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

Fraction of the entire distance that was covered by Luke more than Simon 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 Simon ran in the first 30 min
= 4 x 1365
= 5460 m

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

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

Fraction of the journey that Luke 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 Luke would take to complete
= 9555 ÷ 273
= 35 min

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

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

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

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

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

Answer(s): 93%