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 \boxed{} weeks, pete and sophie will each have completed \boxed{} missions.

Explanation:

Step1: Define Variables

Let \( w \) be the number of weeks until Sophie catches up to Pete. Let \( P(w) \) be the number of missions Pete has completed after \( w \) weeks, and \( S(w) \) be the number of missions Sophie has completed after \( w \) weeks.

Step2: Write Equations for Missions

For Pete: He starts with 27 missions and completes 3 per week, so \( P(w) = 27 + 3w \).
For Sophie: She starts with 13 missions and completes 5 per week, so \( S(w) = 13 + 5w \).

Step3: Set Equations Equal (Catch Up)

When Sophie catches up, \( P(w) = S(w) \):
\( 27 + 3w = 13 + 5w \)

Step4: Solve for \( w \)

Subtract \( 3w \) from both sides: \( 27 = 13 + 2w \)
Subtract 13 from both sides: \( 14 = 2w \)
Divide by 2: \( w = 7 \)

Step5: Find Missions Completed

Substitute \( w = 7 \) into either \( P(w) \) or \( S(w) \). Using \( P(w) \):
\( P(7) = 27 + 3(7) = 27 + 21 = 48 \)

Answer:

In \(\boldsymbol{7}\) weeks, Pete and Sophie will each have completed \(\boldsymbol{48}\) missions.