Here is a square with some lines drawn inside (Fig. 8.4). What fraction of area of the whole square does the shaded region occupy?
Answer
Peel the picture one layer at a time, taking the whole square as 1 square unit.
Step 1: the top-right square is a quarter of the whole = 1/4 sq unit
Step 2: the yellow triangle inside it is half of that = 1/2 × 1/4 = 1/8 sq unit
Step 3: the shaded part is 3/4 of that triangle = 3/4 × 1/8
= 3/32 square units
So the shaded region occupies 3⁄32 of the whole square.
Why this method works: “a fraction of a fraction of a fraction” is just a chain of multiplications. Each step shrinks the piece you are looking at, and multiplying the three fractions gives the share of the original whole.
Q2.
What fraction of this yellow triangle is shaded? Are you able to see why?
Answer
3⁄4 of the yellow triangle is shaded.
Joining the midpoints of the two legs to the midpoint of the hypotenuse cuts the triangle into four equal pieces. Three of them are shaded.
Unshaded piece = 1 of the 4 equal pieces = 1/4
Shaded = 1 − 1/4 = 3/4
Why the four pieces are equal: the vertical line from the midpoint of the base and the horizontal line from the midpoint of the vertical side both meet the hypotenuse at its midpoint. They cut the triangle into a small triangle, a square (= two such triangles) and another small triangle — four congruent halves in all.
Q3.
In each of the figures given below, find the fraction of the big square that the shaded region occupies.
Answer
Figure (a): 3⁄8. The shaded band lies between two parallel diagonals.
Big triangle below the main diagonal = 1/2 of the square
Small triangle below the shorter parallel line = 1/2 × 1/2 × 1/2 = 1/8
Shaded band = 1/2 − 1/8
= 4/8 − 1/8
= 3/8
Figure (b): 1⁄16. Everything happens inside the top-left quarter.
Top-left quarter = 1/4 of the square
The tilted square joining its midpoints = 1/2 × 1/4 = 1/8
The 4 corner triangles together = 1/4 − 1/8 = 1/8, so each = 1/32
Two corner triangles are shaded = 2 × 1/32
= 1/16
Tip: in (b) the shaded pieces are the corner triangles on the right of the tilted square — one at the top, one at the bottom.