After 30 minutes into a race, Julian has run ⁶₁₃ of the route while Reggie has covered only ⁴₁₃ of the distance. Julian runs at the same average speed throughout the race and Reggie’s average speed is 105 m/min slower than Julian’s. If Reggie wants to finish the race at the same time as Julian, 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 Julian ran more than Reggie in first 30 min
= 30 x 105
= 3150 m

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

²₁₃ of the entire distance = 3150 m
¹₁₃ of the entire distance = 3150 ÷ 2 = 1575 m

Distance that Reggie ran in the first 30 min
= 4 x 1575
= 6300 m

Reggie's average speed
= 6300 ÷ 30
= 210 m/min

Julian's average speed
= 210 + 105
= 315 m/min

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

¹₁₃ of the entire distance = 1575 m
⁷₁₃ of the entire distance = 7 x 1575 = 11025 m

Duration that Julian would take to complete
= 11025 ÷ 315
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1575 = 14175 m  

Speed that Reggie needs to take to complete distance at the same time as Julian
= 14175 ÷ 35
= 405 m/min

Increase in Reggie's average speed
= 405 - 210
= 195 m/min

Percentage increase in Reggie's average speed
= ¹⁹⁵₂₁₀ x 100%
≈ 93%

Answer(s): 93%