Let

'x' represent the height of the fence

'y' represent the height of the tree one year ago

A tree next to a fence is 2 feet taller than the fence means:

y = x + 2

One year later the tree grew an additional 0.4 feet and the ratio of the height of the fence to the height of the tree is now 5:8 means:

x : (y + 0.4) = 5 : 8

x / (y + 0.4) = 5 / 8

Since 0.4 = 4/10 = 2/5 we can write:

x / (y + 2/5) = 5 / 8

We have the following system of equations

y = x + 2

x / (y + 2/5) = 5 / 8

From y = x + 2 we get x = 2 - y

We plug x = 2 - y into x / (y + 2/5) = 5 / 8 and get:

(2 - y) / (y + 2/5) = 5 / 8

........

Click here to see the step by step solution

........

x = 1.08 ft

The fence is approximately 1.08 feet tall.