To solve this problem, we need to determine how long it would take the second pump to fill the tank on its own. The given equation representing the combined filling rates of the pumps is:
[tex]\[
\frac{1}{150} + \frac{1}{x} = \frac{1}{60}
\][/tex]
Here, [tex]\(\frac{1}{150}\)[/tex] represents the rate at which the first pump fills the tank (since it takes 150 minutes for the first pump to fill the tank alone). [tex]\(\frac{1}{x}\)[/tex] represents the rate at which the second pump fills the tank on its own, and [tex]\(\frac{1}{60}\)[/tex] represents the combined rate at which both pumps fill the tank together (since together they fill the tank in 60 minutes).
We need to solve for [tex]\(x\)[/tex]. Let's isolate [tex]\(\frac{1}{x}\)[/tex] first:
[tex]\[
\frac{1}{x} = \frac{1}{60} - \frac{1}{150}
\][/tex]
To perform this subtraction, we need a common denominator. The least common multiple of 60 and 150 is 300. Let's rewrite each fraction with a denominator of 300:
[tex]\[
\frac{1}{60} = \frac{5}{300}
\][/tex]
[tex]\[
\frac{1}{150} = \frac{2}{300}
\][/tex]
Now, we can subtract these fractions:
[tex]\[
\frac{1}{x} = \frac{5}{300} - \frac{2}{300} = \frac{3}{300}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{1}{x} = \frac{1}{100}
\][/tex]
Now, take the reciprocal of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 100
\][/tex]
Thus, the second pump, working alone, would fill the tank in 100 minutes. Therefore, the correct answer is:
C. 100 minutes
