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At the office, mice vary inversely with cats, that is, mice = k/cats, for some value of k. When there are 3r-19 cats, there are 2r+1 mice, and when there are 6r-24 mice, there are r-3 cats. Find k.

Aug 10, 2024

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Understanding the Problem

We're given an inverse relationship between mice and cats:

Mice = k/cats

We have two sets of data:

When cats = 3r - 19, mice = 2r + 1

When mice = 6r - 24, cats = r - 3

We need to find the value of k.

Setting up the Equations

Using the given information, we can create two equations:

2r + 1 = k / (3r - 19)

6r - 24 = k / (r - 3)

Solving for k

We can solve either equation for k and then substitute it into the other equation. Let's solve the first equation for k:

k = (2r + 1)(3r - 19)

Now, substitute this expression for k in the second equation:

6r - 24 = [(2r + 1)(3r - 19)] / (r - 3)

Simplify the equation:

(6r - 24)(r - 3) = (2r + 1)(3r - 19)

6r^2 - 42r + 72 = 6r^2 - 35r - 19

7r = 91

r = 13

Now, substitute r = 13 back into either of the original equations to find k. Let's use the first equation:

k = (213 + 1)(313 - 19)

k = (27)(20)

k = 540

Therefore, the value of k is 540.

Aug 10, 2024