Question

A rectangular garden has dimensions of 100 feet by 10 feet. Next season, the garden is decreased to 25
of its size. What is the scale of the drawing for the new garden shown below?
A rectangle that is 8 inches long and four-fifths of an inch wide.

in. =
ft

Answer 1

To determine the scale of the drawing for the new garden, we first need to find the area of the original garden and then calculate the dimensions of the new garden after a reduction in size.

1. **Calculate the area of the original garden:**
\[
\text{Area} = \text{Length} \times \text{Width} = 100 \text{ ft} \times 10 \text{ ft} = 1000 \text{ sq ft}
\]

2. **Calculate the size of the garden after it is decreased to \( \frac{1}{4} \) of its original size:**
\[
\text{New Area} = \frac{1}{4} \times 1000 \text{ sq ft} = 250 \text{ sq ft}
\]

3. **Determine the dimensions of the new garden:**
Since the shape remains rectangular, we can assume it keeps the same ratio of dimensions as the original. The original ratios of length to width are 10:1.

Let:
- new length = \( 10x \)
- new width = \( x \)

The area of the new garden can be expressed as:
\[
\text{New Area} = (10x)(x) = 10x^2 = 250 \text{ sq ft}
\]

Solving for \( x \):
\[
10x^2 = 250 \\
x^2 = 25 \\
x = 5 \text{ ft}
\]

So, the new dimensions are:
- New Length = \( 10x = 10 \times 5 = 50 \text{ ft} \)
- New Width = \( x = 5 \text{ ft} \)

4. **Scale of the drawing:**
The drawing shows a rectangle that is 8 inches long and \( \frac{4}{5} \) inch wide:
- Drawing Length = 8 inches
- Drawing Width = \( \frac{4}{5} \text{ inches} = 0.8 \text{ inches} \)

Now, we calculate the scale. First, convert the dimensions of the new garden from feet to inches (1 foot = 12 inches):
\[
\text{New Length in inches} = 50 \text{ ft} \times 12 \, \text{inches/ft} = 600 \text{ inches} \\
\text{New Width in inches} = 5 \text{ ft} \times 12 \, \text{inches/ft} = 60 \text{ inches}
\]

5. **Calculate the scale:**
For the length:
- Drawing length = 8 inches
- Actual length = 600 inches
\[
\text{Scale} = \frac{\text{Drawing Length}}{\text{Actual Length}} = \frac{8 \text{ inches}}{600 \text{ inches}} = \frac{1}{75}
\]

For the width:
- Drawing width = 0.8 inches
- Actual width = 60 inches
\[
\text{Scale} = \frac{0.8 \text{ inches}}{60 \text{ inches}} = \frac{1}{75}
\]

Both calculations yield the same scale. Therefore, the scale of the drawing for the new garden is:
\[
\text{Scale} = \frac{1 \text{ inch}}{75 \text{ ft}}
\]

So to answer the question:
\[
\text{in} = \frac{1}{75} \text{ ft}
\]