After 35 minutes into a race, Neave has run ⁷₁₃ of the route while Paul has covered only ⁴₁₃ of the distance. Neave runs at the same average speed throughout the race and Paul’s average speed is 78 m/min slower than Neave’s. If Paul wants to finish the race at the same time as Neave, 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 Neave ran more than Paul in first 35 min
= 35 x 78
= 2730 m

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

³₁₃ of the entire distance = 2730 m
¹₁₃ of the entire distance = 2730 ÷ 3 = 910 m

Distance that Paul ran in the first 35 min
= 4 x 910
= 3640 m

Paul's average speed
= 3640 ÷ 35
= 104 m/min

Neave's average speed
= 104 + 78
= 182 m/min

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

¹₁₃ of the entire distance = 910 m
⁶₁₃ of the entire distance = 6 x 910 = 5460 m

Duration that Neave would take to complete
= 5460 ÷ 182
= 30 min

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

⁹₁₃ of the entire distance = 9 x 910 = 8190 m  

Speed that Paul needs to take to complete distance at the same time as Neave
= 8190 ÷ 30
= 273 m/min

Increase in Paul's average speed
= 273 - 104
= 169 m/min

Percentage increase in Paul's average speed
= ¹⁶⁹₁₀₄ x 100%
≈ 163%

Answer(s): 163%