Show that the equation \(\frac{1}{60} = \frac{2x+50}{x(x+50)}\) is equivalent to \(x^2 - 70x - 3,\!000 = 0\) for all values of \(x\) not equal to 0 or -50. Explain each step as you rewrite the original equation.

Kiran jogs at a speed of 6 miles per hour when there are no hills. He plans to jog up a mountain road, which will cause his speed to decrease by \(r\) miles per hour. Which expression represents the time, \(t\), in hours it will take him to jog 8 miles up the mountain road?

The rational function \(g(x) = \frac{x+10}{x}\) can be rewritten in the form \(g(x) = c + \frac{r}{x}\), where \(c\) and \(r\) are constants. Which expression is the result?

For each equation below, find the value(s) of \(x\) that make it true.

1. \(10 = \frac{1+7x}{7+x}\)
2. \(0.2=\frac{6+2x}{12+x}\)
3. \(0.8= \frac{x}{0.5+x}\)
4. \(3.5=\frac{4+2x}{0.5-x}\)

A softball player has had 8 base hits out of 25 at bats for a current batting average of \(\frac{8}{25}=.320\).

How many consecutive base hits does she need if she wants to raise her batting average to .400? Explain or show your reasoning.