After 35 minutes into a race, Ethan has run ⁷₁₃ of the route while Eric has covered only ⁴₁₃ of the distance. Ethan runs at the same average speed throughout the race and Eric’s average speed is 90 m/min slower than Ethan’s. If Eric wants to finish the race at the same time as Ethan, 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 Ethan ran more than Eric in first 35 min
= 35 x 90
= 3150 m

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

³₁₃ of the entire distance = 3150 m
¹₁₃ of the entire distance = 3150 ÷ 3 = 1050 m

Distance that Eric ran in the first 35 min
= 4 x 1050
= 4200 m

Eric's average speed
= 4200 ÷ 35
= 120 m/min

Ethan's average speed
= 120 + 90
= 210 m/min

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

¹₁₃ of the entire distance = 1050 m
⁶₁₃ of the entire distance = 6 x 1050 = 6300 m

Duration that Ethan would take to complete
= 6300 ÷ 210
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1050 = 9450 m  

Speed that Eric needs to take to complete distance at the same time as Ethan
= 9450 ÷ 30
= 315 m/min

Increase in Eric's average speed
= 315 - 120
= 195 m/min

Percentage increase in Eric's average speed
= ¹⁹⁵₁₂₀ x 100%
≈ 163%

Answer(s): 163%