Problem 57 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Sale Price Patty paid \(\$ 35\) for a purse on sale for \(\$ 10\) off the original price. What was the original price of the purse?

Step-by-Step Solution

Answer

The original price was \(\$ 45\).

Step 1: Understand the Problem

Recognize that Patty bought a purse on sale. She paid \(\$ 35\). This amount represents the discounted price.

Step 2: Identify the Discount

Notice that the purse was on sale for \(\$ 10\) off the original price. This discount is subtracted from the original price to get the sale price.

Step 3: Set Up the Equation

Let the original price of the purse be \(x\). According to the problem, \(x - 10 = 35\).

Step 4: Solve the Equation

Add \(10\) to both sides of the equation to solve for \(x\).\(x - 10 + 10 = 35 + 10\),\(x = 45\).

Step 5: Verify the Solution

Substitute \(x\) back into the context of the problem to check: \(45 - 10\) should equal \(35\). This verifies the solution.

Key Concepts

discount calculation

When dealing with discounts, we need to understand the original price and the amount deducted from it. In Patty's case, the purse she bought had a \(10 discount. This means that \)10 was subtracted from the original price to get the sale price. To find the original price, we add the discount back to the sale price. Using this method simplifies the calculation and helps in understanding how discounts affect prices.

setting up equations

A crucial part of solving word problems in algebra involves setting up an equation. For Patty's problem, we need to translate the words into a mathematical expression. Let’s use the variable \(x\) to represent the original price of the purse. The problem tells us that the sale price (\(35) is the original price minus the discount (\)10). So, we can set this up as follows:

• \( x - 10 = 35 \)

By creating an equation, we can then solve for \(x\). This involves performing algebraic operations to isolate the variable, making the problem more manageable.

verifying solutions

After solving the equation, it’s important to verify the solution to ensure it’s correct. We found the original price of Patty's purse to be \$45. Verification means substituting this value back into the original context of the problem. Let’s check:

• Original price: \$45
• Discount: \$10

Subtract the discount from the original price to get the sale price:
\(45 - 10 = 35\). Since this matches the given sale price, our solution is verified and correct. Verification helps confirm that our answer makes sense in the context of the problem.