# RE3 | google/gemini-3.6-flash | run 1

## Prompt

If 3 workers build a wall in 12 hours, how long would 4 workers take to build the same wall working at the same rate? Show your work.

## Response

Four workers working at the same individual rate would take 9 hours to build the wall. Because the work rate per individual remains constant, the number of workers and the total time required are inversely proportional. The total labor needed to complete the wall is 36 worker-hours.

## Step-by-Step Calculation

**Method 1: Worker-Hours Concept**

**1. Calculate Total Work Required**
Multiply the initial number of workers by the hours taken to find the total worker-hours needed for the job:
\[ \text{Total Work} = 3 \text{ workers} \times 12 \text{ hours} = 36 \text{ worker-hours} \]

**2. Solve for Time with New Worker Count**
Divide the total worker-hours by the new team size:
\[ \text{Time} = \frac{36 \text{ worker-hours}}{4 \text{ workers}} = 9 \text{ hours} \]

**Method 2: Inverse Proportion Formula**

Because work done is constant, worker count (\( W \)) and time (\( T \)) follow an inverse relationship:
\[ W_1 \times T_1 = W_2 \times T_2 \]

Substitute the known values into the equation:
\[ 3 \times 12 = 4 \times T_2 \]
\[ 36 = 4 T_2 \]
\[ T_2 = \frac{36}{4} = 9 \text{ hours} \]

Would you like to calculate how much time it would take if additional workers were added or if the total workload changed?
