IMU Question 3 It has been found the generally the time, \( t \), taken to a puzzle varies directly with the number of pieces, \( n \), and inversely with the number of people, \( p \), working on it. If it takes 4 people 36 minutes to solve and 180 piece puzzle, how long will it take 6 people to solve a 420 piece puzzle?

To solve this problem, we can use the relationship given in the prompt: \( t \propto \frac{n}{p} \). This means that the time \( t \) is directly proportional to the number of pieces \( n \) and inversely proportional to the number of people \( p \). From the initial condition: \[ t = k \cdot \frac{n}{p} \] where \( k \) is the constant of proportionality. For 4 people working on 180 pieces in 36 minutes: \[ 36 = k \cdot \frac{180}{4} \] \[ 36 = k \cdot 45 \] \[ k = \frac{36}{45} = \frac{4}{5} \] Now, we want to find out how long it will take 6 people to solve a 420 piece puzzle. Plugging in the values: \[ t = \frac{4}{5} \cdot \frac{420}{6} \] \[ t = \frac{4}{5} \cdot 70 \] \[ t = \frac{280}{5} = 56 \] So, it will take 6 people 56 minutes to solve the 420 piece puzzle.