After 35 minutes into a race, Zane has run ⁷₁₃ of the route while Sam has covered only ⁴₁₃ of the distance. Zane runs at the same average speed throughout the race and Sam’s average speed is 84 m/min slower than Zane’s. If Sam wants to finish the race at the same time as Zane, 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 Zane ran more than Sam in first 35 min
= 35 x 84
= 2940 m

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

³₁₃ of the entire distance = 2940 m
¹₁₃ of the entire distance = 2940 ÷ 3 = 980 m

Distance that Sam ran in the first 35 min
= 4 x 980
= 3920 m

Sam's average speed
= 3920 ÷ 35
= 112 m/min

Zane's average speed
= 112 + 84
= 196 m/min

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

¹₁₃ of the entire distance = 980 m
⁶₁₃ of the entire distance = 6 x 980 = 5880 m

Duration that Zane would take to complete
= 5880 ÷ 196
= 30 min

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

⁹₁₃ of the entire distance = 9 x 980 = 8820 m  

Speed that Sam needs to take to complete distance at the same time as Zane
= 8820 ÷ 30
= 294 m/min

Increase in Sam's average speed
= 294 - 112
= 182 m/min

Percentage increase in Sam's average speed
= ¹⁸²₁₁₂ x 100%
≈ 163%

Answer(s): 163%