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## 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 travelling a certain distance, they were just travelling 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?