Problem 95 - College Algebra

Question

Your computer store is having an incredible sale. The price on one model is reduced by \(40 \% .\) Then the sale price is reduced by another \(40 \% .\) If \(x\) is the computer's original price, the sale price can be represented by $$(x-0.4 x)-0.4(x-0.4 x).$$ a. Factor out \((x-0.4 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions: With a \(40 \%\) reduction followed by a \(40 \%\) reduction, is the computer selling at \(20 \%\) of its original price? If not, at what percentage of the original price is it selling?

Step-by-Step Solution

Answer

After the two 40% reductions, the computer is not selling at 20% of its original price, but at 36% of the original price.

Step 1: Factoring

First, factor out \((x-0.4x)\) from each term. This gives: \[(x-0.4x) - 0.4(x-0.4x) = (1-0.4)x - 0.4(1-0.4)x.\] So, the equation simplifies to \((0.6)x - 0.4(0.6)x\).

Step 2: Simplification

Then simplify this to obtain the sale price of the computer as a percentage of the original price. This gives:\[(0.6)x - 0.4(0.6)x = (0.6)x - (0.24)x = (0.36)x.\]

Step 3: Computing the Percentage of Original Price

The final sale price of the computer, \((0.36)x\), represents 36% of the original price, because 0.36 is the decimal equivalent of 36%.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is key to solving problems like the one presented. An algebraic expression is a mathematical phrase that includes numbers, variables (like \(x\)), and operational symbols (such as +, –, ×, and ÷). In the problem,

• \(x\) represents the original price of a computer
• This expression, \((x - 0.4x) - 0.4(x - 0.4x)\), shows how the price is reduced twice.

The first part, \((x - 0.4x)\), represents the first 40% discount from the original price \(x\). To understand how this works,

• consider the expression \(0.4x\), which calculates 40% of \(x\). This part is subtracted from \(x\) to find the first reduced price.

For the second discount, the part \(-0.4(x - 0.4x)\) takes 40% off the new reduced price \((x - 0.4x)\). Carefully breaking down such expressions helps to accurately represent and understand changes in values.

Percentage Reduction

Percentage reduction is a common concept used to understand how much a value decreases. Here, you encounter a double percentage reduction.Initially, the computer's price is discounted by 40%. This leaves us with 60% of the original price, represented by \((0.6)x\). When applying the second 40% reduction to the reduced price,

• calculate 40% of the new price \((0.6)x\), which is \(0.4(0.6)x = 0.24x\).

Subtracting this value from the reduced price leaves us with \((0.36)x\), indicating that only 36% of the original price remains. The key takeaway is that percentage reductions on already reduced amounts decrease the value significantly more than a single reduction. Two 40% reductions do not equate to 80% in total, but rather result in a compounded effect, which in this example, results in just 36% of the original price left.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact form, which is easier to interpret. Starting with the expression: \((x-0.4x)-0.4(x-0.4x)\),

• initially, you factor out the \((x-0.4x)\) part leading to: \((0.6)x - 0.4(0.6)x\).
• You then simplify further by performing the multiplication \(0.4 \times 0.6\) to get \(0.24\), yielding: \((0.6)x - (0.24)x\).

Finally, subtract to find the remaining percentage: \((0.36)x\). Hence, the expression \((0.36)x\) shows the final simplified sale price as a percentage of the original.This exercise demonstrates simplifying complex expressions by factoring, multiplying, and subtracting step-by-step, helping ensure clarity and correctness in reaching the solution.