Question 24123
For this problem, you'll need to apply the concept of weighted averages.

Let x = the number of hours driven by mother. We can find the weighted average of father's speed plus mother's speed as follows:
The weighted average speed is given as: 58 mph.

58 mph = (father's distance + mother's distance)/total trip time.
Father's distance is: 55 mph(4 hrs)
Mother's distance is: 60 mph(x hrs)
The total trip time is (father's driving time) 4 hours plus (mother's driving time) x hours or (4+x) hours. Now you can write the equation for the weighted average speed for the trip which is given as 58 mph.

{{{58 = (55(4) + 60(x))/(4+x)}}} Simplify and solve for x. Multiply both sides by (4+x)
{{{58(4+x) = 220 + 60x}}} Simplify.
{{{232 + 58x = 220 + 60x}}} Subtract 58x from both sides.
{{{232 = 220 + 2x}}} Subtract 220 from both sides.
{{{12 = 2x}}} Divide both sides by 2.
{{{x = 6}}}

So, mother's driving time is 6 hours.
Father's diving time is 4 hhours.
The total trip time is 6 hrs + 4 hrs = 10 hours.

Check:

Father's distance is 55 mph(4 hours) = 220 miles.
Mother's distance is 60 mph(6 hours) = 360 miles.
Total distance is 220 miles + 360 miles = 580 miles.

Total distance driven = weighted average speed (58 mph) X total driving time (10 hours).
Total distance driven = 58 mph X 10 hours = 580 miles.