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

Fraction of the entire distance that was covered by Daniel more than Billy 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 Billy ran in the first 35 min
= 4 x 1190
= 4760 m

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

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

Fraction of the journey that Daniel 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 Daniel would take to complete
= 7140 ÷ 238
= 30 min

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

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

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

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

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

Answer(s): 163%