During a trip, Francine traveled x percent of the total distance at an

To find Francine's average speed for the entire trip in terms of x, we need to consider the distance traveled at each speed.

Let's denote the distance traveled at 40 miles per hour as D1 and the distance traveled at 60 miles per hour as D2. The total distance is the sum of these two distances.

The average speed is calculated by dividing the total distance by the total time taken. Since the time is equal to the distance divided by the speed, we can express the average speed as:

Average speed = (D1 + D2) / (D1/40 + D2/60)

To determine the values of D1 and D2, we need to consider the percentage x.

If x percent of the total distance is traveled at 40 miles per hour, then (100 - x) percent is traveled at 60 miles per hour.

D1 = (x/100) * Total distance
D2 = [(100 - x)/100] * Total distance

Substituting these values into the equation for average speed:

Average speed = [(x/100) * Total distance + [(100 - x)/100] * Total distance] / [(x/100) * Total distance / 40 + [(100 - x)/100] * Total distance / 60]

Simplifying the equation:

Average speed = (x + 100 - x) / [(x/40) + (100 - x)/60]= 100 / [(x/40) + (100 - x)/60] = 100 / [(3x + 2(100 - x))/120] = 100 / [(3x + 200 - 2x)/120] = 100 / [(200 + x)/120] = 12000 / (200 + x)

Therefore, Francine's average speed for the entire trip, in terms of x, is 12000 / (200 + x)