Erica's average rate going uphill was 80% of her average rate going downhill over the same route, and it took Erica 10 minutes longer to go up the hill than to go down the hill. What was the total number of minutes Erica spent going uphill and downhill?

distance = rate * time

so, since the distance covered is the same both ways,

.8r * (t+10) = rt
.8rt + 8r = rt
8r = .2rt
8 = .2t
t = 40

so, total time = 40+10 + 40 = 90 minutes

To solve this problem, we can use the concept of average speed and the formula for finding the time taken.

Let's assume Erica's average rate going downhill is "x" miles per hour.

Her average rate going uphill would be 80% of her average rate going downhill, which is (80/100)x = 0.8x miles per hour.

Now, let's find the time taken for each leg of the trip:

Time taken going downhill = Distance / Rate = D / x (where D is the distance)

Time taken going uphill = Distance / Rate = D / (0.8x)

We are given that Erica took 10 minutes longer to go uphill than to go downhill. Since time is measured in the same units (hours), we need to convert 10 minutes to hours:

10 minutes = 10/60 = 1/6 hours

Therefore, our equation becomes:

D / (0.8x) = D / x + 1/6

To solve for D, the distance, we can multiply both sides of the equation by the least common denominator (LCD) of 6x:

6D = 6D / (0.8x) + x

Multiplying through by 6x:

6D * 6x = 6D * 6D / (0.8x) + x * 6x

36Dx = 6D * 6 + 6x^2

Rearranging the terms:

36Dx - 36D - 6x^2 = 0

We can simplify this equation by dividing both sides by 6:

6Dx - 6D - x^2 = 0

This is a quadratic equation. To solve it, we can factor or use the quadratic formula:

(x - 6)(6D + x) = 0

From this equation, we have two possibilities:

1) x - 6 = 0, which gives us x = 6 miles per hour.

2) 6D + x = 0, which gives us 6D = -x. We know that rates and distances must be positive, so this solution is not valid.

Therefore, Erica's average rate going downhill is 6 miles per hour.

Now, let's find the total time Erica spent going uphill and downhill:

Time taken going downhill = Distance / Rate = D / 6

Time taken going uphill = Distance / Rate = D / (0.8 * 6) = D / 4.8

Adding the times together:

Total time = D / 6 + D / 4.8

To simplify this equation, we need to find the least common denominator (LCD) of 6 and 4.8, which is 24:

Total time = (4D + 5D) / 24 = 9D / 24 = 3D / 8

We know that Erica's total time is equal to 10 minutes longer than her time going downhill:

Total time = Time taken going downhill + 10/60 (converted to hours)
3D / 8 = D / 6 + 1/6

To solve for D, we can multiply both sides of the equation by 24:

3D * 24 / 8 = D * 24 / 6 + 24 / 6
9D = 4D + 4

Subtracting 4D from both sides:

9D - 4D = 4
5D = 4

Dividing both sides by 5:

D = 4/5

So, the distance Erica traveled (D) is 4/5 of a mile.

Now, let's calculate the total time Erica spent going uphill and downhill:

Total time = 3D / 8
= 3 * (4/5) / 8
= 12/40
= 3/10
= 0.3 hours

To convert hours to minutes, we multiply by 60:

Total time = 0.3 * 60
= 18 minutes

Therefore, Erica spent a total of 18 minutes going uphill and downhill.