Problem 61 - Algebra A Combined Function

Question

Bill Shaughnessy and his son Billy can clean the house together in 4 hours. When the son works alone, it takes him an hour longer to clean than it takes his dad alone. Find how long to the nearest tenth of an hour it takes the son to clean alone.

Step-by-Step Solution

Answer

It takes the son approximately 8.5 hours to clean the house alone.

Step 1: Set Up Equations for Work Rate

Let the time it takes for the dad to clean the house alone be \( x \) hours. Therefore, his work rate is \( \frac{1}{x} \) of the house per hour. For the son, it takes \( x + 1 \) hours to clean the house alone, so his work rate is \( \frac{1}{x+1} \) of the house per hour.

Step 2: Express Combined Work Rate

When Bill and his son work together, they can clean the house in 4 hours. Their combined work rate is \( \frac{1}{4} \) of the house per hour. Therefore, the equation becomes \[ \frac{1}{x} + \frac{1}{x+1} = \frac{1}{4} \].

Step 3: Solve Equation for x

Multiply both sides by \( 4x(x+1) \) to eliminate the fractions: \[ 4(x+1) + 4x = x(x+1) \]. This simplifies to \( 4x + 4 + 4x = x^2 + x \). Rearrange it to form a quadratic equation: \[ x^2 - 7x - 4 = 0 \].

Step 4: Use the Quadratic Formula

Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \), where \( a = 1, b = -7, c = -4 \).\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]\[ x = \frac{7 \pm \sqrt{49 + 16}}{2} \]\[ x = \frac{7 \pm \sqrt{65}}{2} \].

Step 5: Calculate the Positive Root

Using the '+' in the quadratic formula, calculate \( x \):\[ x = \frac{7 + \sqrt{65}}{2} \].Calculate \( \sqrt{65} \approx 8.06 \), so:\[ x = \frac{7 + 8.06}{2} \approx 7.53 \].

Step 6: Determine the Son's Time

Since \( x \) is the dad's time and the son takes 1 hour longer, the son's time is \( x + 1 = 7.53 + 1 \approx 8.53 \) hours.

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool in solving quadratic equations like those found in work rate problems. Quadratic equations often come in the form \[ ax^2 + bx + c = 0 \]. To solve them, the quadratic formula is used: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. It helps find the values of \( x \) that make the equation true. In work rate problems, a quadratic equation often emerges when solving for time or rates. By identifying the coefficients \( a \), \( b \), and \( c \), you can apply the formula to find the needed solution. It simplifies the process, reducing the complexity of manually factoring or estimating possible solutions. Ensuring the discriminant \( b^2 - 4ac \) is positive is key, as it indicates real solutions exist.

Combined Work Rate

In many real-world scenarios, like cleaning a house, teamwork can change the output rate. When considering the combined work rate of two or more individuals, we express their collective ability to complete a task. The equation ties individual work rates together to find the total rate. If one person can do a task in \( t_1 \) hours with a rate of \( \frac{1}{t_1} \) and another in \( t_2 \) hours with \( \frac{1}{t_2} \), their combined work rate is \( \frac{1}{t_1} + \frac{1}{t_2} \). For Bill and his son, their combined work rate was \( \frac{1}{4} \) of the house per hour. This combined rate of \( \frac{1}{x} + \frac{1}{x+1} = \frac{1}{4} \) was used to set up the equation reflecting their teamwork's efficiency.

Algebraic Equations

Algebraic equations are the backbone of mathematical problem-solving. They express relationships in a way that allows you to solve for unknown variables, like the time it takes for Bill and Billy to clean. An equation is set up based on the information given: equations can be linear, quadratic, or of higher degrees. By translating a word problem into an algebraic equation, mathematical tools and formulas can be applied to find solutions. In this exercise, several algebraic manipulations were performed, including clearing fractions by multiplying both sides of the equation by the product of the denominators, and rearranging terms to form a standard quadratic equation \( x^2 - 7x - 4 = 0 \). Such operations are vital in isolating and solving for variables.

Problem-Solving Steps

Solving math problems like work rate exercises involves a structured approach:

• Start by understanding the problem statement thoroughly.
• Identify what you are being asked to find and label any unknowns.
• Set up equations based on the given information and known formulas.
• Simplify equations and use appropriate methods, like factoring or applying formulas, to solve them.

In this exercise, steps involved setting up individual work rates, expressing a combined work rate, forming a quadratic equation, then solving using the quadratic formula. Following each problem-solving step ensures you do not miss any vital information and reach an accurate solution efficiently. Always verify your result to confirm its accuracy within the context of the problem.