Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Painting a Room Hazel can paint a room in 2 hours less time than her friend Gwyneth. Working together, they can paint the room in 2.4 hours. How long does it take each woman to paint the room by herself?

Short answer

Gwyneth: about 3.41 hours. Hazel: about 1.41 hours.

Step-by-step solution

Define Variables

Let's define the time it takes Gwyneth to paint the room by herself as \( x \) hours. According to the problem, Hazel can paint the room in 2 hours less time than Gwyneth, so it will take her \( x-2 \) hours.

Determine Rates of Work

The rate at which Gwyneth paints the room is \( \frac{1}{x} \) rooms per hour. The rate at which Hazel paints the room is \( \frac{1}{x-2} \) rooms per hour.

Setup the Equation

When working together, their combined rate is the sum of their individual rates. Together, they can paint the room in 2.4 hours, so their combined rate is \( \frac{1}{2.4} \). Thus, the equation is:\[ \frac{1}{x} + \frac{1}{x-2} = \frac{1}{2.4} \]

Find a Common Denominator

To combine the fractions on the left-hand side, find a common denominator:\[ \frac{(x-2) + x}{x(x-2)} = \frac{2x - 2}{x^2 - 2x} \]

Set Up Equation for Solving

Convert \( \frac{1}{2.4} \) to a fraction and solve:\[ \frac{2}{x^2 - 2x} = \frac{1}{2.4} \]

Solve for \( x \)

Cross multiply to set up an equation:\[ 2 = \frac{x^2 - 2x}{2.4} \]Multiply both sides by 2.4 to clear the fraction:\[ 4.8 = x^2 - 2x \]Rearrange to form a standard quadratic equation:\[ x^2 - 2x - 4.8 = 0 \]

Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -2 \), and \( c = -4.8 \):\[ x = \frac{2 \pm \sqrt{4 + 19.2}}{2} \]\[ x = \frac{2 \pm \sqrt{23.2}}{2} \]\[ x = 1 \pm \sqrt{5.8} \]

Determine Meaningful Solution

Since time cannot be negative, take the positive root:\[ x = 1 + \sqrt{5.8} \]This is approximately 3.41 hours. So, it takes Gwyneth about 3.41 hours.

Calculate Hazel's Time

Hazel's time is \( x - 2 \):\[ x - 2 = 1 + \sqrt{5.8} - 2 \]\[ x - 2 \approx 1.41 \] hours.

Key concepts

Variable Definition

To solve work and time problems, start by defining variables to represent the unknown values. For this problem, define the time it takes for Gwyneth to paint a room alone as \(x\) hours. Hazel is faster, and she takes 2 hours less, so her time to paint the room alone would be \(x - 2\) hours. These variables help us translate the problem into mathematical equations we can solve.

Rate of Work

The rate of work refers to how much work is done per unit of time. For work and time problems, we often express the rate of work as a fraction of the job done per hour. Gwyneth's rate of work, working alone, is \(\frac{1}{x}\) rooms per hour, since she takes \(x\) hours to paint one room. Hazel’s rate of work is \(\frac{1}{x-2}\) rooms per hour, as she completes the room faster than Gwyneth. When they work together, their combined rate is the sum of their individual rates. In this problem, their combined rate is \(\frac{1}{2.4}\) because they complete the room in 2.4 hours together.

Solving Quadratic Equations

Next, we set up the equation to find the unknown time \(x\). From Step 4, we find the common denominator and combine the fractions: \(\frac{2x - 2}{x^2 - 2x} = \frac{1}{2.4}\). To solve this, cross-multiply to get: \(2 = \frac{x^2 - 2x}{2.4}\). Multiply both sides by 2.4: \(4.8 = x^2 - 2x\), which rearranges to \(x^2 - 2x - 4.8 = 0\). This is a standard quadratic equation \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -2\), and \(c = -4.8\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values: \(x = \frac{2 \pm \sqrt{23.2}}{2}\) which simplifies to \(x = 1 \pm \sqrt{5.8}\). Since time cannot be negative, choose the positive solution: \(x = 1 + \sqrt{5.8}\). This approximately equals 3.41 hours.

Finding Common Denominator

Combining fractions is crucial in many algebra problems. Here, we need a common denominator to add Gwyneth's and Hazel's rates. For their rates \(\frac{1}{x}\) and \(\frac{1}{x-2}\), the common denominator is their product: \(x(x-2)\). Rewrite each rate with this common denominator: \(\frac{x-2 + x}{x(x-2)} = \frac{2x-2}{x^2 - 2x}\). This allows us to set up the equation \(\frac{2(x-1)}{x^2 - 2x} = \frac{1}{2.4}\). After simplifying and solving the quadratic equation, we find that \(x = 3.41\) hours (approx). Hazel's time is then \(x - 2\): 3.41 - 2 = 1.41 hours.