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

Fraction of the entire distance that was covered by Charlie more than Ethan 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 Ethan ran in the first 30 min
= 4 x 1575
= 6300 m

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

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

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

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

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

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

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

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

Answer(s): 93%