Problem 5 - Precalculus: Mathematics for Calculus

Question

A painter paints a wall in \(x\) hours, so the fraction of the wall that she paints in 1 hour is ____.

Step-by-Step Solution

Answer

The fraction painted in 1 hour is \(\frac{1}{x}\).

Step 1: Understand the Total Task

The total task is painting the entire wall, which takes the painter a full duration of \(x\) hours. In this problem, this duration is 100% or the whole task.

Step 2: Determine the Fraction of Work per Hour

Since the entire wall is painted in \(x\) hours, the amount of work completed in one hour is a fraction of the total work. To find this fraction, we divide 1 (representing the whole wall) by \(x\).

Step 3: Calculate the Fraction

So, the fraction of the wall that is painted in one hour is \(\frac{1}{x}\). This represents part of the task completed in one hour if the entire wall takes \(x\) hours to paint.

Key Concepts

time and work problems

When you encounter time and work problems, they often involve determining how long it will take a person or group to complete a certain task. These types of problems require you to understand the relationship between the time taken to complete the task and the rate at which the work is done.

Usually, problems are presented with a single worker, or multiple workers, and involve finding either the total time taken to finish the task or the portion of the task completed in a specific time frame.

The key to solving these problems is to break down the total task into smaller, more manageable parts, often represented as fractions of the whole.

rate of work

The concept of the rate of work is essential to solving time and work problems. It refers to the amount of work completed per unit of time, like one hour or one day. In our painter example, the rate of work is how much of the wall is painted in one hour.

If a task takes a worker 4 hours to complete, the rate of work is \( \frac{1}{4} \) of the task per hour. Similarly, if the entire task takes \( x \) hours, the work rate is \( \frac{1}{x} \) of the task per hour.

Knowing the rate of work helps you predict how much time it will take for a single worker, or a group of workers, to complete the entire task.

task division in mathematics

Task division in mathematics helps to understand and manage how complex tasks are divided into smaller sections. When you divide tasks, you can easily assign portions of the task to multiple workers or allocate specific periods to complete each fraction of the work.

In the painter problem, dividing the painting into hourly segments allows one to calculate exactly what fraction of the wall is completed each hour. This is crucial when you need to deal with multiple workers who might be working simultaneously or taking turns, as it simplifies the calculation of total work done by different people over time.

• This concept helps in optimizing and coordinating efforts efficiently.
• It allows for a clear understanding of progress in complex tasks.
• Makes it easier to apply changes if a variable such as worker count or availability changes.