A pet store has only cats and dogs. The ratio of the number of cats to the number of dogs in the pet store is 2:3 : 1/4 of the cats and 1/2 of the dogs wear collars. If there are 48 animals that wear collars, how many total animals are there in the pet store?
The Correct Answer and Explanation is:
Let’s break down the problem step-by-step:
Problem Recap:
- The pet store has only cats and dogs.
- The ratio of the number of cats to dogs is 2:3.
- 1/4 of the cats wear collars.
- 1/2 of the dogs wear collars.
- The total number of animals wearing collars is 48.
- We need to find the total number of animals in the store.
Step 1: Define variables
Let the number of cats be CC and the number of dogs be DD.
Given the ratio of cats to dogs: CD=23 ⟹ C=23D\frac{C}{D} = \frac{2}{3} \implies C = \frac{2}{3} D
Step 2: Calculate animals wearing collars
- Number of cats wearing collars = 14C\frac{1}{4} C
- Number of dogs wearing collars = 12D\frac{1}{2} D
The total number of animals wearing collars is given as 48: 14C+12D=48\frac{1}{4} C + \frac{1}{2} D = 48
Step 3: Substitute C=23DC = \frac{2}{3} D into the collar equation
14×23D+12D=48\frac{1}{4} \times \frac{2}{3} D + \frac{1}{2} D = 48
Simplify: 212D+12D=48\frac{2}{12} D + \frac{1}{2} D = 48 16D+12D=48\frac{1}{6} D + \frac{1}{2} D = 48
Convert 12\frac{1}{2} to sixths: 16D+36D=48\frac{1}{6} D + \frac{3}{6} D = 48 46D=48\frac{4}{6} D = 48
Simplify 46\frac{4}{6} to 23\frac{2}{3}: 23D=48\frac{2}{3} D = 48
Step 4: Solve for DD
D=48×32=72D = 48 \times \frac{3}{2} = 72
Step 5: Find CC
C=23×72=48C = \frac{2}{3} \times 72 = 48
Step 6: Find total animals
Total=C+D=48+72=120\text{Total} = C + D = 48 + 72 = 120
Answer:
The pet store has 120 animals in total.
Detailed Explanation:
This problem involves a ratio and fractional parts applied to a group of animals. First, understanding the ratio of cats to dogs, 2:3, tells us that for every 2 cats, there are 3 dogs. We represent this relationship with variables CC (cats) and DD (dogs), expressing cats as a fraction of dogs: C=23DC = \frac{2}{3}D.
Next, the problem mentions that a certain fraction of these animals wear collars: 1/4 of cats and 1/2 of dogs. We use this information to write an equation relating the number of animals wearing collars to the total number of animals: 14C+12D=48\frac{1}{4}C + \frac{1}{2}D = 48.
By substituting the ratio expression for cats (C=23D)(C = \frac{2}{3}D) into the collar equation, we reduce the problem to an equation in terms of DD only. Simplifying this equation involves converting fractions to a common denominator and combining terms.
Solving the simplified equation gives the exact number of dogs D=72D = 72. Then, using the ratio again, we find the number of cats C=48C = 48. Adding these together provides the total number of animals in the store, which is 120.
This approach ensures the problem is tackled methodically—using algebra and fraction manipulation—resulting in a clear and accurate solution.