After 30 minutes into a race, Justin has run ⁶₁₃ of the route while Sam has covered only ⁴₁₃ of the distance. Justin runs at the same average speed throughout the race and Sam’s average speed is 35 m/min slower than Justin’s. If Sam wants to finish the race at the same time as Justin, 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 Justin ran more than Sam in first 30 min
= 30 x 35
= 1050 m

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

²₁₃ of the entire distance = 1050 m
¹₁₃ of the entire distance = 1050 ÷ 2 = 525 m

Distance that Sam ran in the first 30 min
= 4 x 525
= 2100 m

Sam's average speed
= 2100 ÷ 30
= 70 m/min

Justin's average speed
= 70 + 35
= 105 m/min

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

¹₁₃ of the entire distance = 525 m
⁷₁₃ of the entire distance = 7 x 525 = 3675 m

Duration that Justin would take to complete
= 3675 ÷ 105
= 35 min

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

⁹₁₃ of the entire distance = 9 x 525 = 4725 m  

Speed that Sam needs to take to complete distance at the same time as Justin
= 4725 ÷ 35
= 135 m/min

Increase in Sam's average speed
= 135 - 70
= 65 m/min

Percentage increase in Sam's average speed
= ⁶⁵₇₀ x 100%
≈ 93%

Answer(s): 93%