Data Sufficiency Analysis: Finding Black Cows

The question asks for the exact number of black cows in a herd. We are given two statements and need to determine if either statement alone, or both statements together, are sufficient to answer this question.

Analyzing Statement 1: Total Number of Cows

Statement 1 says:

1. There are in total 60 cows.

This statement gives us the total size of the herd. Let's denote the total number of cows by $N$. According to Statement 1, $N = 60$.

However, this statement does not provide any information about the proportion or number of cows that are black. We know the total herd size, but we don't know how many of them are black. It could be 10 black cows, 30 black cows, 60 black cows, or any other number between 0 and 60.

Therefore, Statement 1 alone is not sufficient to determine the number of black cows.

Analyzing Statement 2: Percentage of Black Cows

Statement 2 says:

2. 40% of them are black.

This statement gives us the percentage of cows that are black, relative to the total number of cows in the herd. Let's denote the number of black cows by $B$. According to Statement 2, the number of black cows is 40% of the total number of cows.

In mathematical terms, this can be written as:

$$B = 40\% \text{ of } N$$

or

$$B = \frac{40}{100} \times N$$

or

$$B = 0.40 \times N$$

This statement provides the relationship between the number of black cows ($B$) and the total number of cows ($N$), but it does not give us the value of $N$. Without knowing the total number of cows, we cannot calculate the exact number of black cows.

For example, if there were 100 cows total, there would be $0.40 \times 100 = 40$ black cows. But if there were 50 cows total, there would be $0.40 \times 50 = 20$ black cows. Since the total number of cows ($N$) is unknown from Statement 2 alone, we cannot find a single, definite number for the black cows ($B$).

Therefore, Statement 2 alone is not sufficient to determine the number of black cows.

Analyzing Statements 1 and 2 Together

Now, let's consider both statements together.

From Statement 1, we know the total number of cows:

• $N = 60$

From Statement 2, we know the percentage of black cows:

• The number of black cows ($B$) is 40% of the total number of cows ($N$).

Using the information from both statements, we can substitute the value of $N$ from Statement 1 into the relationship from Statement 2:

$$B = 40\% \text{ of } N$$

Substitute $N = 60$:

$$B = 40\% \text{ of } 60$$

Calculate the value:

$$B = \frac{40}{100} \times 60$$

$$B = 0.40 \times 60$$

$$B = 24$$

By using both statements together, we are able to find a unique and definite number for the black cows, which is 24.

Therefore, both statements taken together are sufficient to answer the question.

Conclusion on Sufficiency

We found that:

• Statement 1 alone is not sufficient.
• Statement 2 alone is not sufficient.
• Both Statement 1 and Statement 2 together are sufficient.

This leads to the conclusion that both statements together are necessary and sufficient, while neither is sufficient on its own.

Revision Table: Key Concepts

Let's quickly review the core ideas in this data sufficiency problem about the cow herd.

• Data Sufficiency: A type of question where you determine if the given data (in statements) is enough to answer the question asked. You don't necessarily have to find the answer, just whether it's possible to find it.
• Sufficiency: A statement or set of statements is sufficient if it provides enough information to yield a single, unique answer to the question.
• Not Sufficient: A statement or set of statements is not sufficient if it does not provide enough information, or if it leads to multiple possible answers.

Additional Information: Solving Data Sufficiency Problems

Here’s a general approach to tackle data sufficiency questions like the one involving the black cows:

1. Understand the Question: Clearly identify what specific value or information the question is asking for.
2. Analyze Statement 1 Alone: Assume Statement 1 is true and see if you can answer the question using only the information from Statement 1. Ignore Statement 2 completely during this step. If you can find a unique answer, Statement 1 is sufficient.
3. Analyze Statement 2 Alone: Now, assume Statement 2 is true and see if you can answer the question using only the information from Statement 2. Ignore Statement 1 completely. If you can find a unique answer, Statement 2 is sufficient.
4. Analyze Both Statements Together: If neither Statement 1 alone nor Statement 2 alone is sufficient, consider both statements together. See if the combined information allows you to find a unique answer to the question. If yes, then both statements together are sufficient.
5. Evaluate Options: Based on your analysis, choose the option that correctly describes the sufficiency of the statements.

In this specific cow herd question, we followed these steps and concluded that only both statements combined were sufficient to determine the number of black cows.