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 puzzle, we can use the relationship described in the question. The time \( t \) varies directly with the number of pieces \( n \) and inversely with the number of people \( p \). We can express this as: \[ t = k \cdot \frac{n}{p} \] Where \( k \) is a constant. First, we need to find the value of \( k \) using the information provided for the first scenario: 4 people take 36 minutes to solve a 180-piece puzzle. Plugging in these values: \[ 36 = k \cdot \frac{180}{4} \] This simplifies to: \[ 36 = k \cdot 45 \] Now, we can solve for \( k \): \[ k = \frac{36}{45} = \frac{4}{5} \] Now that we have \( k \), we can use this constant to find out how long it will take for 6 people to solve a 420-piece puzzle. We set up the equation again using the new parameters: \[ t = \frac{4}{5} \cdot \frac{420}{6} \] Calculating \( \frac{420}{6} \): \[ \frac{420}{6} = 70 \] Now plugging this back into the equation: \[ t = \frac{4}{5} \cdot 70 = 56 \] So, it will take 6 people 56 minutes to solve a 420-piece puzzle.