After 40 minutes into a race, Archie has run ⁸₁₁ of the route while Tim has covered only ²₁₁ of the distance. Archie runs at the same average speed throughout the race and Tim’s average speed is 96 m/min slower than Archie’s. If Tim wants to finish the race at the same time as Archie, by how many percent should he increase his average speed by for the remaining part of the race? Round off your answer to the nearest whole number.
Extra distance that Archie ran more than Tim in first 40 min
= 40 x 96
= 3840 m

Fraction of the entire distance that was covered by Archie more than Tim in the first 40 min
= ⁸₁₁ - ²₁₁
= ⁶₁₁

⁶₁₁ of the entire distance = 3840 m
¹₁₁ of the entire distance = 3840 ÷ 6 = 640 m

Distance that Tim ran in the first 40 min
= 2 x 640
= 1280 m

Tim's average speed
= 1280 ÷ 40
= 32 m/min

Archie's average speed
= 32 + 96
= 128 m/min

Fraction of the journey that Archie needed to run to complete the race
= 1 - ⁸₁₁
= ³₁₁

¹₁₁ of the entire distance = 640 m
³₁₁ of the entire distance = 3 x 640 = 1920 m

Duration that Archie would take to complete
= 1920 ÷ 128
= 15 min

Fraction of the journey that Archie needed to run to complete the race
= 1 - ²₁₁
= ⁹₁₁

⁹₁₁ of the entire distance = 9 x 640 = 5760 m  

Speed that Tim needs to take to complete distance at the same time as Archie
= 5760 ÷ 15
= 384 m/min

Increase in Tim's average speed
= 384 - 32
= 352 m/min

Percentage increase in Tim's average speed
= ³⁵²₃₂ x 100%
≈ 1100%

Answer(s): 1100%