After 35 minutes into a race, Elijah has run ⁷₁₃ of the route while Liam has covered only ⁴₁₃ of the distance. Elijah runs at the same average speed throughout the race and Liam’s average speed is 48 m/min slower than Elijah’s. If Liam wants to finish the race at the same time as Elijah, 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 Elijah ran more than Liam in first 35 min
= 35 x 48
= 1680 m

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

³₁₃ of the entire distance = 1680 m
¹₁₃ of the entire distance = 1680 ÷ 3 = 560 m

Distance that Liam ran in the first 35 min
= 4 x 560
= 2240 m

Liam's average speed
= 2240 ÷ 35
= 64 m/min

Elijah's average speed
= 64 + 48
= 112 m/min

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

¹₁₃ of the entire distance = 560 m
⁶₁₃ of the entire distance = 6 x 560 = 3360 m

Duration that Elijah would take to complete
= 3360 ÷ 112
= 30 min

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

⁹₁₃ of the entire distance = 9 x 560 = 5040 m  

Speed that Liam needs to take to complete distance at the same time as Elijah
= 5040 ÷ 30
= 168 m/min

Increase in Liam's average speed
= 168 - 64
= 104 m/min

Percentage increase in Liam's average speed
= ¹⁰⁴₆₄ x 100%
≈ 163%

Answer(s): 163%