After 30 minutes into a race, Jack has run ⁶₁₃ of the route while Luke has covered only ⁴₁₃ of the distance. Jack runs at the same average speed throughout the race and Luke’s average speed is 91 m/min slower than Jack’s. If Luke 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 Luke in first 30 min
= 30 x 91
= 2730 m

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

²₁₃ of the entire distance = 2730 m
¹₁₃ of the entire distance = 2730 ÷ 2 = 1365 m

Distance that Luke ran in the first 30 min
= 4 x 1365
= 5460 m

Luke's average speed
= 5460 ÷ 30
= 182 m/min

Jack's average speed
= 182 + 91
= 273 m/min

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

¹₁₃ of the entire distance = 1365 m
⁷₁₃ of the entire distance = 7 x 1365 = 9555 m

Duration that Jack would take to complete
= 9555 ÷ 273
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1365 = 12285 m  

Speed that Luke needs to take to complete distance at the same time as Jack
= 12285 ÷ 35
= 351 m/min

Increase in Luke's average speed
= 351 - 182
= 169 m/min

Percentage increase in Luke's average speed
= ¹⁶⁹₁₈₂ x 100%
≈ 93%

Answer(s): 93%