Find the ratio of A's speed to B's speed.

Question

Two men, A and B, start at the same point, walk in opposite directions, and reach their respective destinations, X and Y each in one hour. Had they exchanged destinations from the start, A would have arrived at Y 35 minutes after B would have arrived at X.

Answer 1

distance = rate x time

 

Ra is man A's walking speed

Rb is man B's walking speed

 

AX is the distance from A to X

BY is the distance from B to Y

 

AX = Ra * 1 = Ra

 

BY = Rb*1 = Rb

 

 

35 minutes = 35/60 hr = 7/12 hour

 

AX = Rb * t  and BY = Ra * (t+7/12)  where t is the time it takes man B to go from point A to X

 

But from the previous equations:

 

Ra = AX = Rb * t  and Rb = BY = Ra * (t + 7/12) <--- please label these equations ALPHA and BETA respectively

 

Substituting the first equation into the second

 

  Rb = (Rb * t) ( t + 7/12)

 

 Rb = (Rb * t^2 ) + (7/12)(Rb)(t)

 

 0 = (Rb * t^2 ) + (7/12)(Rb)(t) - Rb  

 

 Rb factors out and cancels

 

0 = t^2 + (7/12)t - 1

 

0 = 12t^2 + 7t - 12 <--- multiplies the entire equation by 12

 

0 = (  4 t    -     3)( 3 t  +    4   )

 

3t + 4 = 0 results in negative time values.

 

4 t - 3 = 0 ---> t = 3/4

 

Substituting this into equations ALPHA and BETA in bold above

 

Ra = AX = Rb * (3/4)  and Rb = BY = Ra * (3/4 + 7/12)

 

Ra = (3/4) * Rb

 

Rb = Ra * 4/3  <-----  3/4 + 7/12 = 9/12 + 7/12 = 16/12 = 4/3

 

These two equations are really the same, which shows the ratios of their speeds.