If the price of a certain product is first decreased by 35% and then increased by 20%, then what is the net change in the price of the product?
22%
Calculating Net Percentage Change in Price
This question asks us to find the overall percentage change in the price of a product after it undergoes two consecutive percentage changes: first a decrease, and then an increase.
Let's break down the problem step-by-step to find the net change in the price of the product.
Step 1: Assume an Initial Price
To make calculations easy, let's assume the initial price of the product is \(P = 100\) units. Using 100 as the base makes percentage calculations straightforward.
Step 2: Calculate Price After the First Change (Decrease)
The price is first decreased by 35%. A 35% decrease means the new price will be \(100\% - 35\% = 65\%\) of the original price.
Price after decrease \(P'\) is calculated as:
\(P' = P \times (1 - \text{Percentage Decrease})\)
\(P' = 100 \times (1 - 0.35)\)
\(P' = 100 \times 0.65\)
\(P' = 65\)
So, after a 35% decrease, the price becomes 65 units.
Step 3: Calculate Price After the Second Change (Increase)
Next, the price is increased by 20%. This increase is applied to the *new* price, which is 65 units. A 20% increase means the final price will be \(100\% + 20\% = 120\%\) of the price after the decrease.
Final price \(P''\) is calculated as:
\(P'' = P' \times (1 + \text{Percentage Increase})\)
\(P'' = 65 \times (1 + 0.20)\)
\(P'' = 65 \times 1.20\)
To calculate \(65 \times 1.20\):
- \(65 \times 1 = 65\)
- \(65 \times 0.20 = 65 \times \frac{1}{5} = \frac{65}{5} = 13\)
- \(P'' = 65 + 13 = 78\)
So, after the 20% increase on the decreased price, the final price is 78 units.
Step 4: Determine the Net Change
The initial price was 100 units, and the final price is 78 units. The net change in price is the final price minus the initial price.
\(\text{Net Change} = P'' - P\)
\(\text{Net Change} = 78 - 100\)
\(\text{Net Change} = -22\)
The negative sign indicates a decrease. The price has decreased by 22 units from the original 100 units.
Step 5: Calculate the Net Percentage Change
The net percentage change is the net change expressed as a percentage of the original price.
\(\text{Net Percentage Change} = \frac{\text{Net Change}}{\text{Original Price}} \times 100\%\)
\(\text{Net Percentage Change} = \frac{-22}{100} \times 100\%\)
\(\text{Net Percentage Change} = -22\%\)
The net change in the price of the product is a decrease of 22%. The question asks for the net change as a percentage, and the options provide the magnitude of this change.
Summary of Changes
| Description | Calculation | Price |
|---|---|---|
| Initial Price | 100 | |
| After 35% Decrease | \(100 \times (1 - 0.35)\) | 65 |
| After 20% Increase | \(65 \times (1 + 0.20)\) | 78 |
| Net Change | \(78 - 100\) | -22 |
| Net Percentage Change | \(\frac{-22}{100} \times 100\%\) | -22% |
The net change is a 22% decrease in the price of the product.
Revision Table: Price Change Concepts
| Concept | Formula (starting with value V) | Explanation |
|---|---|---|
| Percentage Increase | New Value = \(V \times (1 + \text{increase %}/100)\) | Adding a percentage of the original value to the original value. |
| Percentage Decrease | New Value = \(V \times (1 - \text{decrease %}/100)\) | Subtracting a percentage of the original value from the original value. |
| Net Change after Multiple Changes | Apply changes sequentially. Final Value vs Initial Value. | The base for the second change is the result of the first change, not the original value. |
Additional Information: Successive Percentage Changes
When a value undergoes successive percentage changes, the changes are compounded. This means the second percentage change is calculated on the value resulting from the first change, not on the original value. This is why simply adding or subtracting the percentages (\(-35\% + 20\% = -15\%\)) does not give the correct net change.
In our case, a 35% decrease followed by a 20% increase is not equivalent to a 15% decrease. The base for the 20% increase is smaller than the original base, leading to a different overall effect.
The general formula for two successive percentage changes (\(r_1\) and \(r_2\), expressed as decimals) is:
\(\text{Net Change Factor} = (1 + r_1) \times (1 + r_2)\)
\(\text{Net Percentage Change} = ((1 + r_1) \times (1 + r_2) - 1) \times 100\%\)
For this problem: \(r_1 = -0.35\) (35% decrease) and \(r_2 = +0.20\) (20% increase).
\(\text{Net Change Factor} = (1 - 0.35) \times (1 + 0.20)\)
\(\text{Net Change Factor} = 0.65 \times 1.20\)
\(\text{Net Change Factor} = 0.78\)
This means the final price is 0.78 times the original price, or 78% of the original price.
The net percentage change is \((0.78 - 1) \times 100\% = -0.22 \times 100\% = -22\%\). This confirms our step-by-step calculation.
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