Find the total number of squares in a rectangle having 8 rows and 15 columns.

Options

A. 345
B. 456
C. 567
D. 606

Answer

To find the total number of squares in a rectangle with 8 rows and 15 columns, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Dimensions**:
We have a rectangle with 8 rows and 15 columns.

2. **Understanding Square Sizes**:
A square can be of different sizes. The possible sizes of squares that can fit in the rectangle range from 1x1 up to the largest square that can fit, which is 8x8 (since there are only 8 rows).

3. **Calculate Squares of Each Size**:
- For a **1x1 square**:
The number of 1x1 squares = (Number of rows) × (Number of columns) = 8 × 15 = 120.
- For a **2x2 square**:
The number of 2x2 squares = (Number of rows - 1) × (Number of columns - 1) = 7 × 14 = 98.
- For a **3x3 square**:
The number of 3x3 squares = (Number of rows - 2) × (Number of columns - 2) = 6 × 13 = 78.
- For a **4x4 square**:
The number of 4x4 squares = (Number of rows - 3) × (Number of columns - 3) = 5 × 12 = 60.
- For a **5x5 square**:
The number of 5x5 squares = (Number of rows - 4) × (Number of columns - 4) = 4 × 11 = 44.
- For a **6x6 square**:
The number of 6x6 squares = (Number of rows - 5) × (Number of columns - 5) = 3 × 10 = 30.
- For a **7x7 square**:
The number of 7x7 squares = (Number of rows - 6) × (Number of columns - 6) = 2 × 9 = 18.
- For an **8x8 square**:
The number of 8x8 squares = (Number of rows - 7) × (Number of columns - 7) = 1 × 8 = 8.

4. **Totaling All Squares**:
Now, we add all the squares calculated:
\[ \text{Total squares} = 120 + 98 + 78 + 60 + 44 + 30 + 18 + 8 \]
- Adding these values step-by-step:
- 120 + 98 = 218
- 218 + 78 = 296
- 296 + 60 = 356
- 356 + 44 = 400
- 400 + 30 = 430
- 430 + 18 = 448
- 448 + 8 = 456

5. **Final Answer**:
The total number of squares in the rectangle is **456**.