Introduction / Context:
Data sufficiency questions test whether the given statements provide enough information to answer a question uniquely, without necessarily requiring the exact numerical values. Here, we have four bags A, B, C and D, and we need to determine which bag is the lightest in weight using the information in two statements. Understanding how to combine inequalities and compare objects logically is the key concept being tested in this problem.
Given Data / Assumptions:
- We have four bags: A, B, C and D.
- Question: Which one of these four bags is the lightest?
- Statement 1: B is heavier than A (weight of B > weight of A).
- Statement 2: A is lighter than both C and D (weight of A < weight of C and weight of A < weight of D).
- We assume all weights are well-defined and that comparisons are accurate.
Concept / Approach:
The idea in data sufficiency is to check each statement alone and then in combination. A statement is sufficient if, using only that statement (and the question), we can uniquely identify the lightest bag. If a statement leaves multiple possibilities open, it is not sufficient. If combining statements gives a unique answer, then the pair of statements is sufficient together.
Step-by-Step Solution:
Step 1: Analyse Statement 1 alone. It says B is heavier than A. This only compares A and B, and says nothing about C and D. So among all four bags, we cannot be sure who is lightest, because C or D could be lighter than A. Thus, Statement 1 alone is not sufficient.
Step 2: Analyse Statement 2 alone. It says A is lighter than C and D. There is still no information about B compared with A. B could be heavier than A or lighter than A. Therefore we cannot determine whether A is the lightest or whether B might be lighter. Statement 2 alone is also not sufficient.
Step 3: Combine Statements 1 and 2. From Statement 1, we know B > A. From Statement 2, we know C > A and D > A. Therefore, A is lighter than B, C and D. So bag A is definitely the lightest among all four bags.
Step 4: Since we now have a unique answer (A is the lightest bag) only when we use both statements together, the pair of statements is sufficient while each one alone is not.
Verification / Alternative check:
Take an example set of weights that satisfies both statements: A = 1 kg, B = 2 kg, C = 3 kg, D = 4 kg. Here, B > A and C, D are both heavier than A. It is clear that A is the lightest.
Try to construct any configuration satisfying both statements where A is not the lightest. It is impossible, because both B, C and D are all strictly heavier than A by the given conditions.
Why Other Options Are Wrong:
Option A (Statement 2 alone is sufficient) is wrong because Statement 2 tells us only that A is lighter than C and D. Without a comparison between A and B, B might still be the lightest.
Option C (Statement 1 alone is sufficient) is wrong because knowing only that B is heavier than A tells us nothing about C and D. Either C or D could be lighter than A.
Option D (Statements 1 and 2 together are not sufficient) is wrong because, when combined, the statements force all of B, C and D to be heavier than A, making A uniquely the lightest bag.
Common Pitfalls:
A common error is to focus only on the items directly mentioned in each statement and ignore that other items might still be lighter or heavier.
Another mistake is to confuse data sufficiency with actually computing a numerical value. Here we do not need the exact weights; we only need to know which bag is definitely the lightest.
Some test-takers also forget to check the combined effect of both statements when each one alone seems insufficient.
Final Answer:
The bags' weights can be uniquely compared only when both statements are used together, which show that A is lighter than B, C and D. Hence, Statements 1 and 2 together are sufficient to answer the question.
Discussion & Comments