Runs 2x rate paddles, bikes 8 km/hr faster than runs, skis 7 km/hr faster than paddles. Ski rate?

Question

Kate runs twice the rate she paddles. She bikes 8 km/hr faster than she runs and skis 7 km/hr faster than she paddles. How fast does she ski?

The quadrathlon consists of a 10 km ski, a 8 km run, a 8 km paddle, and a 20 km bike ride.

 When she completed the course, Kate noticed that her total time running and paddling was exactly the same as her total time skiing and biking.

 

So far have figured out the following equations:

2p = r

r+8 = b

p+7 = s

r+p = s+b

Answer 1

The first three equations are good, but the last one isn't.

distance = speed * time

time = distance / speed

The time she spends skiing is 10/s

The time she spends running is 8/r

The time she spends paddling is 8/p

The time she spends biking is 20/b

Saying that her total time running and paddling is the same as skiing and biking means this:

(8/r) + (8/p) = (10/s) + (20/b)

Let's multiply both sides by bprs to get rid of the fractions:

8bps + 8brs = 10bpr + 20prs

Now use the other three equations:

r = 2p

b = 2p + 8

s = p + 7

And substitute them into 8bps + 8brs = 10bpr + 20prs

8(2p+8)(p)(p+7) + 8(2p+8)(2p)(p+7) = 10(2p+8)(p)(2p) + 20(p)(2p)(p+7)

8p(2p^2 + 8p + 14p + 56) + 16p(2p^2 + 8p + 14p + 56) = 20(p^2)(2p+8) + 40(p^2)(p+7)

8p(2p^2 + 8p + 14p + 56) + 8p(2)(2p^2 + 8p + 14p + 56) = 20(p^2)(2p+8) + 20(p^2)(2)(p+7)

8p(2p^2 + 22p + 56 + 4p^2  + 44p + 112) = 20(p^2)(2p + 8 + 2p + 14)

8p(6p^2 + 66p + 168) = 20(p^2)(4p + 22)

48p(p^2 + 11p + 24) = 40(p^2)(2p + 11)

6(p^2 + 11p + 24) = 5p(2p + 11)

6p^2 + 66p + 168 = 10p^2 + 55p

-4p^2 + 11p +168 = 0

quadratic formula

p = (-11 +- sqrt( (11^2) - 4(-4)(168) ) / 2(-4)

p = (-11 +- sqrt(121 + 2688) ) / (-8)

p = (-11 +- sqrt(2809) ) / (-8)

p = (-11 +- 53) / (-8)

p = -64 / (-8) or 42 / (-8)

p = 8 or p = -21 / 4

But it's speed, so it's not negative (she isn't paddling backwards), so we can throw out p = -21 / 4

p = 8 km/hr

s = p + 7

s = 15 km/hr