After 30 minutes into a race, Paul has run ⁶₁₃ of the route while Peter has covered only ⁴₁₃ of the distance. Paul runs at the same average speed throughout the race and Peter’s average speed is 98 m/min slower than Paul’s. If Peter 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 Peter in first 30 min
= 30 x 98
= 2940 m

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

²₁₃ of the entire distance = 2940 m
¹₁₃ of the entire distance = 2940 ÷ 2 = 1470 m

Distance that Peter ran in the first 30 min
= 4 x 1470
= 5880 m

Peter's average speed
= 5880 ÷ 30
= 196 m/min

Paul's average speed
= 196 + 98
= 294 m/min

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

¹₁₃ of the entire distance = 1470 m
⁷₁₃ of the entire distance = 7 x 1470 = 10290 m

Duration that Paul would take to complete
= 10290 ÷ 294
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1470 = 13230 m  

Speed that Peter needs to take to complete distance at the same time as Paul
= 13230 ÷ 35
= 378 m/min

Increase in Peter's average speed
= 378 - 196
= 182 m/min

Percentage increase in Peter's average speed
= ¹⁸²₁₉₆ x 100%
≈ 93%

Answer(s): 93%