For the solution, click "Read More" below.
There are different ways to reason through these problems. including by using the formula for a geometric series. Here are some ideas for reasoning through the problems without using the formula.
Problem #1
One-third of the square is shaded.
One-third of the square is shaded.
For this square we can focus one quarter since they differ only by a rotation.
Notice that we can think of this piece as a series of L-shaped pieces that get smaller and smaller. While the L's are all different in size, they each have the same fraction shaded.
Since 1/3 of each "L" is shaded, then 1/3 of the whole square is shaded.
Problem #2
Two-thirds of the square is shaded.
Notice that two-fourths are shaded completely, and then 1/3 of the other two-fourths are each 1/3 shaded for the same reason explained above. Adding the parts we get 1/2 + (1/3 x 1/2) = 2/3
Problem #3
Two-thirds of the square is shaded.
Notice that we can break down the square into Ls where each one is 2/3 shaded:
So the whole thing is 2/3 shaded.
Problem #4
The square is half shaded. Notice that for every triangle that is shaded, there is a corresponding triangle that is not shaded. Since the two parts are equal, it is half shaded.
Problem #5
This square is 7/12 shaded.
Each quarter differs only by a rotation, so let's focus on one quarter.
This square, like some of the others, is made of nested Ls. The largest one, which is 3/4 of the square, is 2/3 shaded.
The others, which together form 1/4 of the square, are all together 1/3 shaded, because of reasoning given above in problem 1.
All together we get (2/3 x 3/4) + (1/3 x 1/4) = 7/12.
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