Problem 70 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. A survey asked 505 companies whether they would continue to match their employees' contributions to their \(401 \mathrm{k}\) retirement plans. Find the number of companies that will continue to match the contributions. Three out of five employers maintain \(401(\mathrm{k})\) match despite economic crisis. (Source: www.americanbenefitscouncil.org, March 17, 2009)

Step-by-Step Solution

Answer

303 companies

Step 1: Understand the Given Information

The problem states that 505 companies were surveyed. Out of the companies surveyed, three out of five are maintaining their 401(k) match contributions.

Step 2: Set Up a Proportion

We know that 3 out of 5 employers are maintaining the match. This fraction can be written as \( \frac{3}{5} \). We need to find out how many companies this represents out of the 505 surveyed. Let this number be \( x \).

Step 3: Write the Proportion

We can set up the proportion as follows: \[ \frac{3}{5} = \frac{x}{505} \]

Step 4: Solve for x

To find \( x \), cross-multiply and solve the equation: \[ 3 \times 505 = 5 \times x \] \[ 1515 = 5x \] \[ x = \frac{1515}{5} \] \[ x = 303 \]

Step 5: Interpret the Result

The solution indicates that 303 out of the 505 companies surveyed will continue to match their employees' 401(k) retirement plan contributions.

Key Concepts

Solving Proportions

Proportions are equations that show two ratios are equivalent. They are fundamental in many areas of algebra and real-life applications. To solve a proportion, you first set up the equation to equal something we can work with. For instance, if we know that 3 out of 5 companies maintain their 401(k) contributions, and we need to find out how many out of 505 companies would do the same, we can set up a proportion: \[ \frac{3}{5} = \frac{x}{505} \] Here, \[ \frac{3}{5} \] is the ratio we know, and \[ \frac{x}{505} \] is the ratio we want to find. By setting these two ratios equal, we’re leveraging the power of proportions to solve for the unknown. Once the proportion is set, you can solve it using cross-multiplication.

Cross-Multiplication

Cross-multiplication is a method used to solve proportions. It involves multiplying across the equals sign in a diagonal pattern. For the proportion \[ \frac{3}{5} = \frac{x}{505} \], you would cross-multiply by multiplying 3 by 505 and 5 by x. This gives the equation: \[ 3 \times 505 = 5 \times x \] which simplifies to: \[ 1515 = 5x \] To find x, you then divide both sides by 5: \[ x = \frac{1515}{5} \] and this results in: \[ x = 303 \] So, 303 companies out of 505 are expected to continue matching 401(k) contributions. Cross-multiplication simplifies the process of solving proportions and is a crucial skill in algebra.

Interpreting Survey Data

Surveys are a common method to gather data about preferences, behaviors, or facts about a group. Interpreting survey data correctly is important to make informed decisions. For example, in the given problem, we know that 505 companies were surveyed, and 3 out of 5 maintain their 401(k) contributions. This information helps us predict larger trends. By setting up and solving a proportion, we determined that approximately 303 out of the 505 companies will continue their contributions. This tells us something about company behavior trends during economic challenges. Proper interpretation of survey data involves understanding ratios, setting up proportions, solving for unknowns, and validating the results to ensure accuracy.

Retirement Plan Contributions

Retirement plan contributions, particularly to 401(k) plans, are significant for employees planning their future. Companies may match these contributions, enhancing the employee’s retirement savings. In challenging economic times, not every company will maintain this match. Knowing how many companies continue these contributions can aid in understanding the broader economic landscape. For instance, if a survey shows that 3 out of 5 companies are maintaining their 401(k) matches, and we find that 303 out of 505 companies (via proportion calculations) still provide this benefit, it gives employees reassurance or helps policymakers make decisions. Retirement plan contributions reflect a company's financial stability and commitment to employee welfare.