After 35 minutes into a race, Owen has run ⁵₁₁ of the route while Ethan has covered only ³₁₁ of the distance. Owen runs at the same average speed throughout the race and Ethan’s average speed is 96 m/min slower than Owen’s. If Ethan wants to finish the race at the same time as Owen, 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 Owen ran more than Ethan in first 35 min
= 35 x 96
= 3360 m

Fraction of the entire distance that was covered by Owen more than Ethan in the first 35 min
= ⁵₁₁ - ³₁₁
= ²₁₁

²₁₁ of the entire distance = 3360 m
¹₁₁ of the entire distance = 3360 ÷ 2 = 1680 m

Distance that Ethan ran in the first 35 min
= 3 x 1680
= 5040 m

Ethan's average speed
= 5040 ÷ 35
= 144 m/min

Owen's average speed
= 144 + 96
= 240 m/min

Fraction of the journey that Owen needed to run to complete the race
= 1 - ⁵₁₁
= ⁶₁₁

¹₁₁ of the entire distance = 1680 m
⁶₁₁ of the entire distance = 6 x 1680 = 10080 m

Duration that Owen would take to complete
= 10080 ÷ 240
= 42 min

Fraction of the journey that Owen needed to run to complete the race
= 1 - ³₁₁
= ⁸₁₁

⁸₁₁ of the entire distance = 8 x 1680 = 13440 m  

Speed that Ethan needs to take to complete distance at the same time as Owen
= 13440 ÷ 42
= 320 m/min

Increase in Ethan's average speed
= 320 - 144
= 176 m/min

Percentage increase in Ethan's average speed
= ¹⁷⁶₁₄₄ x 100%
≈ 122%

Answer(s): 122%