After 35 minutes into a race, Nick has run ⁷₁₃ of the route while Ethan has covered only ⁴₁₃ of the distance. Nick runs at the same average speed throughout the race and Ethan’s average speed is 36 m/min slower than Nick’s. If Ethan wants to finish the race at the same time as Nick, 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 Nick ran more than Ethan in first 35 min
= 35 x 36
= 1260 m

Fraction of the entire distance that was covered by Nick more than Ethan in the first 35 min
= ⁷₁₃ - ⁴₁₃
= ³₁₃

³₁₃ of the entire distance = 1260 m
¹₁₃ of the entire distance = 1260 ÷ 3 = 420 m

Distance that Ethan ran in the first 35 min
= 4 x 420
= 1680 m

Ethan's average speed
= 1680 ÷ 35
= 48 m/min

Nick's average speed
= 48 + 36
= 84 m/min

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

¹₁₃ of the entire distance = 420 m
⁶₁₃ of the entire distance = 6 x 420 = 2520 m

Duration that Nick would take to complete
= 2520 ÷ 84
= 30 min

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

⁹₁₃ of the entire distance = 9 x 420 = 3780 m  

Speed that Ethan needs to take to complete distance at the same time as Nick
= 3780 ÷ 30
= 126 m/min

Increase in Ethan's average speed
= 126 - 48
= 78 m/min

Percentage increase in Ethan's average speed
= ⁷⁸₄₈ x 100%
≈ 163%

Answer(s): 163%