Problem 40 - Algebra 1

Question

You do 4 loads of laundry each week at a launderette where each load costs \(\$ 1.25 .\) You could buy a washing machine that costs \(\$ 400 .\) Washing 4 loads at home will cost about \(\$ 1\) per week for electricity. How many loads of laundry must you do in order for the costs to be equal?

Step-by-Step Solution

Answer

You need to do laundry for 400 weeks for the costs to be the same whether you do it at a launderette or buy a new machine and do it at home.

Step 1: Translate the word problem into a mathematical equation

Let \( x \) be the number of loads of laundry. Each load at the launderette costs $1.25, thus 4 loads cost $5 per week. Thus, doing \( x \) loads of laundry at the launderette will cost \( 5x \). On the other hand, buying a washing machine costs $400 initially and then each load costs $1, thus 4 loads cost $4 per week. Thus, doing \( x \) loads of laundry at home will cost \( 400 + 4x \). To find out when the costs are equal the equation to solve is \( 400 + 4x = 5x \).

Step 2: Simplify the equation

Subtract \( 4x \) from both sides to get \( x = 400 \).

Step 3: Interpret the result

This means that after 400 weeks, the costs of doing laundry at the launderette and at home will be the same.

Key Concepts

Cost Analysis of Doing Laundry

To tackle the question of whether it's cheaper to use a launderette or invest in a washing machine for home use, we perform a cost analysis. This involves comparing the expenses associated with each option. The lauderette charges for each laundry load, which accumulates over time. Conversely, a washing machine requires an upfront cost, but offers lower running costs per load.

• Launderette Costs: Each load costs $1.25, which totals $5 per week for 4 loads. Over multiple weeks, this becomes a significant expense.
• Washing Machine Costs: The initial purchase is $400, plus the variable cost of $1 per week for electricity when performing 4 loads.

Understanding these costs allows us to calculate when the investment in a washing machine breaks even compared to using a launderette. With this analysis, decision-making becomes more straightforward.

Solving Word Problems in Algebra

Word problems can seem daunting, but they often follow a logical structure that makes them easier to tackle once broken down. The key is to translate the situation into a clear mathematical equation, allowing for analysis and solution.

When faced with a word problem, follow these steps:

• Identify Variables: Determine what the unknowns are and represent them using variables, e.g., let "x" be the number of laundry loads.
• Set Up the Equation: Convert the words into a mathematical representation based on given data and relationships. In our exercise, the equation becomes: cost of using launderette equals cost of washing at home.
• Solve the Equation: Employ algebraic techniques such as simplification and isolation of variables to solve for the unknown.

Practicing these steps will improve problem-solving skills and build confidence in handling various algebraic word problems.

Understanding the Laundry Problem

The laundry problem is a classic representation of comparing ongoing expenses versus one-time purchases. It combines elements of practical budgeting and simple algebra to understand real-life cost management.

In this context, consider the following:

• Real-Life Application: Decisions like these occur frequently, where one must weigh daily expenses against a larger single investment.
• Time Factor: The solution reveals that performing 400 loads at home versus a launderette will equate in cost. This break-even point highlights how costs accumulate over time.

By accurately breaking down the problem into solvable steps, one can make informed decisions about personal budget strategies and optimize spending efficiently.