Loto begins at his back door and walks 8 yards east, 6 yards north, 12 yards east, and 5 yards north to the barn door. About how many yards less would he walk if he could walk directly from the back door to the barn door?

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We can make a quick sketch of the path:

Going directly would be the same as going the distance of the hypotenuse of the triangle formed by the path:

The hypotenuse can be found using the pythagorean theorem:

$$ a^2+b^2=c^2 $$ $$ 20^2 + 11^2 = c^2 $$ $$ 400+121=c^2 $$ $$ c=\sqrt{521} \approx 22.8 $$

The original path is:

$$ 8+6+12+5 = 31 $$

The difference:

$$ 31 - 22.8 \approx \boxed{8} $$