Let's go through each part of the question systematically.
### Part 1: Finding the Correct Equation
We need to find the correct equation that represents the given situation. Recall:
- Half of the herd is grazing in the field.
- Three-fourths of the remaining deer are playing nearby.
- The rest, which is 9 deer, are drinking water from the pond.
Let [tex]\( x \)[/tex] be the total number of deer. Therefore:
- Half of the deer are grazing: [tex]\(\frac{x}{2}\)[/tex].
- Remaining deer: [tex]\(x - \frac{x}{2} = \frac{x}{2}\)[/tex].
- Three-fourths of the remaining are playing: [tex]\(\frac{3}{4} \cdot \frac{x}{2} = \frac{3x}{8}\)[/tex].
- Rest are drinking water: 9 deer.
The equation should add up the grazing deer, playing deer, and the drinking deer to equal the total number of deer:
[tex]\[
\frac{x}{2} + \frac{3x}{8} + 9 = x.
\][/tex]
Rewriting with a common denominator:
[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]\[
9 = \frac{x}{8},
\][/tex]
[tex]\[
x = 72.
\][/tex]
So, the correct equation is:
[tex]\[
\frac{x}{2} + \frac{3x}{4} = x - 9.
\][/tex]
Thus, the correct answer is:
(c) [tex]\(\frac{x}{2} + \frac{3x}{4} = x - 9\)[/tex].
### Part 2: Total Number of Deer in the Herd
From our earlier calculation:
[tex]\[
x = 72.
\][/tex]
So, the total number of deer in the herd is:
(b) 72.
### Part 3: Number of Attendants
Recall there is one attendant for every four grazing deer.
Number of grazing deer:
[tex]\[
\frac{x}{2} = \frac{72}{2} = 36.
\][/tex]
Number of attendants:
[tex]\[
\frac{36}{4} = 9.
\][/tex]
So, the number of attendants is:
(b) 9.
### Part 4: Ratio Between Grazing, Playing, and Drinking Deer
- Grazing deer: [tex]\(\frac{x}{2} = 36\)[/tex]
- Playing deer: [tex]\(\frac{3}{4} \cdot \frac{x}{2} = \frac{3}{4} \cdot 36 = 27\)[/tex]
- Drinking water deer: 9
Finding the ratio:
[tex]\[
36 : 27 : 9
\][/tex]
Simplifying the ratio:
[tex]\[
\frac{36}{9} : \frac{27}{9} : \frac{9}{9} = 4 : 3 : 1.
\][/tex]
So, the correct ratio is:
(b) 4 : 3 : 1.
