Understanding Net Price Change with Successive Percentages

This problem involves calculating the overall or net change in the price of a book after it undergoes two consecutive percentage changes: an increase followed by a decrease of the same percentage.

When a quantity is subjected to successive percentage changes, the final result is not simply the sum or difference of the percentages. A 18% increase followed by a 18% decrease does not result in 'no change'. This is because the second percentage change is applied to the new price after the first change, not the original price.

Method 1: Assuming an Initial Price

Let's assume the original price of the book is \(P\).

• Step 1: Calculate the price after the 18% increase.

  An 18% increase means the price becomes \(100\% + 18\% = 118\%\) of the original price.

  New Price = \(P \times \left(1 + \frac{18}{100}\right) = P \times (1 + 0.18) = P \times 1.18\)

• Step 2: Calculate the price after the 18% decrease on the new price.

  An 18% decrease on the new price means the price becomes \(100\% - 18\% = 82\%\) of the new price.

  Final Price = \((P \times 1.18) \times \left(1 - \frac{18}{100}\right) = (P \times 1.18) \times (1 - 0.18) = P \times 1.18 \times 0.82\)

• Step 3: Calculate the final price value.

  \(1.18 \times 0.82 = 0.9676\)

  So, Final Price = \(P \times 0.9676\)

• Step 4: Determine the net change in price.

  Net Change = Final Price - Original Price

  Net Change = \(P \times 0.9676 - P = P(0.9676 - 1) = P(-0.0324)\)

• Step 5: Calculate the net percentage change.

  Net Percentage Change = \(\left(\frac{\text{Net Change}}{\text{Original Price}}\right) \times 100\%\)

  Net Percentage Change = \(\left(\frac{P(-0.0324)}{P}\right) \times 100\% = -0.0324 \times 100\% = -3.24\%\)

The negative sign indicates a decrease. So, the net change is a 3.24% decrease.

Method 2: Using the Successive Percentage Change Formula

For two successive percentage changes, A% and B%, the net percentage change is given by the formula:

\( \text{Net Change} = \left(A + B + \frac{AB}{100}\right)\% \)

In this problem, the first change is an 18% increase, so \(A = +18\).

The second change is an 18% decrease, so \(B = -18\).

Substitute these values into the formula:

\( \text{Net Change} = \left(18 + (-18) + \frac{18 \times (-18)}{100}\right)\% \)

\( \text{Net Change} = \left(0 + \frac{-324}{100}\right)\% \)

\( \text{Net Change} = \left(-\frac{324}{100}\right)\% \)

\( \text{Net Change} = -3.24\% \)

Again, the result is a 3.24% decrease.

Analysis of Options

Let's examine the provided options based on our calculation:

The calculations consistently show a net decrease of 3.24%.

Conclusion on Net Price Change

When an item's price increases by a certain percentage and then decreases by the exact same percentage, there is always a net decrease. This decrease is equal to the square of the percentage change (as a decimal) multiplied by 100% or, using the formula \( \frac{x^2}{100}\%\) where x is the percentage change (18 in this case).

\( \frac{18^2}{100}\% = \frac{324}{100}\% = 3.24\%\)

Since the second change is a decrease on the increased value, it doesn't fully offset the initial increase.

Revision Table: Key Concepts

Additional Information: Applying Percentage Concepts

Understanding successive percentage changes is crucial in many areas, including finance, economics, and business. For example:

• Calculating the final price of an item after a markup and then a discount.
• Determining the overall growth or decline of an investment over multiple periods with varying returns.
• Analyzing changes in population, sales figures, or other metrics over time.

Always remember that percentage changes are relative to the base value at the time of the change. In the case of successive changes, the base value changes after each step.