Problem 131 - Precalculus Essentials

Question

Sharks may be scary, but they are responsible for only three deaths worldwide in 2014 . The world's deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000 ) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails.

Step-by-Step Solution

Answer

So, in thousands, mosquitoes, snails, and snakes cause approximately 755, 200, and 94 deaths, respectively, per year.

Step 1: Define the Variables

Define the variables for this problem: Let \( M \) be the number of deaths by mosquitoes, \( S \) be the number of deaths by snails and \( N \) be the number of deaths by snakes. All values are in thousands.

Step 2: Translate Words into Equations

Translate: 'The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand' means \( M = N + 661 \). 'The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand' implies \( S = N + 106 \). The total deaths are 1049 thousand, thus \( M + S + N = 1049 \).

Step 3: Substitute into Total Deaths Equation

Substitute the two equations \( M = N + 661 \) and \( S = N + 106 \) into the third equation \( M + S + N = 1049 \), we get \[( N + 661) + (N + 106) + N = 1049 \]

Step 4: Solve for N (Snake Deaths)

Solve the equation for N, which gives the number of deaths by snakes. \[ 3N + 767 = 1049 \n N = (1049 - 767) / 3 \n N = 94 \]

Step 5: Solve for M (Mosquito Deaths) and S (Snail Deaths)

Substitute N = 94 into the two equations \( M = N + 661 \) and \( S = N + 106 \) to find the values of M and S. \[ M = 94 + 661 = 755 \] and \[ S = 94 + 106 = 200 \]

Key Concepts

Variables

In mathematics, variables are fundamental components used to represent unknown values within equations. In the context of our shark problem, we define the variables to set a clear understanding of the unknowns we want to solve.

For instance, we use:

• \( M \) for the number of deaths caused by mosquitoes.
• \( S \) for the number of deaths caused by snails.
• \( N \) for the number of deaths caused by snakes.

Variables help in organizing the information and setting the ground for further calculations. Without them, formulating equations becomes more challenging, as they allow us to refer back easily to the quantities being calculated. Using variables makes solving real-world problems much more manageable and systematic.

Equations

Equations are mathematical statements that express the equality of two expressions. They are crucial for translating real-world scenarios into solvable mathematical problems.

In our exercise, we use several equations to describe the relationships between mosquito, snail, and snake deaths:

• The equation \( M = N + 661 \) indicates that mosquito deaths are 661 thousand more than snake deaths.
• The equation \( S = N + 106 \) shows that snail deaths exceed snake deaths by 106 thousand.
• Finally, the sum equation \( M + S + N = 1049 \) represents the total deaths by all three animals.

By setting up these equations, we convert the problem from a linguistic description to a form suitable for mathematical analysis and solution. This is how equations bridge the gap between abstract concepts and their practical applications.

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable, and this solution is substituted into another equation. This approach simplifies the system, making it easier to solve.

In our problem, substitution plays a pivotal role:

• First, we solve \( M = N + 661 \) and \( S = N + 106 \) for \( M \) and \( S \), respectively.
• We then substitute these expressions into the total deaths equation \( M + S + N = 1049 \).

The substituted equation becomes \( (N + 661) + (N + 106) + N = 1049 \), which simplifies to a single equation in terms of \( N \). This method allows us to focus on solving for one variable at a time, streamlining the process of finding the entire solution set.

Linear Equations

Linear equations are equations that form a straight line when graphed, representing relationships with constant rates of change. They typically appear in the form \( ax + b = c \). In the shark problem, each equation is linear.

Each variable represents a quantity, and their relationships are expressed linearly:

• \( M = N + 661 \) is a linear equation showing the relation between mosquito and snake deaths.
• \( S = N + 106 \) is another linear equation relating snail and snake deaths.
• The equation \( M + S + N = 1049 \) is the total linear equation encompassing all three variables.

These equations help depict the problem's structure clearly, allowing us to use techniques like substitution to find specific numerical answers. Understanding linear equations is crucial as they are foundational across many mathematical problems, especially in modeling real-life scenarios like this one.