After 30 minutes into a race, Carl has run ⁶₁₃ of the route while Michael has covered only ⁴₁₃ of the distance. Carl runs at the same average speed throughout the race and Michael’s average speed is 49 m/min slower than Carl’s. If Michael wants to finish the race at the same time as Carl, 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 Carl ran more than Michael in first 30 min
= 30 x 49
= 1470 m

Fraction of the entire distance that was covered by Carl more than Michael 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 Michael ran in the first 30 min
= 4 x 735
= 2940 m

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

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

Fraction of the journey that Carl 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 Carl would take to complete
= 5145 ÷ 147
= 35 min

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

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

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

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

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

Answer(s): 93%