Problem 47 - Precalculus Essentials

Question

After a \(30 \%\) price reduction, you purchase a television for \(\$ 980 .\) What was the television's price before the reduction?

Step-by-Step Solution

Answer

The original price of the television before the 30% discount was $1400.

Step 1: Understand the percentage given

The television was purchased at a 30% price reduction, this translates to paying for 70% of the original price which is $980.

Step 2: Calculate the Original Price

If 70% represents $980, we have to find the value of 100% which represents the original price. We have to apply the formula for percentages: original price is equal to (cost after discount divided by the percent paid) times 100. The formula can be written as follows: original price = (980 divided by 70) times 100.

Step 3: Perform the calculation

We do the calculation by dividing 980 by 70 which gives us 14, then multiply 14 by 100, the result of this calculation gives us 1400. So, before the discount the television cost $1400.

Key Concepts

Understanding Price Reductions

In retail, a price reduction is a decrease in the original price of a product. This is often expressed as a percentage which makes it essential to understand how this percentage impacts the final price you pay. When a television is sold with a 30% reduction, it means you save 30% off the original cost. Therefore, you pay only 70% of the original price.

To put it simply:

• The original price starts at 100%.
• A 30% reduction leaves 70% to be paid.

Understanding these simple reductions will help you make informed purchasing decisions and be aware of the actual cost savings being offered.

Original Price Calculation

Calculating the original price before a discount involves reversing the percentage reduction calculation. This skill is useful when dealing with various discounts and sales. If you buy a television at 70% of its original price and pay \(980, you're essentially working to find out what that 70% is part of.

Here's a step-by-step approach:

• Define the equation: If 70% equals \)980, then you set up the formula to find 100%, which is the original price.
• Use the formula: \( \text{Original Price} = (\frac{\\(980}{70}) \times 100 \)
• Calculate: First divide \)980 by 70 to calculate what 1% of the original price is, getting 14.
• Multiply 14 by 100 to find the full original price, resulting in $1,400.

This method enables you to quickly determine the original price of discounted items by working backwards from the final purchase price.

Precalculus Problem Solving

Working with percentages in price calculation is a common precalculus problem that requires analytical thinking and step-by-step problem-solving skills.

In these price calculation problems:

• Identify the known percentage and the resulting value it represents after changes (like price reductions).
• Set up an equation using the known value and the percentage to find the unknown original value.
• Apply algebraic manipulation to solve for the unknown, often involving division and multiplication.

Precalculus problems like this engage mathematical reasoning and real-world application, enhancing your problem-solving techniques. By understanding and practicing these problems, you build a strong foundation in analytical thinking and mathematics, applicable to various real-life situations beyond discounts and retail shopping.
Gaining fluency in these concepts aids not only in exams but also in day-to-day financial literacy.