The length and the breadth of a rectangle are made to increase and decrease, respectively, by 8% and 10%. What is the percentage increase or decrease in its area?
Decrease by 2.8%
Understanding Rectangle Area Percentage Change
This problem asks us to find the overall percentage change in the area of a rectangle when its length and breadth undergo specific percentage changes: an increase in length and a decrease in breadth.
The area of a rectangle is calculated using the formula: Area = Length × Breadth.
We can solve this problem using two methods: by assuming initial values or by using the formula for successive percentage changes.
Method 1: Using Base Values
Let's assume the original length of the rectangle is \(L\) and the original breadth is \(B\).
The original area is \(A = L \times B\).
Calculating New Dimensions:
- The length is increased by 8%. New length \(L'\) will be: \[L' = L + 8\%\text{ of } L = L + \frac{8}{100}L = L\left(1 + \frac{8}{100}\right) = 1.08L\]
- The breadth is decreased by 10%. New breadth \(B'\) will be: \[B' = B - 10\%\text{ of } B = B - \frac{10}{100}B = B\left(1 - \frac{10}{100}\right) = 0.90B\]
Calculating New Area:
The new area \(A'\) is the product of the new length and new breadth:
\[A' = L' \times B' = (1.08L) \times (0.90B)\] \[A' = (1.08 \times 0.90) \times (L \times B)\] \[A' = 0.972 \times A\]So, the new area is 0.972 times the original area.
Calculating Percentage Change in Area:
The change in area is \(A' - A\):
\[\text{Change in Area} = 0.972A - A = (0.972 - 1)A = -0.028A\]Since the change is negative, the area has decreased.
The percentage change in area is calculated as:
\[\text{Percentage Change} = \frac{\text{Change in Area}}{\text{Original Area}} \times 100\%\] \[\text{Percentage Change} = \frac{-0.028A}{A} \times 100\%\] \[\text{Percentage Change} = -0.028 \times 100\%\] \[\text{Percentage Change} = -2.8\%\]This means the area decreases by 2.8%.
Method 2: Using Successive Percentage Change Formula
When a quantity is affected by two successive percentage changes, say \(x\%\) and \(y\%\), the overall percentage change is given by the formula:
\[\text{Overall Change} = \left(x + y + \frac{xy}{100}\right)\%\]In this problem:
- Percentage increase in length, \(x = +8\%\)
- Percentage decrease in breadth, \(y = -10\%\)
Substitute these values into the formula:
\[\text{Overall Change} = \left(8 + (-10) + \frac{8 \times (-10)}{100}\right)\%\] \[\text{Overall Change} = \left(8 - 10 + \frac{-80}{100}\right)\%\] \[\text{Overall Change} = \left(-2 - 0.8\right)\%\] \[\text{Overall Change} = -2.8\%\]The negative sign indicates a decrease. Therefore, the area decreases by 2.8%.
Conclusion on Rectangle Area Change
Both methods show that when the length of a rectangle increases by 8% and the breadth decreases by 10%, the area of the rectangle decreases by 2.8%.
Revision Table: Rectangle Area Calculations
| Concept | Formula/Calculation | Explanation |
|---|---|---|
| Original Area | \(A = L \times B\) | Product of original length and breadth. |
| New Length (8% increase) | \(L' = L \times (1 + 0.08) = 1.08L\) | Original length plus 8% of original length. |
| New Breadth (10% decrease) | \(B' = B \times (1 - 0.10) = 0.90B\) | Original breadth minus 10% of original breadth. |
| New Area | \(A' = L' \times B' = (1.08L) \times (0.90B) = 0.972LB\) | Product of new length and new breadth. |
| Percentage Change in Area | \(\frac{A' - A}{A} \times 100\%\) or \(\left(x + y + \frac{xy}{100}\right)\%\) | Ratio of area change to original area, expressed as a percentage. Successive percentage change formula simplifies calculation. |
Additional Information on Percentage Changes
Percentage change is a way to express how much a quantity changes relative to its original value. A positive percentage change indicates an increase, while a negative percentage change indicates a decrease.
- An increase of \(p\%\) means multiplying the original value by \((1 + \frac{p}{100})\).
- A decrease of \(p\%\) means multiplying the original value by \((1 - \frac{p}{100})\).
The successive percentage change formula is useful for problems where a quantity changes multiple times consecutively or when two factors multiplying together undergo percentage changes, like in area calculations (Length × Breadth) or product price changes.
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