After 35 minutes into a race, Jack has run ⁷₁₃ of the route while Daniel has covered only ⁴₁₃ of the distance. Jack runs at the same average speed throughout the race and Daniel’s average speed is 108 m/min slower than Jack’s. If Daniel wants to finish the race at the same time as Jack, 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 Jack ran more than Daniel in first 35 min
= 35 x 108
= 3780 m

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

³₁₃ of the entire distance = 3780 m
¹₁₃ of the entire distance = 3780 ÷ 3 = 1260 m

Distance that Daniel ran in the first 35 min
= 4 x 1260
= 5040 m

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

Jack's average speed
= 144 + 108
= 252 m/min

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

¹₁₃ of the entire distance = 1260 m
⁶₁₃ of the entire distance = 6 x 1260 = 7560 m

Duration that Jack would take to complete
= 7560 ÷ 252
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1260 = 11340 m  

Speed that Daniel needs to take to complete distance at the same time as Jack
= 11340 ÷ 30
= 378 m/min

Increase in Daniel's average speed
= 378 - 144
= 234 m/min

Percentage increase in Daniel's average speed
= ²³⁴₁₄₄ x 100%
≈ 163%

Answer(s): 163%