In the diagram on the right, the area of the circle is 2/3 the area of the square. The area of the triangle is 1/5 the area of the circle.
a) Work out 2/3 x 1/5
b) What fraction of the square is shaded?
a) 2/3 x 1/5 = 2/15
b) First, we need to find the area of the square. Let's assume that the diameter of the circle is the same as the side length of the square. The area of the circle is then (pi)(d/2)^2 = (pi)(1/2)^2 = pi/4. Since the area of the circle is 2/3 the area of the square, we can set up the equation (2/3)(area of square) = pi/4. Solving for the area of the square, we get (area of square) = (3/2)(pi/4) = (3/8)pi.
Now, we can find the area of the shaded region, which is the area of the square minus the area of the circle plus the area of the triangle. The area of the triangle is (1/5)(area of circle) = (1/5)(pi/4) = (pi/20).
Therefore, the area of the shaded region is (3/8)pi - pi/4 + pi/20 = (7/40)pi.
The area of the square is (3/8)pi, so the fraction of the square that is shaded is:
(7/40)pi / (3/8)pi = (7/40) / (3/8) = 7/15.
Therefore, 7/15 of the square is shaded.
a) To work out 2/3 x 1/5, we multiply the numerators together (2 x 1) to get 2, and multiply the denominators together (3 x 5) to get 15. So 2/3 x 1/5 = 2/15.
b) To find the fraction of the square that is shaded, we need to know the fraction of the total area that the shaded region represents. Since the area of the circle is 2/3 of the area of the square and the area of the triangle is 1/5 of the area of the circle, we can calculate the total area shaded by multiplying the fractions together: 2/3 x 1/5 = 2/15.
Therefore, the shaded fraction of the square is 2/15.