Shaded Part: Three Corners
Read the puzzle
A square is split into four equal quarters, each a 2 by 2 grid. In three quarters a diagonal cuts the quarter in half and one half is gold: top-left (upper-right half), bottom-left (lower-left half) and bottom-right (upper-right half). The top-right quarter has no gold. What fraction of the whole square is gold?
What fraction of the square is gold?
Split the square into four quarters. How much of each quarter is gold?
3/8
Each triangle is half a quarter: 3 × 1/8 = 3/8.
How to solve it
Work one quarter at a time instead of the whole square.
The square is cut into four equal quarters, so each quarter is 1/4 of the whole. In each gold quarter a diagonal runs from one corner to the opposite corner, and a diagonal always splits a square into two equal triangles. So the gold triangle in that quarter is half of 1/4, which is 1/8 of the whole square.
Three quarters carry a gold triangle and the top-right quarter carries none. That gives 3 x 1/8 = 3/8.
Check with the small grid squares: the big square is 4 x 4 = 16 small squares. Each quarter has 4 of them, and each gold triangle covers half, so 2 small squares. Three triangles cover 3 x 2 = 6, and 6/16 = 3/8.
The common mistake is to answer 3/4, by counting three shaded quarters and forgetting that each one is only half gold. Another is 3/16, from counting only the 3 small squares that are wholly gold and ignoring the cut halves.
Takeaway: when a shape is built from repeated pieces, find the fraction of one piece and multiply by how many there are.
