Find the area of rectangle ABCD.

I. The perimeter of rectangle ABCD is 4.5 times its breadth.

II. The difference between the squares of the adjacent sides of rectangle ABCD is 5.4 times its breadth.

Based on the above statements, choose the correct option regarding sufficiency of data to find the area.

Question

Find the area of rectangle ABCD.

I. The perimeter of rectangle ABCD is 4.5 times its breadth.

II. The difference between the squares of the adjacent sides of rectangle ABCD is 5.4 times its breadth.

Based on the above statements, choose the correct option regarding sufficiency of data to find the area.

Accepted Answer

**Solution:**

1. Let the breadth of the rectangle be *b* and the length be *l*.

2. From statement (I):
   - Perimeter = 2(l + b) = 4.5b
   - So, 2(l + b) = 4.5b
   - Simplifying, l + b = 2.25b
   - Therefore, l = 2.25b - b = 1.25b

Here, length is expressed in terms of breadth, but we do not know the actual value of *b*. Hence, statement (I) alone is insufficient to find the area (which is l × b).

3. From statement (II):
   - Difference between squares of adjacent sides = l² - b² = 5.4b
   - Using the identity l² - b² = (l - b)(l + b), we get:
     (l - b)(l + b) = 5.4b

Without knowing either *l* or *b*, this equation alone cannot determine both variables. So, statement (II) alone is insufficient.

4. Using both statements together:
   - From (I), l + b = 2.25b
   - Substitute into (II): (l - b)(2.25b) = 5.4b
   - Dividing both sides by b (b ≠ 0): (l - b)(2.25) = 5.4
   - So, l - b = 5.4 / 2.25 = 2.4

5. Now we have two equations:
   - l + b = 2.25b
   - l - b = 2.4

From the first, l = 2.25b - b = 1.25b

Substitute into second:
   - 1.25b - b = 2.4
   - 0.25b = 2.4
   - b = 2.4 / 0.25 = 9.6

Then, l = 1.25 × 9.6 = 12

6. Calculate area:
   - Area = l × b = 12 × 9.6 = 115.2

**Answer:** Both statements together are required to find the area; neither alone suffices.

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