remaining percentage as decimal}}$$.
Where remaining percentage = $100\% \text{discount percentage}$.
Common Questions
How do you find the original price if you know the sale price and discount percent?
Divide the sale price by (1 − discount rate as a decimal). For a 25% discount: original = sale price ÷ 0.75.
If an item sells for $60 after a 25% discount, what was the original price?
The sale price is 75% of the original. Divide: $60 ÷ 0.75 = $80. The original price was $80.
Why do you use (1 − discount rate) in this calculation?
A 25% discount means the buyer pays 100% − 25% = 75% of the original price. The sale price is that 75%, so dividing by 0.75 reverses the discount.
How do you find the original price if a 15% tax brings the total to $46?
The total is 115% of the original. Divide: $46 ÷ 1.15 = $40. The original price was $40.
What equation models the relationship between original price, discount, and sale price?
Sale price = Original price × (1 − discount rate). To find original price: Original price = Sale price ÷ (1 − discount rate).
What common mistake do students make when finding the original price?
Subtracting the discount percent from the sale price directly, rather than dividing the sale price by the multiplier factor (1 − rate).