After 30 minutes into a race, Archie has run ⁶₁₃ of the route while Fabian has covered only ⁴₁₃ of the distance. Archie runs at the same average speed throughout the race and Fabian’s average speed is 63 m/min slower than Archie’s. If Fabian 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 Fabian in first 30 min
= 30 x 63
= 1890 m

Fraction of the entire distance that was covered by Archie more than Fabian in the first 30 min
= ⁶₁₃ - ⁴₁₃
= ²₁₃

²₁₃ of the entire distance = 1890 m
¹₁₃ of the entire distance = 1890 ÷ 2 = 945 m

Distance that Fabian ran in the first 30 min
= 4 x 945
= 3780 m

Fabian's average speed
= 3780 ÷ 30
= 126 m/min

Archie's average speed
= 126 + 63
= 189 m/min

Fraction of the journey that Archie needed to run to complete the race
= 1 - ⁶₁₃
= ⁷₁₃

¹₁₃ of the entire distance = 945 m
⁷₁₃ of the entire distance = 7 x 945 = 6615 m

Duration that Archie would take to complete
= 6615 ÷ 189
= 35 min

Fraction of the journey that Archie needed to run to complete the race
= 1 - ⁴₁₃
= ⁹₁₃

⁹₁₃ of the entire distance = 9 x 945 = 8505 m  

Speed that Fabian needs to take to complete distance at the same time as Archie
= 8505 ÷ 35
= 243 m/min

Increase in Fabian's average speed
= 243 - 126
= 117 m/min

Percentage increase in Fabian's average speed
= ¹¹⁷₁₂₆ x 100%
≈ 93%

Answer(s): 93%