Initial length of a rectangluar box is 20cm. This box is remade such that its length is increased by 30% but its breadth is reduced by 20% if area is increased by 100 `cm^(2)` then find new area of box?

Options

A. `2400 cm^(2)`
B. `2200 cm^(2)`
C. `2500 cm^(2)`
D. `2600 cm^(2)`

Answer

To solve the problem step by step, we will follow the given information and perform the necessary calculations.

### Step 1: Determine the initial dimensions of the box.
- The initial length of the rectangular box is given as **20 cm**.
- Let the initial breadth be **x cm**.

### Step 2: Calculate the new length after the increase.
- The length is increased by **30%**.
- New length = Initial length + (30% of Initial length)
- New length = \( 20 + \frac{30}{100} \times 20 \)
- New length = \( 20 + 6 = 26 \) cm.

### Step 3: Calculate the new breadth after the decrease.
- The breadth is reduced by **20%**.
- New breadth = Initial breadth - (20% of Initial breadth)
- New breadth = \( x - \frac{20}{100} \times x \)
- New breadth = \( x - 0.2x = 0.8x \)
- New breadth = \( \frac{80}{100} \times x = \frac{4x}{5} \) cm.

### Step 4: Calculate the initial area of the box.
- Area = Length × Breadth
- Initial Area = \( 20 \times x \) cm².

### Step 5: Calculate the new area of the box.
- New Area = New Length × New Breadth
- New Area = \( 26 \times \frac{4x}{5} \)
- New Area = \( \frac{104x}{5} \) cm².

### Step 6: Set up the equation based on the increase in area.
- We know that the area is increased by **100 cm²**.
- New Area = Initial Area + 100
- \( \frac{104x}{5} = 20x + 100 \).

### Step 7: Solve for x.
- Multiply through by 5 to eliminate the fraction:
\( 104x = 100x + 500 \).
- Rearranging gives:
\( 104x - 100x = 500 \)
- \( 4x = 500 \)
- \( x = \frac{500}{4} = 125 \) cm.

### Step 8: Calculate the new area using the value of x.
- Substitute x back into the new area formula:
- New Area = \( 26 \times \frac{4 \times 125}{5} \)
- New Area = \( 26 \times \frac{500}{5} \)
- New Area = \( 26 \times 100 = 2600 \) cm².

### Final Answer:
The new area of the box is **2600 cm²**.

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