For the integers m, n, r, and s, if m + n = 250 and m > n, is (m – r) > (s – n)? (1) 250 > r + s (2) m + r + s= 375

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(D) Statement (1) tells us that 250 > r + s. Since the question statement tells us that m + n = 250, we can determine that m + n > r + s.

Now, let us manipulate this inequality to see whether it is equivalent to the inequality in the question: (m + n) > (r + s) m > (r + s) – n (m – r) > (s – n)

This is exactly what we were looking for. We can answer the question using Statement (1), hence it is sufficient.

Statement (2) tells us that m + r + s = 375. Because we know that m + n = 250 and m > n, m must be greater than 125. Subtracting 125 from 375 yields 250, so if m is greater than 125, then r + s must be smaller than 250. We are now left with the same inequality that we were given in Statement (1), which can be manipulated to show that (m – r) > (s – n). So Statement (2) is also sufficient.

Since both statements are sufficient alone, the correct answer is choice (D). -------------

The two statements should never contradict themselves: 1) 250 > r + s 2) We know m > 125 If I substitute 126 for m in the statement (2) then r + s = 249 I thought r + s were greater than 250.

There is no contradiction in this question. Statement (1) tells us that (r + s) is smaller than 250. When you plug in m = 126 into statement (2) and get r + s = 249, it doesn't contradict with statement (1) since 249 is smaller then 250.

Furthermore, it is shown in the explanation that statement (2) actually implies statement (1).

There is NO necessity in defining the integers to be positive. In fact, any of the integers r, s or n can be zero or negative. However, the statement implies that m is greater than 125, so it CAN NOT be zero or negative.

Furthermore, let's take a look on the explanation "backwards". Let us analyse the desired inequality first:

m – r > s – n we easily transform it into: m + n > s + r The question statement defines m + n = 250. Lets plug it in. So the original inequality is equivalent to: 250 > s + r

That is exactly the first statement. Therefore the first statement is sufficient by itself. If we use the second statement by itself, then we can plug s + r = 375 – m into the inequality: 250 > 375 – m m > 125

That is true, because from the basic statement we know that m > n, so m + m > n + m = 250. 2m > 250 m > 125

Therefore each statement by itself is sufficient.

As you see, we never required any of the numbers to be positive (except of m, which follows from the basic statement).

To evaluate statement (2) you are using the information provided in statement (1), otherwise you wouldn't get that m must be > 125 ... therefore the answer would be A ... unless you have another way to solve ...

To evaluate statement (2) you are using the information provided in statement (1)

We do NOT use the first statement to evaluate the second one, though statement (2) implies that statement (1) holds true (as we reason that in the solution).

Quote:

otherwise you wouldn't get that m must be > 125

The two facts from the base statement (m + n = 250 and m > n) imply that m > 125. We do NOT use any other statement for that.

m > n Let's add m to the both sides: m + m > m + n Let's plug in m + n = 250: m + m > 250 2m > 250 Then divide the both sides by 2: m > 125.

You may once again go over the proof for the second statement alone to see that we do NOT use the first statement initially, but reason the same inequality along the way based on the question statement + statement (2) ONLY.

Using JUST statement (2) and the original information we have the following facts:

1. m, n, r, and s are integers. 2. m + n = 250 3. m > n 4. m + r + s = 375

Combining the facts # 3 and #2 we get

m + n < m + m 250 < 2m 125 < m

Then we can plug in the formula for m from the fact #4: 125 < 375 – (r + s) r + s < 250

Then we plug in the formula for 250 from the fact #2: r + s < m + n s – n < m – r

Therefore the second statement by ITSELF is sufficient to answer the question.

I was thrown off by the parentheses around (m-r) and (s-n) and it lead me to believe that I couldn't manipulate the inequality by adding and subtracting the variables...I thought I had to keep (m-r) together since it was referring to the resulting value of m-r. Is that the reason they put parentheses, just to possibly trick you or do they actually mean anything?

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