After 35 minutes into a race, Tom has run ⁷₁₃ of the route while Sam has covered only ⁴₁₃ of the distance. Tom runs at the same average speed throughout the race and Sam’s average speed is 108 m/min slower than Tom’s. If Sam wants to finish the race at the same time as Tom, 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 Tom ran more than Sam in first 35 min
= 35 x 108
= 3780 m

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

³₁₃ of the entire distance = 3780 m
¹₁₃ of the entire distance = 3780 ÷ 3 = 1260 m

Distance that Sam ran in the first 35 min
= 4 x 1260
= 5040 m

Sam's average speed
= 5040 ÷ 35
= 144 m/min

Tom's average speed
= 144 + 108
= 252 m/min

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

¹₁₃ of the entire distance = 1260 m
⁶₁₃ of the entire distance = 6 x 1260 = 7560 m

Duration that Tom would take to complete
= 7560 ÷ 252
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1260 = 11340 m  

Speed that Sam needs to take to complete distance at the same time as Tom
= 11340 ÷ 30
= 378 m/min

Increase in Sam's average speed
= 378 - 144
= 234 m/min

Percentage increase in Sam's average speed
= ²³⁴₁₄₄ x 100%
≈ 163%

Answer(s): 163%