The ratio of the length to the width of a rectangular courtyard floor is 2:1, and the perimeter of the courtyard is 348 meters. Approximately how many rectangular tiles, each measuring 4m × 3m, are needed to cover the entire floor?

The correct answer is 561.

To find the total number of rectangular tiles needed to cover the courtyard floor, we need to find the dimensions (length and width) of the floor from its perimeter and ratio, compute the total area of the floor, and then divide that area by the area of a single tile.

Step 1: Find the dimensions of the courtyard floor
Let the width of the rectangular courtyard be $x$ meters.
Since the ratio of length to width is 2:1, the length of the floor is $2x$ meters.

The formula for the perimeter of a rectangle is:

$\text{Perimeter}=2\times (\text{Length}+\text{Width})$

Substitute the given perimeter of 348 meters into the equation:
$348=2\times (2x+x)$
$348=2\times 3x$
$348=6x$
$x=\frac{348}{6}=58$

Therefore, the dimensions of the courtyard floor are:
Width = $58\text{meters}$
Length = $2\times 58=116\text{meters}$

Step 2: Calculate the total area of the courtyard floor
$\text{Area of floor}=\text{Length}\times \text{Width}$
$\text{Area of floor}=116\times 58=6728{\text{m}}^2$

Step 3: Calculate the area of one tile
Each rectangular tile measures 4m × 3m.
$\text{Area of one tile}=4\times 3=12{\text{m}}^2$

Step 4: Calculate the number of tiles needed
Dividing the total floor area by the area of a single tile:
$\text{Number of tiles}=\frac{\text{Area of floor}}{\text{Area of one tile}}$
$\text{Number of tiles}=\frac{6728}{12}\approx 560.67$

Rounding up to ensure complete coverage, approximately 561 tiles are needed.