the number of red marbles varies inversely as the square of the number of blue marbles. when there were four reds, there were 20 blues. How many reds would there be if there were only four blues?

100

r = k/b^2 , r is red, b is blue

4 = k/400
k = 1600
then r = 1600/b^2

when b = 4
r = 1600/16
= 100

To solve this problem, we can use the inverse variation equation:

red marbles ∝ 1/(blue marbles)^2

We are given that when there were four red marbles, there were 20 blue marbles. Let's assign variables to represent the number of red and blue marbles:

Let r be the number of red marbles, and b be the number of blue marbles.

We can set up the inverse variation equation with the given information:

r ∝ 1/b^2

Now we can solve for the constant of variation by plugging in the values from the given information:

4 ∝ 1/20^2

Now, let's solve for the constant of variation:

4 ∝ 1/400

To find the constant of variation, we can cross-multiply:

4 * 400 = 1

1600 = 1

Now, let's use this constant of variation to find the number of red marbles when there are only four blue marbles:

r ∝ 1/b^2

r ∝ 1/4^2

r ∝ 1/16

Now, let's solve for the number of red marbles:

r = 1/16

r = 0.0625

Therefore, there would be approximately 0.0625 red marbles if there were only four blue marbles.

To solve this problem, we can set up a proportion based on the given information.

Let's say the number of red marbles is represented by "r" and the number of blue marbles is represented by "b".

According to the problem, the number of red marbles varies inversely as the square of the number of blue marbles. This can be written as:

r ∝ 1/b^2

We can rewrite this proportion as:

r = k/b^2

where "k" is a constant of variation.

Now, let's use the given information to find the value of "k".

When there were four red marbles, there were 20 blue marbles. So we can substitute these values into the equation:

4 = k/20^2

Simplifying the equation:

4 = k/400

To isolate "k", we can multiply both sides of the equation by 400:

k = 4 * 400

k = 1600

Now that we have the value of "k", we can use it to determine how many red marbles there would be if there were only four blue marbles.

r = 1600/4^2

r = 1600/16

r = 100

Therefore, if there were only four blue marbles, there would be 100 red marbles.