After 30 minutes into a race, Owen has run ⁶₁₃ of the route while Rael has covered only ⁴₁₃ of the distance. Owen runs at the same average speed throughout the race and Rael’s average speed is 63 m/min slower than Owen’s. If Rael wants to finish the race at the same time as Owen, 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 Owen ran more than Rael in first 30 min
= 30 x 63
= 1890 m

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

²₁₃ of the entire distance = 1890 m
¹₁₃ of the entire distance = 1890 ÷ 2 = 945 m

Distance that Rael ran in the first 30 min
= 4 x 945
= 3780 m

Rael's average speed
= 3780 ÷ 30
= 126 m/min

Owen's average speed
= 126 + 63
= 189 m/min

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

¹₁₃ of the entire distance = 945 m
⁷₁₃ of the entire distance = 7 x 945 = 6615 m

Duration that Owen would take to complete
= 6615 ÷ 189
= 35 min

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

⁹₁₃ of the entire distance = 9 x 945 = 8505 m  

Speed that Rael needs to take to complete distance at the same time as Owen
= 8505 ÷ 35
= 243 m/min

Increase in Rael's average speed
= 243 - 126
= 117 m/min

Percentage increase in Rael's average speed
= ¹¹⁷₁₂₆ x 100%
≈ 93%

Answer(s): 93%