Question

The diagram shows a sequence of squares, decreasing in size and alternating between white and grey. Each new square is drawn by connecting the midpoints of the sides of the larger square that precedes it. If this process is continued indefinitely, then the fraction of the area that will ultimately be grey is

• A. 1/4
• B. 1/3
• C. 5/16
• D. 2/5

Answer 1

The correct option is B

1/3

Each square is half the area of the square just bigger than it (which is the other colour), so it is one-quarter of the area of the next larger square of the same colour. Taking the area of the whole figure to be 1, we see that the area of the four largest grey triangles is $\frac{1}{4}$, and each set of four grey triangles is % of the area of the previous one. Thus the total grey area is $\frac{1}{4}$ + $\frac{1}{4^2}$ + $\frac{1}{4^3}$+ ........=$\frac{\frac{1}{4}}{(1-\frac{1}{4})}$ = $\frac{1}{3}$