Problem 18 - Prealgebra

Question

Solve each of these problems using the method developed in this section. When buying some of today's newest electronic gadgets, good things come to those who wait. When Apple released its new iPhone in the summer of \(2007,\) an \(8 \mathrm{GB}\) model sold for \(\$ 499 .\) In July 2008 Apple released its new iPhone \(3 \mathrm{G}\). The \(8 \mathrm{GB}\) model sells for \(\$ 199 .\) What is the percent decrease in price for this new model? Round your answer to the nearest tenth of percent.

Step-by-Step Solution

Answer

The percent decrease is 60.1%.

Step 1: Identify the Original and New Prices

Determine the original price of the iPhone which is $499 and the new price of the iPhone which is $199.

Step 2: Calculate the Difference in Price

Subtract the new price from the original price to find the difference: \(499 - 199 = 300\).

Step 3: Compute the Percent Decrease

Use the formula for percent decrease: \(\text{Percent decrease} = \frac{\text{Difference in price}}{\text{Original price}} \times 100\%\). Substitute in the values: \(\frac{300}{499} \times 100\%\).

Step 4: Solve for the Percent Decrease

Calculate the expression: \(\frac{300}{499} \times 100\% = 60.1202\%\).

Step 5: Round to the Nearest Tenth of a Percent

Round \(60.1202\%\) to the nearest tenth to obtain \(60.1\%\).

Key Concepts

Price Difference

Understanding price difference is essential when dealing with changes in pricing, especially in real-world situations like shopping or comparing costs. The price difference is simply the amount by which one price is higher or lower than another. In our example, this involves calculating how much less the iPhone 3G costs compared to its original release price. To find the price difference, merely subtract the new price from the original price.

For the iPhone example, the original price was $499 and the new price is $199. By subtracting the new price from the original price:

• Original Price: $499
• New Price: $199
• Price Difference: $499 - $199 = $300

This simple subtraction gives us the price difference of $300.

Original and New Price

Identifying the original and new prices is usually the first step when calculating a percent decrease or increase. The original price is the cost before any reductions or changes. Meanwhile, the new price is what you pay after reductions have been applied. In this context, it's crucial to correctly identify both to proceed with further calculations.

In the given exercise, the original price of the iPhone was $499. This figure reflects the cost when the phone was first released on the market. On the other hand, the new price, which is $199, represents the cost after the newer model was released.

Having these numbers laid out helps facilitate the rest of the calculations, such as determining how much the price has decreased in percentage terms.

Percent Calculation

To express changes in price as a percentage, the percent calculation is your tool of choice. This allows for easy comparison regardless of the original amounts. We use the percent calculation to determine the percentage decrease between the original and new prices.

The formula for percent decrease is:

• Percent Decrease = \( \frac{\text{Difference in price}}{\text{Original price}} \times 100\% \)

So, with our example:

• Difference in Price: \(300
• Original Price: \)499

Now plug these values into the formula: \[ \text{Percent Decrease} = \frac{300}{499} \times 100\% \approx 60.1202\% \]
Thus, the iPhone price decreased by approximately 60.1202%.

Rounding Decimals

Rounding decimals is a practical mathematical skill useful for simplifying numbers, making them easier to understand and use in communication. When a number has many decimal places, rounding it to a few makes it more straightforward.

In percent decrease problems, rounding helps simplify results. For our exercise, the calculated percent decrease was 60.1202%. Generally, this would be cumbersome to communicate directly. Consequently, we round to the nearest tenth of a percent for clarity.

When rounding 60.1202% to the nearest tenth of a percent:

• Identify the tenth place: 1 (in 60.1202)
• Look at the next digit: 2 (in the hundredth place)
• Since 2 is less than 5, it does not affect the tenth place.

Therefore, the rounded value is 60.1%.