Let's tackle this problem step-by-step.
### Step 1: Calculate the Volume of a Pack of Gum
To calculate the volume of a single pack of gum, we multiply the given dimensions:
- Length of gum pack: [tex]\(\frac{1}{5} \text{ ft}\)[/tex]
- Width of gum pack: [tex]\(\frac{1}{10} \text{ ft}\)[/tex]
- Thickness of gum pack: [tex]\(\frac{1}{50} \text{ ft}\)[/tex]
The volume [tex]\(V_{\text{gum}}\)[/tex] of a single gum pack is:
[tex]\[ V_{\text{gum}} = \left(\frac{1}{5}\right) \text{ ft} \times \left(\frac{1}{10}\right) \text{ ft} \times \left(\frac{1}{50}\right) \text{ ft} = \frac{1}{2500} \text{ ft}^3 = 0.0004 \text{ ft}^3 \][/tex]
### Step 2: Calculate the Volume of the Gymnasium
Next, we need to calculate the volume of the gymnasium given its dimensions:
- Length of gymnasium: 100 ft
- Width of gymnasium: 80 ft
- Height of gymnasium: 50 ft
The volume [tex]\(V_{\text{gym}}\)[/tex] of the gymnasium is:
[tex]\[ V_{\text{gym}} = 100 \text{ ft} \times 80 \text{ ft} \times 50 \text{ ft} = 400000 \text{ ft}^3 \][/tex]
### Step 3: Calculate How Many Packs of Gum Fit Inside the Gymnasium
To determine how many packs of gum can fit inside the gymnasium, we need to find the ratio of the volume of the gymnasium to the volume of a single pack of gum:
[tex]\[ \text{Number of packs} = \frac{V_{\text{gym}}}{V_{\text{gum}}} = \frac{400000 \text{ ft}^3}{0.0004 \text{ ft}^3} \][/tex]
Calculating the above ratio gives us:
[tex]\[ \frac{400000}{0.0004} = 1000000000 = 1 \times 10^9 \][/tex]
### Step 4: Analyzing the Given Options
Now we compare our result to the given options:
1. [tex]\(\frac{4 \times 10^5}{4 \times 10^{-4}} = 1 \times 10^9\)[/tex]
2. [tex]\(\frac{4 \times 10^5}{4 \times 10^{-4}} = 1 \times 10^1\)[/tex]
3. [tex]\(\frac{4 \times 10^6}{4 \times 10^{-4}} = 1 \times 10^{10}\)[/tex]
4. [tex]\(\frac{4 \times 10^6}{4 \times 10^{-4}} = 1 \times 10^2\)[/tex]
By inspecting option 1:
[tex]\[ \frac{4 \times 10^5}{4 \times 10^{-4}} \][/tex]
We can calculate as follows:
[tex]\[ \frac{4 \times 10^5}{4 \times 10^{-4}} = 4 \times 10^5 \div 4 \times 10^{-4} = 10^5 \times 10^4 = 10^{9} = 1 \times 10^9 \][/tex]
This matches our result exactly.
So, the correct equation that she could have written is:
[tex]\[ \boxed{\frac{4 \times 10^5}{4 \times 10^{-4}} = 1 \times 10^9} \][/tex]
