# A woman cycles 8 mi/hr faster than she runs. Every morning she cycles 4 mi and runs 2 1/2 mi, for a total of one hour of exercise. How fast does she run?

Oct 16, 2015

We need to figure out how much time she spends cycling and walking each morning, then figure how her speed from that

#### Explanation:

Let's get this into a more math-y format. First of all, we want to know her running speed. Let's call that $x$

$x$ = running speed (miles / hour)

Let's call her cycling speed $y$...

$y$ = cycling speed (miles / hour)

So, she cycles for 4 miles, and runs for 2.5 miles

4 miles $\div y$ miles / hour is how long it takes her to cycle for 4 miles

2.5 miles $\div x$ miles / hour is how long it takes her to run for 2.5 miles

We know that this whole process takes 1 hour:

$\frac{4}{y} + \frac{2.5}{x} = 1$

Get rid of those fractions by multiplying both sides by $\left(x\right) \left(y\right)$ (the lowest common denominator of 4 and 2.5):

$4 x + 2.5 y = x y$

From the question, we know that her cycling speed is 8 miles / hour faster than her running speed. So, we can say that

$y = x + 8$

Let's replace $y$ in our equation, then:

$4 x + 2.5 \left(x + 8\right) = x \left(x + 8\right)$

$4 x + 2.5 x + 20 = {x}^{2} + 8 x$

Combine like terms:

$20 = {x}^{2} + 1.5 x$

And get this into the form of a quadratic equation:

${x}^{2} + 1.5 x - 20 = 0$

Plug our numbers into the quadratic formula, which is

Where $a = 1 , b = 1.5 \mathmr{and} c = - 20$

From that, we find that

$x = 3.78$

OR

$x = - 5.28$

We know that this woman cannot run -5.28 miles per hour (she can't run at a negative speed), so

her running speed ($x$) must be 3.78 miles/hour, and her cycling speed (8 miles/hour faster) is 11.78 miles/hour

Let's check:

2.5 miles, at 3.78 miles/hour would take 0.661 hours

4 miles, at 11.78 miles/hour would take 0.339 hours

For a total of 1 hour!