Answer:
Either 200 km or 160 km.
Step-by-step explanation:
Distance = speed × time
Let x = the biker's normal speed
Then
(1) d = 4x
For the return trip,
Time = distance/speed
(2) 4.5 = 100/x + (d -100)/(x - 10)
4.5x = 100/x + (4x -100)/(x - 10) Substituted (1) into (2)
4.5x = 100 + x(4x -100)/(x - 10) Multiplied each side by x
4.5x(x - 10) = 100(x - 10) + x(4x -100) Multiplied each side by (x - 10)
4.5x² - 45x = 100x - 1000 + 4x² -100x Removed parentheses
4.5x² - 45x = -1000 + 4x² Cancelled terms
0.5x² - 45x = -1000 Subtracted 4x² from each side
0.5x² - 45x +1000 = 0 Added 1000 to each side
x² - 90x +2000 = 0 Multiplied each side by 2
(x - 50)(x - 40) = 0 Factored the quadratic
x - 50 = 0 x - 40 = 0 Applied zero product theorem
x = 50 x = 40
Substitute in (1)
d = 4 × 50 = 200 km; d = 4 × 40 = 160 km
The distance between the two cities is either 200 km or 160 km.
Check:
x = 50 x = 40
4.5 = 100/50 + (200 - 100)/(50 - 10) 4.5 = 100/40 + (160 - 100)/(40 - 10)
4.5 = 2 + 100/40 4.5 = 2.5 + 60 /30
4.5 = 2 + 2.5 4.5 = 2.5 + 2
4.5 = 4.5 4.5 = 4.5
OK
