Problem 54 - College Algebra

Question

An animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?

Step-by-Step Solution

Answer

The shelter has 150 cats, 130 dogs, and 70 rabbits.

Step 1: Define the Variables

Let the number of cats be denoted by \( c \), the number of dogs by \( d \), and the number of rabbits by \( r \).

Step 2: Translate the Problem into Equations

From the problem statement, we know that the shelter has a total of 350 animals. This can be expressed as:\[ c + d + r = 350 \]Additionally, the number of rabbits is 5 less than half the number of cats:\[ r = \frac{1}{2}c - 5 \]Finally, there are 20 more cats than dogs:\[ c = d + 20 \]

Step 3: Substitute and Simplify

Substitute the expressions for \( r \) and \( d \) into the first equation:\[ c + (c - 20) + \left(\frac{1}{2}c - 5\right) = 350 \]Combine like terms:\[ 2c - 20 + \frac{1}{2}c - 5 = 350 \]

Step 4: Solve for Cats

Rearrange and simplify the equation from Step 3:\[ 2.5c - 25 = 350 \]Add 25 to both sides:\[ 2.5c = 375 \]Divide both sides by 2.5:\[ c = 150 \]

Step 5: Solve for Dogs and Rabbits

Using the expression \( c = d + 20 \), substitute \( c = 150 \) to find \( d \):\[ 150 = d + 20 \]\[ d = 130 \]Using the expression \( r = \frac{1}{2}c - 5 \), substitute \( c = 150 \) to find \( r \):\[ r = \frac{1}{2}(150) - 5 = 75 - 5 = 70 \]

Step 6: Summarize the Solution

There are 150 cats, 130 dogs, and 70 rabbits at the shelter.

Key Concepts

Solving Equations

Solving equations is all about finding the values of unknown variables that satisfy an equation or a system of equations. In this animal shelter problem, you begin by defining your variables, which helps in structuring the information you have. Here, we used variables:

• \( c \) for the number of cats,
• \( d \) for the number of dogs,
• \( r \) for the number of rabbits.

Next, you express relationships or conditions as equations. For instance, the total number of animals provides one equation. Other conditions like how rabbits and cats relate provide additional equations. Once you have these equations, you solve them using algebraic methods such as:

• Substitution
• Combining like terms
• Rearranging terms to isolate the variable

In this case, we eventually find that there are 150 cats, helping us easily find the numbers for dogs and rabbits once the relationships between these animals are used.

System of Equations

A system of equations consists of multiple equations that are analyzed together because they share one or more variables. Solving a system means finding the values that satisfy all equations simultaneously. In this problem, our system utilizes three equations:\[\begin{align*}c + d + r &= 350 \r &= \frac{1}{2}c - 5 \c &= d + 20 \end{align*}\] These equations give us a complete picture of how the numbers of animals relate to each other: cats, dogs, and rabbits are interconnected. The goal is to find a consistent set of values for \( c, d, \) and \( r \).To solve this system, you start by substituting expressions from one equation into another, which eliminates variables step by step. For example, substituting the expressions for rabbits and dogs into the total number of animals equation drastically simplifies the problem to finding just one unknown, \( c \). This makes the problem more manageable, leading you to a solution for the entire system.

Problem Solving Strategies

Algebra word problems can initially seem challenging because they require more than just number crunching; you're deciphering a situation, identifying the math behind it, and formulating a plan. One effective problem-solving strategy is breaking down the problem: Start by clearly defining the unknowns and expressing them as variables. In this shelter scenario, setting the variables straight helped in crafting the equations necessary for a solution. Next, translate the language of the problems into mathematical statements or equations. This involves carefully reading the problem to ensure you've captured all the critical relationships and conditions into equations. Here, identifying that rabbits are 5 less than half the number of cats gives us a vital equation. Finally, systematically solve the equations using algebraic techniques such as substitution or elimination. Always verify your solutions to ensure they make sense within the context of the problem, helping prevent errors like forgotten conditions or misinterpretations. With practice, these strategies will allow you to tackle even more complex algebra word problems efficiently.