Let's analyze and solve each part of the problem step by step.
### 1. Establishing the Correct Equation
Given:
- Half of the herd is grazing.
- Three-fourths of the remaining are playing.
- The rest 9 are drinking water.
If [tex]\( x \)[/tex] represents the total number of deer in the herd:
- The number of grazing deer is [tex]\( \frac{x}{2} \)[/tex].
- The remaining deer after grazing is [tex]\( x - \frac{x}{2} = \frac{x}{2} \)[/tex].
- Three-fourths of these remaining deer are playing: [tex]\( \frac{3}{4} \times \frac{x}{2} = \frac{3x}{8} \)[/tex].
- The rest, which is 9 deer, are drinking water.
These parts add up to the total number of deer, [tex]\( x \)[/tex]:
[tex]\[ \frac{x}{2} + \frac{3x}{8} + 9 = x \][/tex]
Let's match this equation with the options:
(a) [tex]\(\frac{x}{2} + \frac{3x}{4} = x + 9\)[/tex] [tex]\( \neq\)[/tex]
(b) [tex]\(\frac{1}{2} + \frac{3x}{8} = x\)[/tex] \[tex]\( \neq\)[/tex]
(c) [tex]\(\frac{x}{2} + \frac{3x}{4} = x - 9\)[/tex] \[tex]\( =\)[/tex]
(d) [tex]\(\frac{x}{2} + \frac{3x}{4} = x - 9\)[/tex] \[tex]\( =\)[/tex]
Answer: (c) and (d) are the same and correct.
### 2. Finding the Total Number of Deer
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ \frac{x}{2} + \frac{3x}{8} + 9 = x \][/tex]
Clear the fractions by multiplying everything by 8:
[tex]\[ 4x + 3x + 72 = 8x \][/tex]
Combine like terms:
[tex]\[ 7x + 72 = 8x \][/tex]
Subtract [tex]\( 7x \)[/tex] from both sides:
[tex]\[ 72 = x \][/tex]
Answer: The total number of deer is 72. Therefore, the correct option is (b) 72.
### 3. Number of Attendants
We are given that there is one attendant for every four grazing deer.
- The number of grazing deer is [tex]\( \frac{72}{2} = 36 \)[/tex].
- With one attendant for every four grazing deer:
[tex]\[ \text{Attendants} = \frac{36}{4} = 9 \][/tex]
Answer: The number of attendants is 9. Therefore, the correct option is (b) 9.
### 4. Ratio of Grazing, Playing, and Drinking Deer
Now, we determine the number of deer in each category:
- Grazing: [tex]\( \frac{72}{2} = 36 \)[/tex]
- Playing: [tex]\( \frac{3}{4} \times \frac{72}{2} = \frac{3}{4} \times 36 = 27 \)[/tex]
- Drinking water: 9
The ratio of grazing to playing to drinking is:
[tex]\[ 36 : 27 : 9 \][/tex]
Simplifying by dividing each term by their greatest common divisor (GCD):
[tex]\[ GCD(36, 27, 9) = 9 \][/tex]
[tex]\[ \frac{36}{9} : \frac{27}{9} : \frac{9}{9} = 4 : 3 : 1 \][/tex]
Answer: Therefore, the ratio is (b) [tex]\( 4 : 3 : 1 \)[/tex].
