Multi-Step Math Problem?

Question

A duathlon consist of a run, a bike ride, and a second run. Use the information below about the average rates of one participant who completed a 55 kilometer duathlon in 2 hours 35 minutes.

Information: 1st run 12km/h

bike 30km/h

2nd run 12km/h

MODEL:Write a verbal model that shows the race distance as the sum of the total running distance and the biking distance

TRANSLATE:Write an equation based on the vebal model

Answer 1

Let x represent the total running distance and y is the total biking distance.  There are two constraints on these distances.  The first being that the total must be 55 km since that is the total distance of the race.  The second is based on the constant speed of the athlete during the different phases and the total time required to finish the race.

We can represent this problem as a system of two linear equations:

x + y = 55 km  This is the total distance traveled.

For our second equation we should note that for any value b; b km/h is equivalent to (1/b) h/km.

((1/12) h/km)x + ((1/30) h/km)y = 2 hours 35 minutes = 2+(35/60) = 31/12

Thus our system is:

(1/12)x + (1/30)y = 31/12

x + y = 55

We can solve by replacing x  in the first equation with x = 55 - y

(1/12)(55-y) + (1/30)y = 31/12

(55/12) -(1/12)y+(1/30)y = 31/12

-(1/20)y = -24/12

y = 40

Plugging y into equation 2 x + 40 = 55  Thus x = 15

We conclude that the athlete ran 15 km and biked for 40 km.