Read the following question and decide which of the given statements is/are sufficient to answer the question. If U, V and W are negative numbers, and Z is a number, then is Z positive? Statements: 1. Z = V - W 2. Z × V = U
2 alone is sufficient while 1 alone is not sufficient to answer the question.
Understanding the Data Sufficiency Question
The question asks whether the number Z is positive, given that U, V, and W are all negative numbers. We need to examine two statements to determine if either statement alone, or both statements together, provide enough information to definitively answer whether Z is positive.
The core task is to figure out the sign of Z based on the given conditions and the information provided in each statement.
Analyzing Statement 1: Z = V - W
Statement 1 says that Z is equal to V minus W. We know that V and W are negative numbers.
Let's consider examples:
- If V = -2 and W = -3, then Z = V - W = (-2) - (-3) = -2 + 3 = 1. In this case, Z is positive.
- If V = -5 and W = -1, then Z = V - W = (-5) - (-1) = -5 + 1 = -4. In this case, Z is negative.
As demonstrated by these examples, the value of Z can be positive or negative depending on the specific negative values of V and W. If V is closer to zero (i.e., greater) than W, then V - W will be positive. If V is further from zero (i.e., smaller) than W, then V - W will be negative.
Therefore, Statement 1 alone is not sufficient to determine whether Z is positive.
Analyzing Statement 2: Z × V = U
Statement 2 gives us the equation \(Z \times V = U\). We are given that U and V are negative numbers.
We can rearrange the equation to solve for Z:
\[ Z = \frac{U}{V} \]We know the sign of U is negative and the sign of V is negative.
Let's consider the rule for dividing signed numbers:
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative
In this case, we have a negative number (U) divided by a negative number (V). According to the rules of division, the result must be positive.
\[ Z = \frac{\text{Negative}}{\text{Negative}} = \text{Positive} \]This means that Z must be a positive number.
Therefore, Statement 2 alone is sufficient to definitively answer that Z is positive.
Evaluating Sufficiency
- Statement 1 alone: Not sufficient. Z could be positive or negative.
- Statement 2 alone: Sufficient. Z must be positive.
Since Statement 2 alone provides a definite answer to the question "is Z positive?", Statement 2 alone is sufficient, while Statement 1 alone is not sufficient.
Conclusion on Data Sufficiency
Based on our analysis, Statement 2 alone is sufficient to answer the question, whereas Statement 1 alone is not sufficient.
| Statement | Is it sufficient to determine if Z is positive? | Reason |
|---|---|---|
| 1. Z = V - W | No | Result (Z) can be positive or negative depending on specific values of V and W. |
| 2. Z × V = U | Yes | \(Z = \frac{U}{V}\). A negative number divided by a negative number is always positive. |
Revision Table: Key Concepts for Data Sufficiency
Reviewing fundamental concepts helps tackle data sufficiency problems.
| Concept | Explanation |
|---|---|
| Data Sufficiency | A question type where you determine if given statements provide enough information to answer a question. |
| Sufficiency | A statement is sufficient if it leads to a single, definite answer to the question. |
| Negative Numbers | Numbers less than zero. Their product or quotient follows specific sign rules. |
| Sign Rules for Multiplication/Division | Same signs result in positive; different signs result in negative. |
Additional Information: Properties of Negative Numbers
Understanding how operations work with negative numbers is crucial.
- Addition: Sum of two negative numbers is negative (e.g., -2 + -3 = -5).
- Subtraction: Subtracting a negative number is the same as adding a positive number (e.g., -2 - (-3) = -2 + 3 = 1).
- Multiplication: Product of two negative numbers is positive (e.g., -2 × -3 = 6). Product of a negative and a positive is negative (e.g., -2 × 3 = -6).
- Division: Quotient of two negative numbers is positive (e.g., -6 ÷ -2 = 3). Quotient of a negative and a positive is negative (e.g., -6 ÷ 2 = -3).
In this data sufficiency problem, the division rule for two negative numbers was key to determining the sign of Z based on Statement 2.
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Important Questions from Data Sufficiency
Consider two Statements and a Question:
Statement 1: Priya is 4 ranks below Seema and is 31 st from the bottom.
Statement 2: Ena is 2 ranks above Seema and is 37" from the bottom.
Question: What is Seema’s rank from the top in the class of 40 students ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question:
Statement 1: Each of A and D is heavier than each of B, E and F, but none of them is the-heaviest.
Statement 2: A is heavier than D, but is lighter than C.
Question: Who is the heaviest among A,B,C, D and E?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question :
Statement-1 : The last day of the month is a Wednesday.
Statement-2 : The third Saturday of the month was the seventeenth day.
Question : What day is the fourteenth of the given month ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider the Question and two Statements given below :
Question : Is xan integer?
Statement-1 : x / 3 is not an integer.
Statement-2 : 3 xis an integer.
Which one of the following is correct in respect of the Question and the Statements?
Consider the Question and two Statements given below:
Question : What is the age of Manisha?
Statement-1: Manisha is 24 years younger than her mother.
Statement-2 : 5 years later, the ages of Manisha and her mother will be in the ratio 3 : 5.
Which one of the following is correct in respect of the Question and the Statements?
