Word problems finding the two rates

Question

Jamie recently drove to visit her parents who live 455 miles away. On her way there her average speed was 24 miles per hour faster than on her way home. If Jamie spent a total of 13 hours driving, find the two rates

Answer 1

Generally correct, but the units could be cleaned up:

Total average speed = (455*2)/13 = 70 mph

Given, her average speed going to the destination was the Return Speed (RS) plus 24 mph...

[455/(RS+24)] + [455/RS] = 13 hours, solve for RS.

 

Factoring out 455 [1/(RS+24) + 1/RS] = 13;

Dividing both sides by 455 and solving the LCD, RS/[RS(RS+24)] + (RS+24)/[RS(RS+24)] = 13/455;

Multiplying both sides by the LCD, RS + (RS+24) = 13/455 * RS(RS+24);

Further simplifying, 2RS+24 = 13/455 * (RS^2 + 24RS)

Multiplying both sides by 455/13 produces 455/13 (2RS+24) = RS^2 + 24RS

So RS^2 + 24RS - (910/13RS + 840) equals RS^2 - 46RS - 840

 

Plugging into the Quadratic Equation, RS = [-(-46) +/- Sqrt (46^2 - 4(1)(-840)] / 2(1)

So, RS = [46 +/- Sqrt (2116 + 3360)] / 2 = (46 +/- 74) / 2...

Therefore RS = 60 mph (we can eliminate 14 mph), so the speed going to the destination was 84 mph?

 

Let's plug those solutions in to verify:

74 mph over 455 miles = 455/74 hours = 5.4167 hours

60 mph over 455 miles = 455/60 hours = 7.5833 hours

Adding up both hours arrives at 13.0 hours. Q.E.D.

 

Answer 2

In 13 hours Jamie traveled a distance of (2*455) mi

Hence,  by rule of proportion, in 1 hour Jamie traveled (2*455)/13 mi = 70 mi

Hence, her average speeds of the 2 journeys to her parents home and coming back to her home  is =70 mi/ per hour

Therefore, the sum of average speeds of  to her parents home and coming back to her home =(2*70) mi per hr =140 mi per hr

On her way there her average speed was 24 miles per hour faster than on her way home

Hence, (140-24)mi per hr= 116 mi per hr is the double of the speed on her way home

Hence her average speed, on her way home was (116/2) mi per hr=58 mi per hr

Ob her to her parents home was =(58+24) mi per hr = 82 mi. per her