If the price of a book first increased by 18% and then decreased by 18%. What is the net change in the price of the book?
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:
| Option | Description | Matches Calculation? |
|---|---|---|
| 18% increase | States the price increased by 18% net. | No |
| 3.24% decrease | States the price decreased by 3.24% net. | Yes |
| No change | States the price returned to its original value. | No |
| 18% decrease | States the price decreased by 18% net. | No |
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
| Concept | Explanation |
|---|---|
| Percentage Change | \( \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\% \) |
| Successive Percentage Change | When two or more percentage changes happen sequentially. The base for the second change is the result of the first change. |
| Net Change Formula | For changes A% and B%: \( \left(A + B + \frac{AB}{100}\right)\% \) (Use + for increase, - for decrease) |
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.
Important Questions from Percentage
Radha saves 25% of her income. If her expenditure increases by 20% and her income increases by 29%, then her savings increase by;
The income of A is 45% more than the income of B and the income of C is 60% less than the sum of the incomes of A and B. The income of D is 20% more than that of C. If the difference between the incomes of B and D is Rs. 13200, then the income (in Rs.) of C is:
The price of cooking oil increased by 25%. Find by how much percentage a family must reduce its consumption in order to maintain the same budget.
The population of a city increased by 30% in the first year and decreased by 15% in the next year. If the present population is 11,050 then population 2 years ago was:
The income of A is 30% less than the income of B and the income of B is 137.5% more than that of C. If the income of A is Rs. 28500 less than that of B, then the income (in Rs.) of C is:
