After 40 minutes into a race, Carl has run ⁸₁₁ of the route while Liam has covered only ²₁₁ of the distance. Carl runs at the same average speed throughout the race and Liam’s average speed is 30 m/min slower than Carl’s. If Liam wants to finish the race at the same time as Carl, 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 Carl ran more than Liam in first 40 min
= 40 x 30
= 1200 m

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

⁶₁₁ of the entire distance = 1200 m
¹₁₁ of the entire distance = 1200 ÷ 6 = 200 m

Distance that Liam ran in the first 40 min
= 2 x 200
= 400 m

Liam's average speed
= 400 ÷ 40
= 10 m/min

Carl's average speed
= 10 + 30
= 40 m/min

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

¹₁₁ of the entire distance = 200 m
³₁₁ of the entire distance = 3 x 200 = 600 m

Duration that Carl would take to complete
= 600 ÷ 40
= 15 min

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

⁹₁₁ of the entire distance = 9 x 200 = 1800 m  

Speed that Liam needs to take to complete distance at the same time as Carl
= 1800 ÷ 15
= 120 m/min

Increase in Liam's average speed
= 120 - 10
= 110 m/min

Percentage increase in Liam's average speed
= ¹¹⁰₁₀ x 100%
≈ 1100%

Answer(s): 1100%