If the length of a rectangle is increased by `62(1)/(2)%` and its breadth is decreased by `7(9)/(13)%`. Find the percentage change in area.

Options

A. a) `30%` increased
B. b) `20%` increased
C. c) `40%` increased
D. d) `50%` increased

Answer

To find the percentage change in the area of a rectangle when the length is increased and the breadth is decreased, we can follow these steps:

### Step 1: Convert the percentage changes to fractions
- The length is increased by \( 62 \frac{1}{2}\% \).
- Convert \( 62 \frac{1}{2}\% \) to an improper fraction:
\[ 62 \frac{1}{2} = \frac{125}{2} \% \]

- The breadth is decreased by \( 7 \frac{9}{13}\% \).
- Convert \( 7 \frac{9}{13}\% \) to an improper fraction:
\[ 7 \frac{9}{13} = \frac{100}{13} \% \]

### Step 2: Calculate the net percentage change in area
The formula for the percentage change in area when one dimension increases and the other decreases is:
\[ \text{Percentage Change} = a + b - \frac{ab}{100} \]
where \( a \) is the percentage increase and \( b \) is the percentage decrease.

Here, we have:
- \( a = \frac{125}{2} \)
- \( b = -\frac{100}{13} \)

Substituting these values into the formula:
\[ \text{Percentage Change} = \frac{125}{2} - \frac{100}{13} - \frac{\left(\frac{125}{2} \cdot \frac{100}{13}\right)}{100} \]

### Step 3: Simplifying the expression
First, calculate \( \frac{125}{2} \cdot \frac{100}{13} \):
\[ \frac{125 \cdot 100}{2 \cdot 13} = \frac{12500}{26} \]

Now, substituting back into the formula:
\[ \text{Percentage Change} = \frac{125}{2} - \frac{100}{13} - \frac{12500}{2600} \]

### Step 4: Find a common denominator
The common denominator for \( 2 \), \( 13 \), and \( 2600 \) is \( 2600 \).

Convert each term:
- \( \frac{125}{2} = \frac{125 \cdot 1300}{2600} = \frac{162500}{2600} \)
- \( \frac{100}{13} = \frac{100 \cdot 200}{2600} = \frac{20000}{2600} \)

Now substitute these back into the formula:
\[ \text{Percentage Change} = \frac{162500}{2600} - \frac{20000}{2600} - \frac{12500}{2600} \]

### Step 5: Combine the fractions
Combine the numerators:
\[ \text{Percentage Change} = \frac{162500 - 20000 - 12500}{2600} = \frac{125000}{2600} \]

### Step 6: Simplify the fraction
Now simplify \( \frac{125000}{2600} \):
\[ \frac{125000 \div 2600}{2600 \div 2600} = \frac{125}{26} \]

### Step 7: Convert to percentage
To convert \( \frac{125}{26} \) to a percentage:
\[ \frac{125}{26} \approx 4.8077 \approx 48.08\% \]

### Final Result
Since the result is positive, the percentage change in area is approximately \( 48.08\% \).