After 35 minutes into a race, Paul has run ⁷₁₃ of the route while Liam has covered only ⁴₁₃ of the distance. Paul runs at the same average speed throughout the race and Liam’s average speed is 96 m/min slower than Paul’s. If Liam wants to finish the race at the same time as Paul, 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 Paul ran more than Liam in first 35 min
= 35 x 96
= 3360 m

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

³₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 3 = 1120 m

Distance that Liam ran in the first 35 min
= 4 x 1120
= 4480 m

Liam's average speed
= 4480 ÷ 35
= 128 m/min

Paul's average speed
= 128 + 96
= 224 m/min

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

¹₁₃ of the entire distance = 1120 m
⁶₁₃ of the entire distance = 6 x 1120 = 6720 m

Duration that Paul would take to complete
= 6720 ÷ 224
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1120 = 10080 m  

Speed that Liam needs to take to complete distance at the same time as Paul
= 10080 ÷ 30
= 336 m/min

Increase in Liam's average speed
= 336 - 128
= 208 m/min

Percentage increase in Liam's average speed
= ²⁰⁸₁₂₈ x 100%
≈ 163%

Answer(s): 163%