Problem 22 - Precalculus

Question

Daniel's Exotic Animal Rescue houses snakes, tarantulas and scorpions. When asked how many animals of each kind he boards, Daniel answered: 'We board 49 total animals, and I am responsible for each of their 272 legs and 28 tails.' How many of each animal does the Rescue board? (Recall: tarantulas have 8 legs and no tails, scorpions have 8 legs and one tail, and snakes have no legs and one tail.)

Step-by-Step Solution

Answer

15 snakes, 21 tarantulas, and 13 scorpions.

Step 1: Define Variables

Let's define variables to represent the number of each type of animal: Let \( s \) be the number of snakes, \( t \) be the number of tarantulas, and \( c \) be the number of scorpions.

Step 2: Create Total Animals Equation

Daniel boards a total of 49 animals. This gives us our first equation: \( s + t + c = 49 \).

Step 3: Create Legs Equation

Tarantulas and scorpions have 8 legs each, while snakes have no legs. Therefore, for the total number of legs: \( 8t + 8c = 272 \).

Step 4: Create Tails Equation

Tarantulas have no tails, scorpions have 1 tail, and snakes have 1 tail. For the total number of tails: \( c + s = 28 \).

Step 5: Solve System of Equations

Now, we solve the system of equations:1. \( s + t + c = 49 \) 2. \( 8t + 8c = 272 \) 3. \( c + s = 28 \)From equation (2), we have: \( t + c = 34 \) (dividing by 8).Using equation (3), we rewrite as \( s = 28 - c \).Substitute \( s = 28 - c \) in equation (1):\[ (28 - c) + t + c = 49 \]\[ 28 + t = 49 \]\[ t = 21 \].Substitute \( t = 21 \) in equation \( t + c = 34 \):\[ 21 + c = 34 \]\[ c = 13 \].Substitute \( c = 13 \) back into \( s = 28 - c \):\[ s = 28 - 13 = 15 \].

Step 6: Verify Solution

Verify by substituting back into each original equation:- Total animals: \( 15 + 21 + 13 = 49 \)- Total legs: \( 8(21) + 8(13) = 168 + 104 = 272 \)- Total tails: \( 13 + 15 = 28 \).All conditions are satisfied, confirming the solution.

Key Concepts

animal counting problems

Animal counting problems are a fun and engaging way to apply algebraic concepts to real-world scenarios. In these puzzles, we need to determine the number of each type of animal based on a set of given conditions, such as the total number of animals or the total number of legs and tails. These problems develop critical thinking by requiring the solver to analyze relationships and constraints in a creative manner.

For instance, in Daniel's Exotic Animal Rescue, we know the total number of animals, legs, and tails. By breaking down the problem into specific questions, like how many animals each attribute belongs to, we shed light on possible solutions. Such exercises enhance our skills in crafting equations and making sense of the involved variables. Each animal contributes differently to legs and tails count, guiding how we build our system of equations.

problem solving in algebra

Algebraic problem solving is all about turning words into mathematical equations and symbols. Let's think of each piece of information given as a clue that helps us form a picture of the whole situation.

When tackling the problem at hand, translating Daniel's statements into equations is our first step. We break down the problem by understanding what's being asked and the facts provided. One tip is to list your known quantities and what you need to find out.

In this case, we carefully interpret the information on animals, legs, and tails, then align them with mathematical expressions. Through practice, this approach hones our ability to dissect problems and find ways to organize data efficiently, leading to a clearer path toward the solution.

linear equations

Linear equations are foundational in algebra and involve expressions where each term is either a constant or the product of a constant and a variable. The exercise presented is a classic example involving three linear equations.

Here are the linear equations in Daniel's animal counting exercise:

• The first equation represents the total number of animals: \( s + t + c = 49 \).


• The second equation accounts for the total number of legs: \( 8t + 8c = 272 \).


• The third equation deals with the total number of tails: \( c + s = 28 \).

A crucial step is solving these equations simultaneously, which involves manipulating them to isolate and find values for each variable. These equations are linear because they have a constant sum, which directly equates to a straight line relationship when plotted on a graph. As such, understanding linear equations is fundamental to breaking down and resolving real-world algebraic problems.

variables in algebra

Variables are symbols, often letters, used to represent unknown values in mathematical equations. In algebra, they serve as placeholders that allow us to create equations and explore their potential solutions.

In our animal counting challenge, we employ variables \( s \), \( t \), and \( c \) for snakes, tarantulas, and scorpions, respectively. These variables help summarize the problem's conditions into a mathematical form that can be methodically solved.

Understanding how variables work facilitates better comprehension of algebraic expressions and equations. They let us transition from qualitative descriptions to quantitative analysis. By solving for each variable, we find precise amounts, transforming abstract queries into concrete answers. Embracing the use of variables in algebra thus promotes a more streamlined and logical problem-solving process.