Solve Algebra Rate Problem: Dragon Walking & Car Driving

[daigo]

Homework Statement

A long dragon that is 100 feet long is walking.
A car starts driving from the rear of the dragon to the head of the dragon, then drives back to the tail of the dragon.
If the car drives three times as fast as the dragon can walk, how many feet has the dragon walked by the time the car has returned to the tail of the dragon?

Homework Equations

distance = rate * time

The Attempt at a Solution

I thought I could put it into a ratio like this:

(r = rate)

x / r = 100 ft / 3r

Cross multiply:

3rx = 100r

Divide by r on both sides:

3x = 100

x = 100/3

But then I figured that neither dragon nor car was traveling a certain distance, they were just traveling at a pace.

I already know how to arrive at the answer, but I don't understand why I've arrived at the answer:

distance = rate * time

(r = rate, t = time)
rt + 100 = 3rt
100 = 2rt
50 = rt

(r = rate, y = time)
4ry = 100
ry = 25

distance = rate(t + y)
distance = rate(t) + rate(y)

(plug in from the solutions of above equations)

distance = 50 + 25

distance = 75

I don't understand from the very first step of how the equations came to be. Is there a different way to solve this? So I can understand it intuitively instead of algebraically?

 

[cepheid]

distance = rate * time

(r = rate, t = time)
rt + 100 = 3rt
100 = 2rt
50 = rt

(r = rate, y = time)
4ry = 100
ry = 25

distance = rate(t + y)
distance = rate(t) + rate(y)

(plug in from the solutions of above equations)

distance = 50 + 25

distance = 75

I don't understand from the very first step of how the equations came to be. Is there a different way to solve this? So I can understand it intuitively instead of algebraically?

This solution is not hard to understand. You're breaking the problem up into two parts. The first part is when the car moves from tail to head, which takes a time "t." If "r" is the walking speed of the dragon, then 3r is the driving speed of the car (this is given in the problem). In time t, the car moves a distance of speed*time = 3rt. HOWEVER, the distance traveled by the car must be equal to the length of the dragon PLUS the distance traveled by the dragon in time t, which is rt. Hence:

100+rt = 3rt

This equation is saying "distance traveled by dragon + length of dragon = distance traveled by car"

Now consider the second part, when the car moves from the head back to the tail, which takes an amount of time "y". In this case, the distance traveled by the car is the length of the dragon MINUS the distance traveled by the dragon. It's minus, because the car has to go less than 100 ft to reach the tail, because the tail is moving towards it at speed r.

The distance traveled by the car is: 3ry
The distance traveled by the dragon is ry

Hence, from what we said above:

3ry = 100 - ry

4ry = 100

Now you know where these equations came from.