Problem 69 - College Algebra

Question

At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55 \(\%\) of the endangered species. Birds accounted for 0.7\(\%\) more than fish, and fish accounted for 1.5\(\%\) more than mammals. What percent of the endangered species came from mammals, birds, and fish?

Step-by-Step Solution

Answer

Mammals: 17.1\%, Fish: 18.6\%, Birds: 19.3\%.

Step 1: Define Variables

Let's define the percentages of endangered species for mammals, fish, and birds as \( M \), \( F \), and \( B \) respectively. From the problem, we know that these three categories make up 55\(\%\) of all endangered species. So, we write:\[ M + F + B = 55 \]

Step 2: Express Birds in Terms of Fish

According to the problem, birds account for 0.7\(\%\) more than fish. Thus, we can express the percentage of birds in terms of fish:\[ B = F + 0.7 \]

Step 3: Express Fish in Terms of Mammals

The problem states that fish account for 1.5\(\%\) more than mammals. So, we express fish in terms of mammals:\[ F = M + 1.5 \]

Step 4: Substitute Expressions

Now, substitute the expressions from Steps 2 and 3 into the equation from Step 1:1. Substitute \( F = M + 1.5 \) into the equation: \[ M + (M + 1.5) + B = 55 \]2. Substitute \( B = F + 0.7 = (M + 1.5) + 0.7 \) into the equation to get: \[ M + (M + 1.5) + (M + 1.5 + 0.7) = 55 \]

Step 5: Simplify and Solve for Mammals

Combine like terms and solve for \( M \):\[ M + M + 1.5 + M + 1.5 + 0.7 = 55 \]\[ 3M + 3.7 = 55 \]Subtract 3.7 from both sides:\[ 3M = 51.3 \]Divide by 3:\[ M = 17.1 \]

Step 6: Solve for Fish

Now, find the percentage of fish using the expression from Step 3:\[ F = M + 1.5 = 17.1 + 1.5 = 18.6 \]

Step 7: Solve for Birds

Finally, find the percentage of birds using the expression from Step 2:\[ B = F + 0.7 = 18.6 + 0.7 = 19.3 \]

Key Concepts

Equation solving

When tackling algebra problems, equation solving is a foundational skill. It allows us to find unknown values, which in this case are the percentages of mammals, fish, and birds on the endangered species list. We begin by translating the problem's conditions into mathematical equations. Start by defining the variables:

• Let \( M \) be the percentage of mammals.
• Let \( F \) be the percentage of fish.
• Let \( B \) be the percentage of birds.

We know from the problem that:\[ M + F + B = 55 \]This equation states that the sum of the percentages of these three groups equals 55%.
To solve for each variable, express each unknown in terms of one variable, simplifying until you can solve for a specific numeric value. By using substitutions from additional information given in the problem, we create a system of equations. This helps isolate one variable by expressing others in its terms, using both the initial conditions and additional contextual information as highlighted above.

Linear equations

Linear equations are equations of the first order, and their graph is a straight line in the coordinate plane. In this exercise, linear equations come into play when expressing the relationships between the different animal categories in terms of percentages. Each step in the process depends on forming and manipulating linear equations. First, notice how relationships between species are expressed:

• Birds account for 0.7% more than fish: \( B = F + 0.7 \)
• Fish account for 1.5% more than mammals: \( F = M + 1.5 \)

These equations are linear because they include variables raised to the first power and can be graphically represented as straight lines. Substitution is the key technique used here. Substituting one equation into the other helps reduce the equations into a simpler form that can be solved step by step, ultimately determining the exact percentages:

Percentage calculations

Understanding how to calculate percentages is crucial in problems like this, which involve expressing part-whole relationships as a fraction of 100. In this exercise, we're calculating what percentage mammals, fish, and birds each represent of the total number of endangered species within the specified groups, which combined make up 55% of all endangered listings.
The step-by-step approach helps break down the task into manageable parts:

• By substituting the values found for each variable, \( M = 17.1 \), \( F = 18.6 \), and \( B = 19.3 \), into their respective placeholder equations, we determine each category's percentage share of the total group.
• This involves rearranging equations and applying basic arithmetic operations like addition and subtraction to isolate variables.

By understanding each species' percentage, we can see how much more or less protection is devoted to each group and track changes over time. Mastery of percentage calculations is essential not just for mathematical problems, but also for real-world applications.