Statements Analysis for Four Numbers

We are given three statements about four numbers, let's call them $a, b, c, d$. We need to determine which combination of statements necessarily leads to another statement being true.

Statement 1: Average of Four Numbers

Statement (S1) says the average of the four numbers is 25.

Mathematically, this means:

Multiplying both sides by 4 gives the sum of the numbers:

Statement 2: Maximum Value Constraint

Statement (S2) states that each number is at most 40.

This can be written as:

Statement 3: Minimum Value Constraint

Statement (S3) states that each number is at least 20.

This can be written as:

Evaluating Statement Implications

Now, let's examine the implications proposed in the options.

Option 1 Analysis: (S1) & (S2) vs (S3)

Can statements (S1) and (S2) guarantee (S3)? (Sum is 100, and numbers are $\le 40$).

Consider the numbers: 10, 30, 30, 30.

• Sum: $10 + 30 + 30 + 30 = 100$. Average is $100 / 4 = 25$. (Satisfies S1)
• Maximum value: All numbers are $\le 40$. (Satisfies S2)
• Minimum value: 10 is less than 20. (Violates S3)

Since we found a case where S1 and S2 are true but S3 is false, (S1) and (S2) do not necessarily imply (S3).

Option 2 Analysis: (S2) & (S3) vs (S1)

Can statements (S2) and (S3) guarantee (S1)? (Numbers are between 20 and 40 inclusive).

Consider the numbers: 20, 20, 20, 20.

• Minimum value: All are $\ge 20$. (Satisfies S3)
• Maximum value: All are $\le 40$. (Satisfies S2)
• Average: $(20+20+20+20)/4 = 80/4 = 20$. This is not 25. (Violates S1)

Since we found a case where S2 and S3 are true but S1 is false, (S2) and (S3) do not necessarily imply (S1).

Option 3 Analysis: (S1) & (S3) vs (S2)

Can statements (S1) and (S3) guarantee (S2)? (Sum is 100, and numbers are $\ge 20$).

Let's use proof by contradiction. Assume (S2) is false, meaning at least one number is greater than 40. Let this number be $a$, so $a > 40$. We can write $a = 40 + x$, where $x > 0$.

From (S1), we have $a+b+c+d = 100$. Substituting $a$:

From (S3), we know that $b \ge 20$, $c \ge 20$, and $d \ge 20$. Therefore, the minimum possible sum for $b+c+d$ is:

So, we must have $60 - x \ge 60$. This implies $-x \ge 0$, which means $x \le 0$.

This contradicts our initial assumption that $x > 0$ (or $a > 40$). Therefore, the assumption that any number is greater than 40 must be false.

This proves that all numbers must be less than or equal to 40. Hence, (S1) and (S3) together necessarily imply (S2).

Option 4 Analysis: (S1) vs (S3)

Does statement (S1) alone imply (S3)? (Average is 25).

Consider the numbers: 10, 30, 30, 30.

• Average is 25 (Satisfies S1).
• However, 10 is less than 20, so (S3) is not satisfied.

Therefore, (S1) does not necessarily imply (S3).

Necessarily Correct Implication

Based on the analysis, the only condition that is necessarily correct is that Statements (S1) and (S3) together imply Statement (S2).