After 30 minutes into a race, Ethan has run ⁶₁₃ of the route while Tim has covered only ⁴₁₃ of the distance. Ethan runs at the same average speed throughout the race and Tim’s average speed is 49 m/min slower than Ethan’s. If Tim 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 Tim in first 30 min
= 30 x 49
= 1470 m

Fraction of the entire distance that was covered by Ethan more than Tim in the first 30 min
= ⁶₁₃ - ⁴₁₃
= ²₁₃

²₁₃ of the entire distance = 1470 m
¹₁₃ of the entire distance = 1470 ÷ 2 = 735 m

Distance that Tim ran in the first 30 min
= 4 x 735
= 2940 m

Tim's average speed
= 2940 ÷ 30
= 98 m/min

Ethan's average speed
= 98 + 49
= 147 m/min

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

¹₁₃ of the entire distance = 735 m
⁷₁₃ of the entire distance = 7 x 735 = 5145 m

Duration that Ethan would take to complete
= 5145 ÷ 147
= 35 min

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

⁹₁₃ of the entire distance = 9 x 735 = 6615 m  

Speed that Tim needs to take to complete distance at the same time as Ethan
= 6615 ÷ 35
= 189 m/min

Increase in Tim's average speed
= 189 - 98
= 91 m/min

Percentage increase in Tim's average speed
= ⁹¹₉₈ x 100%
≈ 93%

Answer(s): 93%