I asked for 7 feet, 4 inches and my friend brought me 4 feet, 7 inches.

If we label what I asked for as A feet and B inches, which is (12 x A + B) inches, then I actually received B feet and A inches, which is (12 x B + A) inches. Which means that:

(5 ÷ 8) x (12 x A + B) = (12 x B + A)

Simplifying gives:

A = (91 ÷ 52) x B

As B is the number of inches, it can only be between 1 and 12 and must give A as a whole number of feet. B = 4 is the only possible value, which means that A = 7. So I originally asked for 7 feet, 4 inches. QED.