In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown.  Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street.  How many more feet than Sarah does Sam run for every lap around the block?

[asy]
draw((0,0)--(10,0));
draw((1,1)--(9,1));
draw((0,0)--(0,10));
draw((0,10)--(10,10));
draw((10,10)--(10,0));
draw((9,1)--(9,9));
draw((9,9)--(1,9));
draw((1,9)--(1,1));
fill((1,1)--(1,9)--(9,9)--(9,1)--cycle,gray);
draw((9,10.5)--(9,11));
draw((10,10.5)--(10,11));
draw((9,10.75)--(10,10.75));
label("20'",(9.5,11),N);
draw((10.5,9)--(11,9));
draw((10.5,10)--(11,10));
draw((10.75,9)--(10.75,10));
label("20'",(11,9.5),E);
label("400'",(5,9),S);
label("400'",(9,5),W);
[/asy]
At each corner of the street, Sam runs 40 more feet than Sarah. Otherwise, Sam runs the same distance as Sarah. Since there are four corners, Sam runs $40\cdot4=\boxed{160}$ more feet than Sarah does.
160
Level 4
Prealgebra
Problem: In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown.  Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street.  How many more feet than Sarah does Sam run for every lap around the block?

[asy]
draw((0,0)--(10,0));
draw((1,1)--(9,1));
draw((0,0)--(0,10));
draw((0,10)--(10,10));
draw((10,10)--(10,0));
draw((9,1)--(9,9));
draw((9,9)--(1,9));
draw((1,9)--(1,1));
fill((1,1)--(1,9)--(9,9)--(9,1)--cycle,gray);
draw((9,10.5)--(9,11));
draw((10,10.5)--(10,11));
draw((9,10.75)--(10,10.75));
label("20'",(9.5,11),N);
draw((10.5,9)--(11,9));
draw((10.5,10)--(11,10));
draw((10.75,9)--(10.75,10));
label("20'",(11,9.5),E);
label("400'",(5,9),S);
label("400'",(9,5),W);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the distance Sam runs compared to Sarah, we first determine the perimeter of the paths they each take. Sarah runs around the inner square, which has a side length of 400 feet. The perimeter of a square is calculated using the formula P = 4s, where s is the side length. Thus, Sarah's distance is P_Sarah = 4 * 400 = 1600 feet.

Next, we consider Sam's path, which runs around the outer square. The outer square is formed by adding the width of the street on both sides of the inner square. Since the street is 20 feet wide, we add 40 feet to each side of the inner square (20 feet on each side). Therefore, the side length of the outer square is 400 + 40 = 440 feet.

Using the same perimeter formula for the outer square, we calculate Sam's distance: P_Sam = 4 * 440 = 1760 feet.

To find how much more Sam runs than Sarah, we subtract Sarah's distance from Sam's distance: 1760 - 1600 = 160 feet.

Thus, for every lap around the block, Sam runs 160 feet more than Sarah. The final answer is 160.<|END_OF_LATENT|> The final answer is $160$.