There were 800 turtles and rabbits in a pet shop. After selling a total of 192 animals, 3/4 of the turtles and 5/6 of the rabbits were left. How many rabbits were there at first?

t+r = 800

(3/4)t + (5/6)r = 800-192

to find r, substitute for t:

(3/4)(800-r) + (5/6)r = 608
(1/12)r + 600 = 608
r/12 = 8
r = 96

To find out the number of rabbits at first, we need to set up a system of equations.

Let's assume the number of turtles at first is represented by 't' and the number of rabbits at first is represented by 'r'.

According to the problem, there were 800 turtles and rabbits in the pet shop, so we can write the equation:

t + r = 800 ---(equation 1)

After selling a total of 192 animals, we are left with:
3/4 of the turtles and 5/6 of the rabbits.

So, the number of turtles left = (3/4)t
The number of rabbits left = (5/6)r

According to the problem, the number of animals left after selling is 800 - 192 = 608. So we can write the second equation:

(3/4)t + (5/6)r = 608 ---(equation 2)

Now we have a system of equations consisting of equations 1 and 2. We can solve this system of equations to find the values of 't' and 'r'.

To solve the system of equations, we can multiply equation 1 by 3, and equation 2 by 8 to eliminate the fractions:

3(t + r) = 3(800)
8[(3/4)t + (5/6)r] = 8(608)

Simplifying these equations gives us:
3t + 3r = 2400 ---(equation 3)
6t + 10r = 4864 ---(equation 4)

Now we can solve this system of equations.

We can multiply equation 3 by 2 and subtract it from equation 4 to eliminate 't':

12t + 20r - (6t + 6r) = 4864 - (4800)
6t + 14r = 64
Subtracting equation 3 from equation 4 gives us:
6t + 10r - (3t + 3r) = 4864 - 2400
3t + 7r = 2464

Now we have a new system of equations:
6t + 14r = 64 ---(equation 5)
3t + 7r = 2464 ---(equation 6)

To solve this system of equations, we can multiply equation 5 by 2 and subtract it from equation 6 to eliminate 't':

6t + 14r - (12t + 28r) = 2464 - (128)
-6t - 14r = 2464 - 128
Adding equation 5 and equation 6 gives us:
-7r = 2336
r = 2336 / -7

Calculating r gives us r ≈ -334.29

However, we cannot have a negative number of rabbits, so this result is not valid. It seems there might be a mistake in the problem statement or calculation.

Please double-check the information provided and confirm the numbers given for the turtles and rabbits in the pet shop.