Final answer:
Renzo will take approximately 10.2857 hours to finish the puzzle by himself. This is calculated by finding the combined rate at which Natalie and Renzo can complete the puzzle, and then isolating Renzo's rate before calculating the time it would take him to complete the puzzle solo.
Step-by-step explanation:
We must find out how long it will take Renzo to finish the puzzle on his own. First, let's define the rates at which Natalie and Natalie and Renzo together can complete the puzzle. Natalie can complete the puzzle in 8 hours, meaning her rate is 1 puzzle per 8 hours, or 1/8 puzzle per hour. When Natalie and Renzo work together, they complete the puzzle in about 4.5 hours, or 1/4.5 puzzle per hour. Now, let's denote Renzo's rate of completing the puzzle by himself as R puzzles per hour.
To find Renzo's rate, we add together the rates of both Natalie and Renzo when they are working together:
1/8 + R = 1/4.5
To solve for R, we subtract 1/8 from both sides:
R = 1/4.5 - 1/8
Finding a common denominator (which is 8 * 4.5 = 36), we have:
R = (8/36) - (4.5/36)
R = (8 - 4.5) / 36
R = 3.5/36
R = 1/10.2857
Hence, Renzo's rate is 1/10.2857 puzzles per hour. To find out how long it will take Renzo to finish one puzzle if he works by himself, we take the reciprocal of his rate:
Time = 1 / R = 10.2857 hours
Therefore, Renzo will take approximately 10.2857 hours to finish the puzzle by himself.