Problem Solving

BTGModeratorVI wrote: ↑

Sun Aug 02, 2020 6:56 am

In a certain animal shelter, the ratio of the number of dogs to the number of cats is 15 to 7. If 12 additional cats were to be taken in by the shelter, the ratio of the number of dogs to the number of cats would be 15 to 11. How many dogs are in the shelter?

A. 15
B. 25
C. 30
D. 45
E. 60

Answer: D
Source: Veritas Prep

Here's a solution that involves two variables

In a certain animal shelter, the ratio of the number of dogs to the number of cats is 15 to 7
Let D = number of dogs in shelter
Let C = number of cats in shelter
We can write: D/C = 15/7
Cross multiply to get: 7D = 15C
Rewrite as: 15C - 7D = 0

If 12 additional cats were to be taken in by the shelter, the ratio of the number of dogs to the number of cats would be 15 to 11
So, C + 12 = RESULTING number of cats in the shelter
We can write: D/(C + 12) = 15/11
Cross multiply to get: 11D = 15(C + 12)
Expand right side to get: 11D = 15C + 180
Rewrite as: 12C - 11D = -180

How many dogs are in the shelter?
We have the following system:
15C - 7D = 0
12C - 11D = -180

Subtract the bottom equation from the top equation to get: 4D = 180
Solve: D = 45

Answer: D

Cheers,
Brent

BTGModeratorVI wrote: ↑

Sun Aug 02, 2020 6:56 am

In a certain animal shelter, the ratio of the number of dogs to the number of cats is 15 to 7. If 12 additional cats were to be taken in by the shelter, the ratio of the number of dogs to the number of cats would be 15 to 11. How many dogs are in the shelter?

A. 15
B. 25
C. 30
D. 45
E. 60

Answer: D

Solution:

We can let the original number of dogs and cats be 15x and 7x, respectively. After the additional cats are taken in, the number of dogs is unchanged, but the number of cats has increased to 7x + 12. We can create the equation for the new ratio as:

15x / (7x + 12) = 15 / 11

15x(11) = 15(7x + 12)

11x = 7x + 12

4x = 12

x = 3

Therefore, there are 15(3) = 45 dogs in the shelter.

Answer: D