How to calculate the speed of the slower horse in the following question?

Two horses start trotting towards each other, one from A to B and another from B to A. They cross each other after one hour and the first horse reaches B, 5/6 hour before the second horse reaches A. If the distance between A and B is 50 km, what is the speed of the slower horse?
Options are as follows:
1) 30 km/h
2) 15 km/h
3) 20 km/h
4) 25 km/h

1 Answer
Mar 1, 2018

3) 20 km/h

Explanation:

We have two different times and distances related by the same rate. The basic equation is #D = RxxT#. The first set is the meeting point, which must be at a point #50 - x# from one side and #x# from the other. Thus, the two rate equations are:
#D_1 = R_1xx1# and #D_2 = R_2xx1#
#50 - x = R_1# and #x = R_2#

The second set are the time equations related to the total distance:
#50 = R_1xxT_1# and #50 = R_2xxT_2#
We also know that #T_1 = T_2 - 5/6# or #T_2 = T_1 + 5/6#

#50 = R_2xx (T_1 + 5/6)# ; #R_2 = 50/(T_1 + 5/6)#
#50 = R_1xxT_1# ; #R_1 = 50/(T_1)#
Pick a number from the options to avoid further math manipulations.
For:
#R_2 = 20#
#20 = 50/(T_1 + 5/6)# ; #20T_1 + 100/6 = 50#
#T_1 = (50 - 100/6)/20 = 1.67#
#50 = R_1xxT_1# ; #50 = R_1xx1.6667#
#R_1 = 30#
Check:
#50 - x = R_1# and #x = R_2#
#50 - x = 30# and #x = 20#
#50 - 20 = 30# CORRECT!

For:
#R_2 = 30#
#30 = 50/(T_1 + 5/6)# ; #30T_1 + 100/6 = 50#
#T_1 = (50 - 100/6)/30 = 1.11#
#50 = R_1xxT_1# ; #50 = R_1xx1.11#
#R_1 = 45#
Check:
#50 - x = R_1# and #x = R_2#
#50 - x = 45# and #x = 30#
#50 - 30 != 45# INcorrect!
You can try out the others.