please help me with my word problem!

Question

The rail line between two cities consists of two segments, one 96km longer than the other. A passenger train averages 60km/hr over the shorter segment, 120km/hr over the longer segment and 100km/hr over the entire trip. How far apart are the cities?

Answer 1

distance=rate*time

d=rt

x=shorter segment

x+96=longer segment

x=60t and x+96=120t so t=x/60 and t=(x+96)/120

find the total time for the trip...

(x/60)+[(x+96)/120]=total time

simplifying...

(2x/120)+[(x+96)/120]=(3x+96)/120 for the total time

total distance=average rate*total time

(x+x+96)=100*(3x+96)/120

2x+96=5*(3x+96)/6 (simplified the right side)

multiply both sides by 6

12x+576=5(3x+96)

12x+576=15x+480

15x-12x=576-480

3x=96

x=96/3

x=32 km for the shorter distance

x+96=128 km for the longer distance

128 km+ 32 km=160 km

the cities are 160 km apart

check; d=rt

160=100t

160/100=t

1.6 hours is the time

32=60t, 32/60=t

128=120t, 128/120=t

32/60+128/120=64/120+128/120=192/120=1.6 hours again for the total time