Let's carefully go through the given problem step by step.
### Step 1: Understanding the Problem
We need to find:
1. The correct equation representing the total number of deer.
2. The total number of deer in the herd.
3. The number of attendants, given that there is one attendant for every four grazing deer.
4. The ratio of the number of deer grazing, the number of deer playing, and the number of deer drinking water.
### Initial Information
- Half of the herd is grazing in the field.
- Three-fourths of the remaining are playing nearby.
- The rest, 9 deer, are drinking water from the pond.
- Let [tex]\( x \)[/tex] represent the total number of deer.
### Step 1: Formulating the Correct Equation
1. Deer Grazing:
[tex]\[
\text{Deer Grazing} = \frac{x}{2}
\][/tex]
2. Deer Playing:
[tex]\[
\text{Remaining deer} = x - \frac{x}{2} = \frac{x}{2}
\][/tex]
Thus, the number of deer playing:
[tex]\[
\text{Deer Playing} = \frac{3}{4} \times \frac{x}{2} = \frac{3x}{8}
\][/tex]
3. Deer Drinking Water:
[tex]\[
\text{Deer Drinking Water} = 9
\][/tex]
By summing these three parts, we get the equation:
[tex]\[
\frac{x}{2} + \frac{3x}{8} + 9 = x
\][/tex]
So, the correct equation from the options given is:
[tex]\[
\boxed{\text{none of the given options correctly represent the equation.}}
\][/tex]
### Step 2: Solving the Equation
To find the total number of deer in the herd, solve the equation:
[tex]\[
\frac{x}{2} + \frac{3x}{8} + 9 = x
\][/tex]
Simplifying the left-hand side:
[tex]\[
\frac{4x}{8} + \frac{3x}{8} + 9 = x
\][/tex]
[tex]\[
\frac{7x}{8} + 9 = x
\][/tex]
[tex]\[
9 = x - \frac{7x}{8}
\][/tex]
[tex]\[
9 = \frac{8x - 7x}{8}
\][/tex]
[tex]\[
9 = \frac{x}{8}
\][/tex]
[tex]\[
x = 72
\][/tex]
Thus, the total number of deer in the herd is:
[tex]\[
\boxed{72}
\][/tex]
### Step 3: Finding the Number of Attendants
Since there is one attendant for every four grazing deer and we know that half of the herd is grazing:
[tex]\[
\text{Deer Grazing} = \frac{72}{2} = 36
\][/tex]
So,
[tex]\[
\text{Number of Attendants} = \frac{36}{4} = 9
\][/tex]
Hence, the number of attendants is:
[tex]\[
\boxed{9}
\][/tex]
### Step 4: Ratio of the Number of Deer
Now, we'll calculate the ratio of deer grazing, deer playing, and deer drinking water:
1. Deer Grazing:
[tex]\[
\frac{72}{2} = 36
\][/tex]
2. Deer Playing:
[tex]\[
\frac{3}{4} \times \frac{72}{2} = \frac{3}{4} \times 36 = 27
\][/tex]
3. Deer Drinking Water:
[tex]\[
9
\][/tex]
Thus, the ratio is:
[tex]\[
\boxed{4:3:1}
\][/tex]
