Each of n cats has 2n fleas. If two cats (and their fleas) are removed, and three fleas are removed from each remaining cat, the total number of fleas remaining would be half the original total number of fleas. What is the value of n?
Please help. I really need help!
2 n^2 fleas to start
2 n^2 - 4 n - 3(n - 2) after removal
n^2 = 2 n^2 - 7 n + 6 ... 0 = n^2 - 7 n + 6
solve the quadratic for n
now:
number of cats --- n
number fleas per cat --- 2n
total number of fleas = 2n^2
after removal episode:
number of cats = n-2
fleas removed from those cats = 2(2n) = 4n
fleas removed from remaining cats = 3(n-2) = 3n - 6
fleas remaining = 2n^2 - 4n - (3n-6)
= 2n^2 - 7n + 6
2n^2 - 7n + 6 = (1/2)(2n^2) = n^2
n^2 - 7n + 6 = 0
(n - 1)(n - 6) = 0
n = 1 or n = 6, but we can't have 1 cat and then remove 2
so n = 6
check:
6 cats each with 12 fleas , total fleas = 72
2 cats removed leaves 4 cats, also 24 fleas were removed
also the remaining 4 cats each had 3 fleas removed
= 12
remaining fleas = 72-24-12 = 36
which is half of the original number of fleas
My answer is correct
Thank you. This helped me a lot.
To solve this problem, let's break it down step by step:
Step 1: Let's assume the original number of cats is n.
Step 2: Each cat has 2n fleas, so the total number of fleas in the beginning is n * 2n = 2n^2.
Step 3: Two cats are removed, which means there are now n - 2 cats remaining.
Step 4: Three fleas are removed from each remaining cat, which means each of the remaining n - 2 cats now has 2n - 3 fleas.
Step 5: The total number of fleas remaining is (n - 2) * (2n - 3).
Step 6: According to the problem, this number is half the original total number of fleas, so we have the equation:
(n - 2) * (2n - 3) = (1/2) * 2n^2.
Simplifying this equation will allow us to find the value of n:
(n - 2) * (2n - 3) = n^2.
2n^2 - 7n + 6 = n^2.
n^2 - 7n + 6 = 0.
Now we have a quadratic equation. Factoring it, we get:
(n - 6) * (n - 1) = 0.
So we have two potential solutions: n - 6 = 0 and n - 1 = 0.
Solving these equations, we find:
n = 6 or n = 1.
However, since we are removing two cats, n cannot be 1, as there would be no cats left. Thus, the only valid solution is n = 6.
Therefore, the value of n is 6.