Question

The time it takes for two people working together to complete a job is given by the formula \(T=\frac{a b}{a+b},\) where \(a\) and \(b\) are the times it takes for the two people to complete the same job individually. Sarah can set up the auditorium for an assembly in 30 min, but when she works with James they can set it up in 10 min. How long would it take James to set it up by himself?

Short answer

James would take 15 minutes to set up the auditorium by himself.

Step-by-step solution

Identify the Given Values

Sarah's time to set up the auditorium: \(a = 30 \text{ min}\).Combined time with James: \(T = 10 \text{ min}\).

Plug Values into the Formula

The formula provided is \(T = \frac{a b}{a+b}\). Plug in the values for \(a\) and \(T\): \[ 10 = \frac{30 b}{30 + b} \]

Solve for James' Time (\(b\))

Multiply both sides of the equation by \(30 + b\) to clear the fraction:\[ 10(30 + b) = 30b \] Distribute 10 on the left side: \[ 300 + 10b = 30b \]Subtract \(10b\) from both sides: \[ 300 = 20b \]Divide both sides by 20 to solve for \(b\): \[ b = \frac{300}{20} \] \[ b = 15 \text{ min} \]

Verify the Solution

Plug \(b = 15 \text{ min}\) back into the formula to ensure correctness: \[ T = \frac{30 \times 15}{30 + 15} = \frac{450}{45} = 10 \text{ min} \] The values are correct.

Key concepts

Time Management

Understanding time management is essential, especially when juggling multiple tasks or working within a limited timeframe. Time management involves planning and controlling how much time to spend on specific activities. Effective time management enables you to accomplish more in a shorter period, lowers stress, and leads to career success.

In our example, Sarah and James are both working to set up an auditorium. Individually, Sarah takes 30 minutes. Knowing how long a task takes can help you allocate your time efficiently. Rather than Sarah doing the entire setup herself, enlisting help allows for better use of time, showing the importance of combined efforts for better time efficiency.

• Plan your tasks: Create a schedule.
• Prioritize: Focus on important tasks first.
• Set Goals: Define clear objectives for each task.

Collaborative Work

Working together often makes tasks more manageable and quicker to complete. Collaborative work brings together different skills and strengths, leading to enhanced productivity and new solutions. In collaborative work, clear communication and well-defined roles are crucial.

In this problem, when Sarah and James work together, they complete the task faster than Sarah would alone. This illustrates the concept of 'synergy,' where the combined effort achieves more than working separately. Effective collaboration includes:

• Setting clear goals.
• Understanding each member's role.
• Fostering good communication.

By working as a team, tasks become less overwhelming, and individual limitations can be balanced with others' strengths.

Rational Equations

Rational equations involve fractions that have polynomials in the numerator and the denominator. They are used to solve problems involving rates and times, like the given exercise. The formula \(T=\frac{a b}{a+b}\) is a rational equation representing the time taken by two people, working together, to complete a task.

Solving these equations typically involves the following steps:

• Identify and write down known values.
• Substitute the known values into the equation.
• Solve for the unknown variable by clearing the fraction.
• Verify your solution by plugging values back into the original equation.

In our case, the rational equation \(10=\frac{30 b}{30+b}\) helped us find that it would take James 15 minutes to set up the auditorium alone. Solving rational equations helps understand relationships and joint efforts in shared workloads.