# 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?

##### 1 Answer

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

Let's call her cycling speed

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:

Get rid of those fractions by multiplying both sides by

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

Let's replace

Combine like terms:

And get this into the form of a quadratic equation:

Plug our numbers into the quadratic formula, which is

Where

From that, we find that

OR

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!