Problem of the Week - Algebra and Up - Nov. 29, 2020 to Dec. 5, 2020

Algebra and Up: 

Question: Nick, Rose, and Leif are decorating for their friend Angie's surprise party. Nick can put up decorations in 1.5 hours, Rose can put up decorations in 2 hours, and Leif can put up decorations in 3.5 hours. How long will it take them to put up decorations if they all work together? Round your answer to the nearest minute. 

Answer: 41 minutes

Solution: We need to combine Nick’s, Rose’s, and Leif’s decorating rates by addition. Choosing a common denominator, Nick’s rate is 1/1.5 = 28/42, Rose’s is 1/2 = 21/42, and Leif’s is 1/3.5 = 12/42. Their combined rate of decorating is 28/42 + 21/42 + 12/42 = 61/42, or 61 parties in 42 minutes. To find their combined rate for 1 party, we set 61/42 equal to 1/x (1 party in x minutes). Using cross products, we find that x = 42/61 of an hour = 42/61 of 60 minutes = 42/61 × 60 minutes ≈ 41 minutes.