After 35 minutes into a race, David has run ⁷₁₃ of the route while Rael has covered only ⁴₁₃ of the distance. David runs at the same average speed throughout the race and Rael’s average speed is 78 m/min slower than David’s. If Rael wants to finish the race at the same time as David, 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 David ran more than Rael in first 35 min
= 35 x 78
= 2730 m

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

³₁₃ of the entire distance = 2730 m
¹₁₃ of the entire distance = 2730 ÷ 3 = 910 m

Distance that Rael ran in the first 35 min
= 4 x 910
= 3640 m

Rael's average speed
= 3640 ÷ 35
= 104 m/min

David's average speed
= 104 + 78
= 182 m/min

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

¹₁₃ of the entire distance = 910 m
⁶₁₃ of the entire distance = 6 x 910 = 5460 m

Duration that David would take to complete
= 5460 ÷ 182
= 30 min

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

⁹₁₃ of the entire distance = 9 x 910 = 8190 m  

Speed that Rael needs to take to complete distance at the same time as David
= 8190 ÷ 30
= 273 m/min

Increase in Rael's average speed
= 273 - 104
= 169 m/min

Percentage increase in Rael's average speed
= ¹⁶⁹₁₀₄ x 100%
≈ 163%

Answer(s): 163%