What is the minimum number of tiles of size 16 by 24 required to form a square by placing the tiles adjacent to one another other?

Options

A. 6
B. 8
C. 11
D. 16

Answer

To find the minimum number of tiles of size 16 by 24 required to form a square by placing the tiles adjacent to one another, we can follow these steps:

### Step 1: Understand the dimensions of the tiles
The dimensions of one tile are given as:
- Length = 24 units
- Breadth = 16 units

### Step 2: Set up the equation for a square
To form a square using these tiles, we need to ensure that the total length and width of the arrangement of tiles are equal. If we place \(X\) tiles horizontally and \(Y\) tiles vertically, the dimensions of the resulting rectangle will be:
- Length = \(24X\)
- Width = \(16Y\)

For the arrangement to form a square, we need:
\[ 24X = 16Y \]

### Step 3: Simplify the equation
We can simplify the equation:
\[ \frac{24X}{16Y} = 1 \]
This simplifies to:
\[ 3X = 2Y \]
From this, we can express \(Y\) in terms of \(X\):
\[ Y = \frac{3}{2}X \]

### Step 4: Determine integer values for \(X\) and \(Y\)
Since \(X\) and \(Y\) must be whole numbers (as they represent the number of tiles), we can find the minimum values for \(X\) and \(Y\) that satisfy the equation \(Y = \frac{3}{2}X\). The smallest integer value for \(X\) that keeps \(Y\) as an integer is \(2\):
- If \(X = 2\), then:
\[ Y = \frac{3}{2} \times 2 = 3 \]

### Step 5: Calculate the dimensions of the square
Now we can calculate the dimensions of the square:
- Length = \(24X = 24 \times 2 = 48\)
- Width = \(16Y = 16 \times 3 = 48\)

Both dimensions are equal, confirming that we can form a square of size \(48 \times 48\).

### Step 6: Calculate the area of the square and the area of one tile
Next, we calculate the area of the square and the area of one tile:
- Area of the square = \(48 \times 48 = 2304\)
- Area of one tile = \(16 \times 24 = 384\)

### Step 7: Determine the number of tiles required
To find the minimum number of tiles required, we divide the area of the square by the area of one tile:
\[ \text{Number of tiles} = \frac{\text{Area of square}}{\text{Area of one tile}} = \frac{2304}{384} = 6 \]

### Conclusion
Thus, the minimum number of tiles required to form a square is **6**.

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