Question

If the length of a rectangle is increased by 5 cm and the breadth is decreased by 5 cm, area of the rectangle remained same. What are the dimensions of the given rectangle?
Statement 1: If the length is decreased by 12 cm and the breadth increased by 8 cm, even then the area is unaltered
Statement 2: Breadth of the rectangle is 4 less than twice its length
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is D

Step 1: Understand the problem.
We are given a rectangle, and its dimensions are such that when the length is increased by 5 cm and the breadth is decreased by 5 cm, the area remains the same. Additionally, two statements are provided to help solve the problem, and we need to determine if either of the statements alone is sufficient to find the dimensions of the rectangle.

Step 2: Set up the equation for the area of the rectangle.
Let the length of the rectangle be \( l \) and the breadth be \( b \). The area of the rectangle is given by: \[ \text{Area} = l \times b \] After increasing the length by 5 cm and decreasing the breadth by 5 cm, the new area remains the same, so: \[ (l + 5) \times (b - 5) = l \times b \] Expanding both sides: \[ l \times b - 5l + 5b - 25 = l \times b \] Simplifying: \[ -5l + 5b = 25 \] Dividing by 5: \[ -l + b = 5 \] So, we have the first equation: \[ b = l + 5 \] Step 3: Analyze Statement 1.
Statement 1 tells us that if the length is decreased by 12 cm and the breadth is increased by 8 cm, the area remains the same. The new equation for the area is: \[ (l - 12) \times (b + 8) = l \times b \] Expanding both sides: \[ l \times b + 8l - 12b - 96 = l \times b \] Simplifying: \[ 8l - 12b = 96 \] Dividing by 4: \[ 2l - 3b = 24 \] Using \( b = l + 5 \) from the previous equation: \[ 2l - 3(l + 5) = 24 \] Expanding: \[ 2l - 3l - 15 = 24 \] Simplifying: \[ -l = 39 \] \[ l = -39 \] This does not make sense, as the length cannot be negative. Thus, Statement 1 is insufficient on its own to determine the dimensions of the rectangle.

Step 4: Analyze Statement 2.
Statement 2 tells us that the breadth of the rectangle is 4 less than twice its length, so: \[ b = 2l - 4 \] Substituting \( b = l + 5 \) into this equation: \[ l + 5 = 2l - 4 \] Solving for \( l \): \[ 5 + 4 = 2l - l \] \[ 9 = l \] So, the length of the rectangle is \( l = 9 \). Substituting this value back into \( b = l + 5 \), we get: \[ b = 9 + 5 = 14 \] Therefore, the dimensions of the rectangle are \( l = 9 \) cm and \( b = 14 \) cm. Statement 2 is sufficient to determine the dimensions of the rectangle.

Step 5: Conclusion.
Statement 2 alone is sufficient to answer the question, and Statement 1 does not provide a valid solution. Thus, the correct answer is that either statement (1) alone or statement (2) alone is sufficient to answer the question.

Final Answer:
The correct answer is (D): either statement (1) alone or statement (2) alone is sufficient to answer the question.