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?

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Answer 1 · 2015-10-16T22:24:34

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

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

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!