Problem 73 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)

Step-by-Step Solution

Answer

Approximately 15,628,571 full-time equivalent workers.

Step 1: Understand the Given Information

Identify the key pieces of information: The death rate is 3.5 workers per 100,000 full-time workers, and in total, 547 workers were killed.

Step 2: Set Up a Proportion

Use the proportion \(\frac{3.5}{100,000} = \frac{547}{x}\) where \(x\) is the number of full-time equivalent workers. This sets up a ratio comparing the known death rate to the total deaths and unknown total workers.

Step 3: Cross Multiply

To solve for \(x\), cross multiply the equation: \(3.5 \times x = 547 \times 100,000\).

Step 4: Solve for x

Divide each side by 3.5 to isolate \(x\): \(x = \frac{547 \times 100,000}{3.5}\).

Step 5: Calculate the Value and Round

Perform the division and round to the nearest whole number: \(x \approx 15,628,571\).

Key Concepts

Cross Multiplication

Cross multiplication is a method used to solve proportions, which are equations where two ratios are set equal to each other. This technique is particularly helpful when dealing with word problems involving ratios. In our exercise, we have the proportion \(\frac{3.5}{100,000} = \frac{547}{x}\).

To cross multiply, we multiply the numerator of one ratio by the denominator of the other ratio: \(3.5 \times x = 547 \times 100,000\). This step eliminates the fractions, making it easier to solve for the unknown variable, which is the number of full-time equivalent workers in this problem.

After cross-multiplying, we end up with a simple algebraic equation to solve.

Solving Equations

Once we have set up our equation through cross-multiplication, the next step is to solve for the variable. In this case, the equation is \(3.5 \times x = 547 \times 100,000\). To isolate \(x\), we need to get rid of the coefficient multiplying it.

We do this by dividing both sides of the equation by 3.5: \(x = \frac{547 \times 100,000}{3.5}\). This step simplifies our equation and allows us to solve for \(x\).

Solving these types of equations requires basic algebraic manipulations, but understanding each step is crucial for accurately finding the solution.

Proportionality

Proportionality is a fundamental concept in algebra that describes the relationship between two quantities that change in a consistent way. A proportion is essentially a statement that two ratios are equal.

In our problem, the ratio of workers killed to full-time equivalent workers is given by \(\frac{3.5}{100,000}\). We are asked to find the total number of workers (\(x\)) given that 547 workers were killed.

Setting up the proportion \(\frac{3.5}{100,000} = \frac{547}{x}\) and solving it allows us to understand how changes in one part of the ratio will affect the other. This helps in solving various real-world problems where understanding proportional relationships is key.

Word Problems

Word problems can often seem intimidating, but they become manageable when broken down into smaller steps. Let's apply this to our given exercise.

We start by identifying key information: the death rate (3.5 per 100,000 workers) and total deaths (547 workers). Next, we set up a proportion that represents the relationship given in the problem: \(\frac{3.5}{100,000} = \frac{547}{x}\).

By solving this proportion through cross-multiplication and basic algebra, we find the solution. Approaching word problems systematically by understanding the provided data, setting up equations, and solving them through established mathematical techniques, can make even complex problems straightforward and solvable.