After 35 minutes into a race, Sam has run ⁷₁₃ of the route while Tom has covered only ⁴₁₃ of the distance. Sam runs at the same average speed throughout the race and Tom’s average speed is 102 m/min slower than Sam’s. If Tom wants to finish the race at the same time as Sam, 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 Sam ran more than Tom in first 35 min
= 35 x 102
= 3570 m

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

³₁₃ of the entire distance = 3570 m
¹₁₃ of the entire distance = 3570 ÷ 3 = 1190 m

Distance that Tom ran in the first 35 min
= 4 x 1190
= 4760 m

Tom's average speed
= 4760 ÷ 35
= 136 m/min

Sam's average speed
= 136 + 102
= 238 m/min

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

¹₁₃ of the entire distance = 1190 m
⁶₁₃ of the entire distance = 6 x 1190 = 7140 m

Duration that Sam would take to complete
= 7140 ÷ 238
= 30 min

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

⁹₁₃ of the entire distance = 9 x 1190 = 10710 m  

Speed that Tom needs to take to complete distance at the same time as Sam
= 10710 ÷ 30
= 357 m/min

Increase in Tom's average speed
= 357 - 136
= 221 m/min

Percentage increase in Tom's average speed
= ²²¹₁₃₆ x 100%
≈ 163%

Answer(s): 163%