A farmer wants to make square shaped vegetable beds. The required squared area of vegetable beds will be prepared from a land which is 28 feet in length and 24 feet in width. a) Find the minimum number of squared vegetable beds that can be prepared from the land without wasting any area. b) Briefly explain the technique you used to solve (a).

To solve this problem, we need to determine the minimum number of square-shaped vegetable beds that can be prepared from the land without wasting any area. Let's break it down step by step.

### Step 1: Determine the area of the land
The area of the land is given by multiplying its length and width:
\[
\text{Area of land} = \text{Length} \times \text{Width} = 28 \times 24 = 672 \, \text{square feet}
\]

### Step 2: Find the largest square size that fits perfectly
The largest square-shaped bed that can fit without any remainder on the land is determined by the **greatest common divisor (GCD)** of the land's dimensions (28 feet and 24 feet). This ensures that the square's sides divide both dimensions evenly, leaving no waste.

The GCD of 28 and 24 is calculated as follows:
- Factors of 28: 1, 2, 4, 7, 14, 28
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors: 1, 2, 4

The greatest common factor is 4. Thus, the largest square that can be used is **4 feet by 4 feet**.

### Step 3: Calculate the number of square beds
The area of each square bed is:
\[
\text{Area of each square bed} = 4 \times 4 = 16 \, \text{square feet}
\]

The number of square beds that can fit into the total area is:
\[
\text{Number of beds} = \frac{\text{Area of land}}{\text{Area of each square bed}} = \frac{672}{16} = 42
\]

### Answer:
**a)** The minimum number of square vegetable beds that can be prepared from the land without wasting any area is **42**.

**b)** The technique used is based on finding the **greatest common divisor (GCD)** of the land’s dimensions to determine the largest possible square size that can completely cover the area without remainder. By dividing the total area by the area of this square, we determined the minimum number of square beds required.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How would the number of square beds change if the dimensions of the land were different?
2. What if the farmer wanted rectangular beds instead of square ones?
3. How would this problem change if the land had an irregular shape?
4. How can the GCD method be applied to other real-world problems?
5. What is the impact of increasing or decreasing the size of each square bed?

**Tip:** When dealing with problems involving area division, always consider the GCD to find the largest possible dimension that fits evenly into all given measurements.

Learn how to calculate the minimum number of square vegetable beds that can be prepared from a land with dimensions 28 feet by 24 feet. The solution involves using the greatest common divisor (GCD) to find the largest square size that fits without waste. Suitable for Grades 7-9.

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