Pete and his sister, Sophie, play the game Space Mission Commander on their video game console. Pete has already completed 27 missions. Sophie has only completed 13 missions. From now on, Pete plans to complete 3 missions per week. Meanwhile, Sophie plans to complete 5 missions per week. At some point, Sophie will catch up to Pete. How many weeks will it take? How many missions each will Pete and Sophie have completed? In ____ weeks, Pete and Sophie will each have completed ____ missions.

Question

Pete and his sister, Sophie, play the game Space Mission Commander on their video game console. Pete has already completed 27 missions. Sophie has only completed 13 missions. From now on, Pete plans to complete 3 missions per week. Meanwhile, Sophie plans to complete 5 missions per week. At some point, Sophie will catch up to Pete. How many weeks will it take? How many missions each will Pete and Sophie have completed?    In ____ weeks, Pete and Sophie will each have completed ____ missions.

Bytelearn · Accepted Answer

Calculate Pete and Sophie's Missions:  Calculate the number of missions Pete and Sophie start with and their weekly completion rates. Pete's initial missions = 27, weekly rate = 3 missions/week. Sophie's initial missions = 13, weekly rate = 5 missions/week.Set Up Catch-Up Equation:  Set up the equation to find when Sophie catches up to Pete. Let $ t $ be the number of weeks it takes for Sophie to catch up. Sophie's total missions = $ 13 + 5t $ Pete's total missions = $ 27 + 3t $ Set the equations equal to find $ t $: $$ 13 + 5t = 27 + 3t $$Solve for t:  Solve for $ t $. Subtract $ 3t $ and 13 from both sides: $$ 5t - 3t = 27 - 13 $$ $$ 2t = 14 $$ $$ t = 7 $$Calculate Total Missions:  Calculate the total missions completed by each after $ t $ weeks. Total missions for Pete = $ 27 + 3 \times 7 = 48 $ Total missions for Sophie = $ 13 + 5 \times 7 = 48 $

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