Question

You can paint a room in 8 hours. Working together, you and your friend can paint the room in just 5 hours. a. Let \(t\) be the time (in hours) your friend would take to paint the room when working alone. Copy and complete the table. \((\) Hint \(:(\) Work done \()=(\) Work rate \() \times(\) Time \())\) b. Explain what the sum of the expressions represents in the last column. Write and solve an equation to find how long your friend would take to paint the room when working alone.

Short answer

The friend would take 5 hours to paint the room alone.

Step-by-step solution

Work Rate Calculation

The problem states that you can paint the room in 8 hours, so your work rate is \(1/8\) of the room per hour. Additionally, the combined rate of you and your friend is \(1/5\) of the room per hour, as the room takes 5 hours to paint when you both work, according to the problem.

Table Completion

In this step, we complete the table substituting the calculated work rates and given times into the formula 'Work done = Work rate × Time'. For you, your work done is \(1/8 × t\). For your friend, the work done is \((1/5 - 1/8) × t\). Since work done by all must be 1 (the whole room), the sum of the two expressions will be \(1/8t + (1/5 - 1/8) × t\). This is equivalent to \(1/8t + (3/40)t\).

Explanation of Result

The sum of the expressions in column 3 represents the total work done when both you and your friend work together for \(t\) hours. This work needs to equal to 1 (full completion), as when total work done by all is combined, the whole task is accomplished.

Formulate and Solve the Equation

Now, to find the time taken by the friend alone to paint the room, we equate the sum of the work rates in the table to 1 (i.e. the whole room). Therefore, \(1/8t + (3/40)t = 1\). Simplifying the equation by finding the common denominator gives \(5/40t + 3/40t = 1\) or \(8/40t = 1\). Dividing both sides by \(8/40\), we find that \(t = 1/(8/40)\) or \(t = 5\) hours. This means your friend would take 5 hours to paint the room alone.

Key concepts

Algebraic Expressions

When dealing with work and rate problems, algebraic expressions are essential tools. These expressions help us represent the work done by different individuals over a period of time. For this painting scenario, we use expressions to denote how much of the room is painted by you and your friend separately.

• Your rate of work can be expressed as \(\frac{1}{8}\), meaning you finish \(\frac{1}{8}\) of the room per hour.
• Similarly, the combined rate with your friend is \(\frac{1}{5}\) of the room per hour.

These fractions are expressions of work rates. When you substitute these into the equation \( \text{Work done} = \text{Work rate} \times \text{Time} \), it results in the total work done by all parties involved.

By equating and solving these algebraic expressions, we find out how work is distributed among the workers, making it essential to find the unknown time an individual would take if working solo.

Time and Work Problems

Time and work problems often present scenarios requiring simultaneous effort to achieve a goal, like painting a room. Understanding these problems involves identifying the "work rate," which determines how much of a task can be completed in a specific time.

• In our example, your individual work rate is \(\frac{1}{8}\), indicating you can paint the room in 8 hours.
• Together with your friend, your combined work rate is \(\frac{1}{5}\), meaning the room is painted in 5 hours.

The challenge often entails finding the time one person takes to complete the task alone. In this exercise, we set up equations to solve for your friend’s time working solo.

These problems improve your ability to analyze situations and establish relationships between time, work rate, and parts of a task completed. They lead to algebraic problem-solving, where setting up and solving equations are critical.

Collaborative Work Equations

Collaborative work equations show how a team effort speeds up task completion. In such problems, each person contributes to the work, and their combined efforts can be expressed in a single equation.

To represent the scenario mathematically:

• First, express each worker's contribution: You at \(\frac{1}{8}t\), and your friend at an unknown rate, but combined with you, equaling \(\frac{1}{5}t\).
• The equation formed is: \(\frac{1}{8}t + \left(\frac{1}{5} - \frac{1}{8}\right)t = 1\), which reflects the complete work done together.

Solving this kind of equation simplifies determining how long each individual would take to finish the job independently.

Understanding such equations helps sharpen analytical skills, making it easier to dissect complex problems into manageable components in any collaborative task setting.