Ajit has red, yellow, green, and blue tiles. All tiles of the same colour have the same dimensions. The red and yellow tiles are squares, and the green and blue tiles are rectangles. Ajit arranges his tiles into the following rectangle so that none of the tiles are overlapping.
If Ajit wanted to use only red squares to make a rectangle with the same dimensions, how many red squares would he need?
The length of a yellow square is twice the length of a red square. This means that if we arrange four red squares, two across and two down, we will form a square with the same dimensions as a yellow square.
The length of the shorter side of a green rectangle is the same as the length of a red square, and the length of the longer side of a green rectangle is the same as the length of a yellow square. Since the length of a yellow square is twice the length of a red square, if we arrange two red squares beside each other, we will form a rectangle with the same dimensions as a green rectangle.
The length of the shorter side of a blue rectangle is the same as the length of a red square, and the length of the longer side of a blue rectangle is the same as the sum of the length of a yellow square plus the length of a red square. Since the length of a yellow square is twice the length of a red square, if we arrange three red squares in a line, we will form a rectangle with the same dimensions as a blue rectangle.
In summary:
\(1\) yellow square \(= 4\) red squares
\(1\) green rectangle \(= 2\) red squares
\(1\) blue rectangle \(= 3\) red squares
We can create a rectangle with the same dimensions as shown in the diagram by replacing each yellow, green, and blue tile with the equivalent number of red squares.
| Tile Colour | Number in Diagram | Equivalent Number of Red Squares |
|---|---|---|
| yellow | \(2\) | \(2 \times 4 = 8\) |
| green | \(3\) | \(3 \times 2 = 6\) |
| blue | \(1\) | \(1 \times 3 = 3\) |
Including the three red squares in the original diagram, we determine that the rectangle can be formed using \(8 + 6 + 3 + 3 = 20\) red squares.