After 30 minutes into a race, Peter has run ⁶₁₃ of the route while Dylan has covered only ⁴₁₃ of the distance. Peter runs at the same average speed throughout the race and Dylan’s average speed is 112 m/min slower than Peter’s. If Dylan wants to finish the race at the same time as Peter, 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 Peter ran more than Dylan in first 30 min
= 30 x 112
= 3360 m

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

²₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 2 = 1680 m

Distance that Dylan ran in the first 30 min
= 4 x 1680
= 6720 m

Dylan's average speed
= 6720 ÷ 30
= 224 m/min

Peter's average speed
= 224 + 112
= 336 m/min

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

¹₁₃ of the entire distance = 1680 m
⁷₁₃ of the entire distance = 7 x 1680 = 11760 m

Duration that Peter would take to complete
= 11760 ÷ 336
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1680 = 15120 m  

Speed that Dylan needs to take to complete distance at the same time as Peter
= 15120 ÷ 35
= 432 m/min

Increase in Dylan's average speed
= 432 - 224
= 208 m/min

Percentage increase in Dylan's average speed
= ²⁰⁸₂₂₄ x 100%
≈ 93%

Answer(s): 93%