After 40 minutes into a race, Paul has run ⁸₁₁ of the route while Daniel has covered only ²₁₁ of the distance. Paul runs at the same average speed throughout the race and Daniel’s average speed is 90 m/min slower than Paul’s. If Daniel wants to finish the race at the same time as Paul, 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 Paul ran more than Daniel in first 40 min
= 40 x 90
= 3600 m

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

⁶₁₁ of the entire distance = 3600 m
¹₁₁ of the entire distance = 3600 ÷ 6 = 600 m

Distance that Daniel ran in the first 40 min
= 2 x 600
= 1200 m

Daniel's average speed
= 1200 ÷ 40
= 30 m/min

Paul's average speed
= 30 + 90
= 120 m/min

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

¹₁₁ of the entire distance = 600 m
³₁₁ of the entire distance = 3 x 600 = 1800 m

Duration that Paul would take to complete
= 1800 ÷ 120
= 15 min

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

⁹₁₁ of the entire distance = 9 x 600 = 5400 m  

Speed that Daniel needs to take to complete distance at the same time as Paul
= 5400 ÷ 15
= 360 m/min

Increase in Daniel's average speed
= 360 - 30
= 330 m/min

Percentage increase in Daniel's average speed
= ³³⁰₃₀ x 100%
≈ 1100%

Answer(s): 1100%