Question

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?

Answer 1

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 `-: y` miles / hour is how long it takes her to cycle for 4 miles

2.5 miles `-: x` miles / hour is how long it takes her to run for 2.5 miles

We know that this whole process takes 1 hour:

`4/y + 2.5/x = 1`

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

`4x + 2.5y = xy`

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:

`4x + 2.5(x+8) = x(x+8)`

`4x + 2.5x + 20 = x^2 + 8x`

Combine like terms:

`20 = x^2 + 1.5x`

And get this into the form of a quadratic equation:

`x^2 + 1.5x - 20 = 0`

Plug our numbers into the quadratic formula, which is

![http://mathbitsnotebook.com/Algebra1/Quadratics/QDquadform.html]()

Where `a=1, b=1.5 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!