Question

Worker A can make a trinket in 4 hours, Worker B can make a trinket in 2 hours. When they work together, how long will it take them to make a trinket?

• A. 6 hours
• B. \(\frac{1}{2}\) hour
• C. \(1\frac{1}{3}\) hours
• D. 3 hours
• E. \(1\frac{1}{2}\) hours

Hint:

For problems where two people work together, a useful shortcut formula is \(T_{Total} = \frac{T_A \times T_B}{T_A + T_B}\), where \(T_A\) and \(T_B\) are the individual times. In this case, \(T_{Total} = \frac{4 \times 2}{4 + 2} = \frac{8}{6} = \frac{4}{3}\) hours.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a work-rate problem. To find the combined time it takes for two workers to complete a task together, we first need to determine their individual rates of work. The combined rate is the sum of their individual rates. The time taken is the reciprocal of the combined rate.
Step 2: Key Formula or Approach:
The rate of work is the amount of work done per unit of time.
\[ \text{Rate} = \frac{\text{Work}}{\text{Time}} \] If two workers have rates \(R_A\) and \(R_B\), their combined rate is \(R_{Total} = R_A + R_B\).
The time it takes to complete one unit of work together is \(T_{Total} = \frac{1}{R_{Total}}\).
Step 3: Detailed Explanation:
First, let's find the individual rates of Worker A and Worker B. The work is "making 1 trinket".
Rate of Worker A (\(R_A\)):
\[ R_A = \frac{1 \text{ trinket}}{4 \text{ hours}} = \frac{1}{4} \text{ trinket/hour} \] Rate of Worker B (\(R_B\)):
\[ R_B = \frac{1 \text{ trinket}}{2 \text{ hours}} = \frac{1}{2} \text{ trinket/hour} \] Next, we find their combined rate by adding their individual rates.
\[ R_{Total} = R_A + R_B = \frac{1}{4} + \frac{1}{2} \] To add these fractions, we find a common denominator, which is 4.
\[ R_{Total} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \text{ trinket/hour} \] This means that working together, they can make \(\frac{3}{4}\) of a trinket in one hour.
Finally, to find the time it takes to make one whole trinket, we take the reciprocal of the combined rate.
\[ T_{Total} = \frac{1}{R_{Total}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \text{ hours} \] To express this as a mixed number:
\[ \frac{4}{3} \text{ hours} = 1\frac{1}{3} \text{ hours} \] Step 4: Final Answer:
Working together, it will take them \(1\frac{1}{3}\) hours to make a trinket.