Probable P’s and Q’s

Question
If p2 – 13p + 40 = q, and p is a positive integer between 1 and 10, inclusive, what is the probability that q < 0?

(A) 1/10
(B) 1/5
(C) 2/5
(D) 3/5
(E) 3/10

Answer
If we factor the right side of the equation, we can come up with a more meaningful relationship between p and q: p2 – 13p + 40 = q so (p – 8)(p – 5) = q. We know that p is an integer between 1 and 10, inclusive, so there are ten possible values for p. We see from the factored equation that the sign of q will depend on the value of p. One way to solve this problem would be to check each possible value of p to see whether it yields a positive or negative q. 

However, we can also use some logic here. For q to be negative, the expressions (p – 8) and (p – 5) must have opposite signs. Which integers on the number line will yield opposite signs for the expressions (p – 8) and (p – 5)? Those integers in the range 5 < p < 8 (notice 5 and 8 are not included because they would both yield a value of zero and zero is a nonnegative integer). That means that there are only two integer values for p, 6 and 7, that would yield a negative q. With a total of 10 possible p values, only 2 yield a negative q, so the probability is 2/10 or 1/5.

The correct answer is B.

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