In a lightning storm, the number of seconds between the flash and the bang varies directly with the number of miles you are from the lightning. It takes 15 seconds for the thunder sound to reach you from lightning that is 3 miles away. If the thunder sound reaches you in 1 second, how far away is the lightning?

15/3 = 1/?

0.2 miles

To solve this problem, we need to use the concept of direct variation. Direct variation states that when two quantities are directly proportional, their ratio remains constant.

Let's denote the number of seconds it takes for the thunder sound to reach you as 't' and the distance to the lightning as 'd'. According to the problem, the number of seconds is directly proportional to the distance: t ∝ d.

In the first scenario given, it takes 15 seconds for the thunder sound to reach you from a lightning that is 3 miles away. This can be represented as:

t_1 = k * d_1

Where t_1 is the time taken (15 seconds), d_1 is the distance (3 miles), and k is the constant of proportionality.

Now, in the second scenario, it takes only 1 second for the thunder sound to reach you. We need to find the distance, which we'll represent as d_2. Using the same formula as before:

t_2 = k * d_2

Given that t_2 is 1 second, we can substitute it into the equation:

1 = k * d_2

Now, we can use the ratios of the two scenarios to find the unknown distance:

t_1 / t_2 = d_1 / d_2

Substituting the given values:

15 / 1 = 3 / d_2

Cross multiplying:

15 * d_2 = 3 * 1

Simplifying further:

15 * d_2 = 3

Dividing both sides by 15:

d_2 = 3 / 15

Therefore, the lightning is 0.2 miles away.