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

Fraction of the entire distance that was covered by Billy 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

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

Fraction of the journey that Billy 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 Billy would take to complete
= 11025 ÷ 315
= 35 min

Fraction of the journey that Billy 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 Billy
= 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%