An animal shelter has 1008 cats, dogs, and rabbits. The ratio of the number of dogs to the number of cats in the animal shelter is 3:7. The ratio of the number of rabbits to the number of dogs is 2:5. How many rabbits are there in the shelter? Show how to solve please.
To solve this problem, we can use a system of equations. Let's assign variables to the number of dogs, cats, and rabbits in the animal shelter.
Let D represent the number of dogs.
Let C represent the number of cats.
Let R represent the number of rabbits.
According to the problem, we have the following information:
1. The ratio of dogs to cats is 3:7. This can be written as D/C = 3/7.
2. The ratio of rabbits to dogs is 2:5. This can be written as R/D = 2/5.
Now, we can set up the system of equations:
Equation 1: D/C = 3/7
Equation 2: R/D = 2/5
To solve the system of equations, we will use the method of substitution.
From Equation 1, we can isolate D by multiplying both sides by C:
D = (3/7)C
Now, substitute this expression for D in Equation 2:
R/(3/7)C = 2/5
To simplify, multiply both sides of the equation by 5/2:
(R/(3/7)C) * (5/2) = 2/5 * (5/2)
After simplifying, we get:
(R/C) * (35/6) = 2/1
Now, cross multiply to solve for R:
R * (35/6) = 2 * C
(R * 35) / 6 = 2 * C
R * 35 = 12 * C
From this equation, we can deduce that 35 divides 12 evenly. Hence, C must be a multiple of 35, say 35k. Substituting C = 35k in this equation:
R * 35 = 12 * 35k
Divide both sides by 35:
R = 12k
This equation shows that R is a multiple of 12. So, to find the number of rabbits, we need to calculate the multiples of 12 that are less than or equal to 1008, the total number of animals.
The multiples of 12 are given by 12, 24, 36, 48, 60, 72, 84, ..., 1008.
Dividing 1008 by 12 gives us 84, which means there are 84 sets of 12 rabbits in the animal shelter. Therefore, the number of rabbits in the shelter is 12 * 84 = 1008.
d : c = 3 : 7
r : d = 2 : 5 or d : r = 5 : 2
then d:c = 15:35 and d:r = 15:6
so d:c:r = 15:6:35
let number of dogs be 15x , number of cats be 6x, and number of rabbits be 35x
then 15x + 6x + 35x = 1008
56x = 1008
x = 18
sub back into my definitions