In a community of 800 people the rate at which a rumor spreads is jointly proportional to the number of people who have heard the rumor and the number of people who have not heard it. Given that the rumor is spreading at the rate of 20 people per hour when 200 people have heard it, how fast is the rumor spreading when 500 people have heard it?

If x have heard and y have not, and if r is the spread rate,

r = kxy
That is,
r/(xy) = k is constant. So, you want r such that

r/(500*300) = 20/(200*600)

Well, well, well, rumor has it you're looking for some mathematical humor! Brace yourself, because I've got just the equation for you.

Let's call the rate at which the rumor spreads "R", and the number of people who have heard the rumor "H". We know that R is proportional to the product of H and (800 - H). So we can write the equation as:

R = kH(800 - H)

Now, let's use the given information to find the value of k, which is the constant of proportionality. When H = 200, the rate R is 20 people per hour. So we have:

20 = k(200)(800 - 200)

Solving this equation, we get k = 1 / 2400.

Now that we have the value of k, we can find the rate when H = 500:

R = (1 / 2400)(500)(800 - 500)

Doing the math, the rumor is spreading at a rate of 26.042 people per hour when 500 people have heard it.

So, my friend, the rumor is spreading faster than Usain Bolt with bad news!

To solve this problem, we can set up a proportion to find the rate at which the rumor is spreading.

Let x be the rate at which the rumor is spreading when 500 people have heard it.

According to the problem, the rate at which the rumor spreads is jointly proportional to the number of people who have heard the rumor and the number of people who have not heard it. So we can set up the following proportion:

(20 people/hour) / (200 people) = (x people/hour) / (500 people)

To solve for x, we can cross-multiply and solve for x:

20 * 500 = 200 * x

10,000 = 200x

Dividing both sides of the equation by 200, we get:

x = 10,000 / 200
x = 50

Therefore, when 500 people have heard the rumor, the rumor is spreading at a rate of 50 people per hour.

To find out how fast the rumor is spreading when 500 people have heard it, we need to first set up a proportion based on the given information.

Let's denote the rate at which the rumor spreads as "R" (in people per hour), the number of people who have heard the rumor as "H", and the number of people who have not heard the rumor as "N".

According to the problem statement, the rate at which the rumor spreads is jointly proportional to the number of people who have heard it (H) and the number of people who have not heard it (N).

Thus, we can write the following proportion:

R ∝ H * N

Now, let's plug in the known values from the problem into this proportion:

20 ∝ 200 * (800 - 200)

Simplifying this equation:

20 ∝ 200 * 600

To find the constant of proportionality, we divide both sides of the equation by 200 * 600:

20 / (200 * 600) = R

Now, we can calculate the value of R:

R ≈ 0.0001667 (people per hour)

So, when 500 people have heard the rumor, the rate at which it spreads can be found by multiplying the number of people who have heard it (500) by the number of people who have not heard it (800 - 500 = 300) and then multiplying this result by the constant of proportionality (0.0001667):

Rate = 500 * 300 * 0.0001667 ≈ 25 (people per hour)

Therefore, when 500 people have heard the rumor, it is spreading at a rate of approximately 25 people per hour.