After 30 minutes into a race, Brandon has run ⁶₁₃ of the route while Rael has covered only ⁴₁₃ of the distance. Brandon runs at the same average speed throughout the race and Rael’s average speed is 112 m/min slower than Brandon’s. If Rael wants to finish the race at the same time as Brandon, 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 Brandon ran more than Rael in first 30 min
= 30 x 112
= 3360 m

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

²₁₃ of the entire distance = 3360 m
¹₁₃ of the entire distance = 3360 ÷ 2 = 1680 m

Distance that Rael ran in the first 30 min
= 4 x 1680
= 6720 m

Rael's average speed
= 6720 ÷ 30
= 224 m/min

Brandon's average speed
= 224 + 112
= 336 m/min

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

¹₁₃ of the entire distance = 1680 m
⁷₁₃ of the entire distance = 7 x 1680 = 11760 m

Duration that Brandon would take to complete
= 11760 ÷ 336
= 35 min

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

⁹₁₃ of the entire distance = 9 x 1680 = 15120 m  

Speed that Rael needs to take to complete distance at the same time as Brandon
= 15120 ÷ 35
= 432 m/min

Increase in Rael's average speed
= 432 - 224
= 208 m/min

Percentage increase in Rael's average speed
= ²⁰⁸₂₂₄ x 100%
≈ 93%

Answer(s): 93%