Given Sale Solve For X

Introduction

When you walk past a store window and see a bright “20 % off” sign, you might instinctively ask yourself, “What was the original price?” or “How much did I actually save?” In many math textbooks and real‑world business contexts, these questions are phrased as “given sale, solve for x.” The phrase solve for x simply means: find the unknown value that makes the equation true. In a sale problem, x often represents the original price, the discount amount, the final sale price, or any other quantity that is hidden behind the discount tag.

Understanding how to solve for x in sale‑related word problems is more than a classroom exercise—it equips you with a practical tool for budgeting, pricing strategies, and everyday financial decisions. Whether you’re a student preparing for an algebra test, a small‑business owner setting promotional rates, or a consumer trying to decode a clearance rack, the ability to translate a percentage discount into a solvable equation is invaluable.

This article will walk you through every facet of the topic, from the basic algebraic concepts that underpin given sale solve for x problems to step‑by‑step methods, real‑world examples, theoretical insights, and common pitfalls. By the end, you’ll not only know how to set up and solve these equations but also appreciate why mastering them can boost both your academic performance and your everyday financial literacy.

Detailed Explanation

What “Given Sale” Means in Algebra

A sale in the mathematical sense is a reduction applied to a listed price, usually expressed as a percentage. The phrase given sale tells you that the discount information is already provided; you don’t have to infer it from context. The unknown you need to find—x—could be the original price before the discount, the amount of money saved, or the final price after the discount, depending on how the problem is framed.

For example, a typical problem might read:

“A jacket is on sale for $78 after a 15 % discount. Find the original price x.”

Here, the **sale

Here, the sale price is $78, and the discount is 15%. To find the original price x, we recognize that the sale price equals the original price minus the discount amount. The discount amount is 15% of x, or 0.15x. Therefore, the equation becomes:

[ x - 0.15x = 78 ]

Simplifying the left side gives (0.85x = 78). Dividing both sides by 0.85 yields (x \approx 91.76). Thus, the original price was about $91.76, and the customer saved $13.76.

Generalizing the Approach

While the unknown x often represents the original price, it can also be the discount rate, the discount amount, or even the sale price itself. The key is to correctly identify what quantity the problem asks for and then relate it to the known values using the fundamental relationship:

[ \text{Sale Price} = \text{Original Price} - \text{Discount} ]

or equivalently,

[ \text{Sale Price} = \text{Original Price} \times (1 - \text{Discount Rate}) ]

Case 1: Solving for the Original Price (x)

If the sale price and discount rate are known, rearrange the second formula:

[ x = \frac{\text{Sale Price}}{1 - \text{Discount Rate}} ]

Example: A smartphone is advertised for $399 after a 20% discount. The original price is (399 / (1 - 0.20)