Problem 43 - Prealgebra

Question

A person-day is a unit of measure that represents one person working for one day. A painting contractor estimates that it will take 24 person-days to paint a house. Write and solve an equation to find how many painters the contractor will need to hire to paint the house in 6 days.

Step-by-Step Solution

Answer

The contractor needs to hire 4 painters.

Step 1: Understand the Problem

The painting contractor needs to complete a task in 24 person-days. The objective is to determine the number of painters required to complete the task in 6 days.

Step 2: Define the Variables

Let's denote the number of painters needed as \( x \). Each painter will work for 6 days. Thus, we must find \( x \) such that the product of \( x \) and the number of days \( (6) \) gives the total person-days \( (24) \).

Step 3: Set Up the Equation

Using the variables defined, the relationship can be expressed as an equation: \[ x \times 6 = 24 \]. This equation represents the total work in person-days.

Step 4: Solve the Equation

To find \( x \), divide both sides of the equation by 6: \[ x = \frac{24}{6} \]. Calculate the division to determine \( x \).

Step 5: Calculate and Interpret the Solution

Perform the division: \[ x = 4 \]. Thus, the contractor needs to hire 4 painters to complete the work in the stipulated 6 days.

Key Concepts

Equation Solving

Equation solving is an invaluable skill when tackling real-world problems, like determining the resources required for a project. In this exercise, we formulated an equation to find the number of painters needed. The process starts by understanding the problem and identifying what is required.
We know the task must be completed in 24 person-days, but the goal is to finish in 6 days. Hence, we need an equation to model this scenario. The key is to express the problem with an equation such as:

• Identify the unknown variable, typically represented by symbols like \( x \).
• Set up an equation ensuring that both sides represent the same value or concept.
• Use mathematical operations to manipulate and solve this equation, maintaining balance by performing the same operations on both sides.

This efficient and systematic approach not only helps in solving the problem at hand but also provides a foundation for tackling more complex issues in mathematics and everyday life.

Multiplication

Multiplication is at the core of calculating how different factors in a problem interact with each other. In this scenario, each painter works for 6 days. The total amount of work done can be represented by a multiplication expression, \( x \times 6 = 24 \).
Understanding multiplication involves knowing how to calculate repeated addition effortlessly:

• Each painter contributes 6 days of work. If there are \( x \) painters, then they collectively contribute \( x \times 6 \) person-days.
• Using multiplication means you are grouping numbers, making it simple to find the total work being done when variables and constants are involved.
• Remember that multiplication can be visualized as creating arrays or groups, which can make abstract concepts more concrete.

Overall, multiplication ensures we quickly quantify how multiple identical tasks sum up in a given scenario.

Division

Division is a mathematical operation that allows us to split a given quantity into equal parts, making it incredibly useful for finding unknown quantities in equations. In our given problem, to solve \( x \times 6 = 24 \), we used division to isolate \( x \).
Here’s how division plays a role:

• To find \( x \), we divide both sides of the equation by 6, yielding \( x = \frac{24}{6} \).
• Division reverses the operation of multiplication, allowing us to determine how many units of one number fit into another.
• This process provides clarity in breaking down total amounts, making division essential for finding averages, rates, and proportions.

This operation is critical in everyday applications, especially when determining allocations of resources and distributing quantities evenly.

Variables

Variables serve as placeholders for unknown values we aim to determine a value for. In mathematical equations, they allow us to write flexible expressions that can be solved to find these unknowns.
In this problem, the variable \( x \) denotes the number of painters needed. Here’s why variables are significant:

• Variables provide a means to express mathematical relationships in an abstract format, allowing us to see the connection between different quantities.
• By assigning a variable like \( x \), we can manipulate the equation using algebraic principles to solve for this unknown quantity.
• They allow for general solutions to specific problems, making them vital for expressing a wide range of scenarios within a simple framework.

The use of variables is a fundamental concept in algebra, enabling us to model and solve real-world problems systematically and efficiently.