Problem 116 - Algebra and Trigonometry

Question

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be modeled by original price, the sale price cafe $$(x-0.3 x)-0.3(x-0.3 x)$$

Step-by-Step Solution

Answer

The final sale price of the plasma television can be calculated using the simplified formula derived in Step 3.

Step 1: Reduction of Original Price

The initial model for the price reduction is \(x - 0.3x\). Here, 'x' is the original price of the television and '0.3x' is the value depicting the initial 30% reduction. By subtracting 0.3x from 'x', we obtain the price after the first discount.

Step 2: Second Price Reduction

The model for the second discount is \((x - 0.3x) - 0.3(x - 0.3x)\). Here, '0.3(x - 0.3x)' represents the additional 30% discount applied on the already discounted price. By subtracting this value from the price after the first discount, we can derive the final sale price.

Step 3: Simplify the Expression

By simplifying the expression obtained in the previous step, we can get the formula for the final sale price in terms of the original price 'x'. This formula will allow us to calculate the sale price for any given original price.

Key Concepts

Discount Calculation

Understanding how discount calculations work is essential when shopping during sales. Discounts are usually given as a percentage, which means a part of the total original price is subtracted. For example, in our exercise, the first discount is 30%. This is calculated as 30% of 'x', the original price, which is represented mathematically as \(0.3x\).

• The first step is to subtract this discount from the original price: \(x - 0.3x\).
• Think of this as calculating 70% of the price, since removing 30% leaves 70% of the original price still to be paid.

Discounts can be applied in series, as shown in the example where a second 30% discount follows the first. This leads to more savings but requires recalculating the discount amount based on the new, already reduced price.

Price Reduction

A price reduction is simply the decrease in the original price, often expressed as a percentage. The key to understanding it is to recognize the base price before each reduction. Each reduction should be calculated successively.

• Initially, the original price 'x' is reduced by 30%, leaving 70% of the original price.
• Then, the new reduced price, which is \(0.7x\), is further reduced by another 30%.
• This second reduction is calculated on \(0.7x\), giving us \(0.3 \times 0.7x = 0.21x\).

Subtract this second discount from the already reduced price to find the final sale price: \(0.7x - 0.21x = 0.49x\).
Two successive 30% reductions do not add up to a total of 60%, but result in a cumulative effect, as each reduction is calculated on a different amount.

Simplifying Expressions

Simplifying algebraic expressions makes it easier to understand and work with pricing calculations. In our exercise, the main expression involves subtracting percentages twice. When we have \( (x - 0.3x) - 0.3(x - 0.3x) \), we perform operations step by step to simplify it.

• Firstly, solve the inner operation: \(x - 0.3x\) results in \(0.7x\).
• Next, calculate the second reduction: \(0.3 \times 0.7x = 0.21x\).
• Subtract \(0.21x\) from \(0.7x\) to simplify the entire expression to \(0.49x\).

This final expression \(0.49x\) tells us that after two 30% reductions, the remaining price is 49% of the original price. Simplifying expressions not only helps with accuracy but also provides a clear understanding of the overall discount effect on the price. This technique allows you to handle financial calculations effectively and confidently.