After 30 minutes into a race, Daniel has run ⁶₁₃ of the route while Jack has covered only ⁴₁₃ of the distance. Daniel runs at the same average speed throughout the race and Jack’s average speed is 105 m/min slower than Daniel’s. If Jack wants to finish the race at the same time as Daniel, 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 Daniel ran more than Jack in first 30 min
= 30 x 105
= 3150 m

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

²₁₃ of the entire distance = 3150 m
¹₁₃ of the entire distance = 3150 ÷ 2 = 1575 m

Distance that Jack ran in the first 30 min
= 4 x 1575
= 6300 m

Jack's average speed
= 6300 ÷ 30
= 210 m/min

Daniel's average speed
= 210 + 105
= 315 m/min

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

¹₁₃ of the entire distance = 1575 m
⁷₁₃ of the entire distance = 7 x 1575 = 11025 m

Duration that Daniel would take to complete
= 11025 ÷ 315
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1575 = 14175 m  

Speed that Jack needs to take to complete distance at the same time as Daniel
= 14175 ÷ 35
= 405 m/min

Increase in Jack's average speed
= 405 - 210
= 195 m/min

Percentage increase in Jack's average speed
= ¹⁹⁵₂₁₀ x 100%
≈ 93%

Answer(s): 93%