AnkitK wrote:
If p^2 – 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
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, which 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.