After 30 minutes into a race, Cole has run ⁶₁₃ of the route while Tom has covered only ⁴₁₃ of the distance. Cole runs at the same average speed throughout the race and Tom’s average speed is 42 m/min slower than Cole’s. If Tom wants to finish the race at the same time as Cole, 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 Cole ran more than Tom in first 30 min
= 30 x 42
= 1260 m

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

²₁₃ of the entire distance = 1260 m
¹₁₃ of the entire distance = 1260 ÷ 2 = 630 m

Distance that Tom ran in the first 30 min
= 4 x 630
= 2520 m

Tom's average speed
= 2520 ÷ 30
= 84 m/min

Cole's average speed
= 84 + 42
= 126 m/min

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

¹₁₃ of the entire distance = 630 m
⁷₁₃ of the entire distance = 7 x 630 = 4410 m

Duration that Cole would take to complete
= 4410 ÷ 126
= 35 min

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

⁹₁₃ of the entire distance = 9 x 630 = 5670 m  

Speed that Tom needs to take to complete distance at the same time as Cole
= 5670 ÷ 35
= 162 m/min

Increase in Tom's average speed
= 162 - 84
= 78 m/min

Percentage increase in Tom's average speed
= ⁷⁸₈₄ x 100%
≈ 93%

Answer(s): 93%