You know area = length × width for whole numbers, and you can multiply fractions. The formula still works when the sides are fractions, and tiling shows why.
Key idea
The area of a rectangle with fraction side lengths is still length × width. Tiling it with small rectangles that are $\frac{1}{b}$ by $\frac{1}{d}$ gives the same answer: count the tiles and multiply by the area of one tile.
Why it works
Cut a unit square into 4 columns and 3 rows. Each tile is $\frac{1}{4}$ by $\frac{1}{3}$, and 12 tiles fill the square, so each tile has area $\frac{1}{12}$. A rectangle $\frac{3}{4}$ long and $\frac{2}{3}$ wide holds 3 tiles across and 2 down: $3 \times 2 = 6$ tiles, an area of $\frac{6}{12}$. That is exactly $\frac{3}{4} \times \frac{2}{3}$.
Worked examples
Worked example: Count the tiles
Find the area of a rectangle $\frac{3}{4}$ ft long and $\frac{2}{3}$ ft wide.
- Tiles $\frac{1}{4}$ by $\frac{1}{3}$ fit: 3 along the length and 2 along the width, so 6 tiles.
- 12 tiles make 1 square foot, so each is $\frac{1}{12}$ square foot.
- Area: $\frac{6}{12} = \frac{1}{2}$ square foot. Same as $\frac{3}{4} \times \frac{2}{3}$.
Answer: $\frac{1}{2}$ square foot
Worked example: Sides longer than 1
Find the area of a rectangle $\frac{5}{3}$ m by $\frac{3}{2}$ m.
- Multiply: $\frac{5}{3} \times \frac{3}{2} = \frac{15}{6}$.
- Simplify: $\frac{15}{6} = \frac{5}{2} = 2\frac{1}{2}$.
- Tiling check: tiles $\frac{1}{3}$ by $\frac{1}{2}$ have area $\frac{1}{6}$, and 5 by 3 of them make 15.
Answer: $2\frac{1}{2}$ square meters
Common mistake
Forgetting that a tile is smaller than a unit square and calling the area “6 square feet” because there are 6 tiles. Check: the rectangle fits inside one square foot, so its area must be less than 1.