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Half of a herd of deer are grazing in the field and three-fourths of the remaining are playing nearby. The rest, 9, are drinking water from the pond.

1. If [tex]x[/tex] represents the total number of deer in the herd, which of the following equations holds true?
(a) [tex]\frac{x}{2}+\frac{3x}{4}=x+9[/tex]
(b) [tex]\frac{1}{2}+\frac{3x}{8}=x+9[/tex]
(c) [tex]\frac{x}{2}+\frac{3x}{4}=x-9[/tex]
(d) [tex]\frac{x}{2}+\frac{3x}{8}=x-9[/tex]

2. The total number of deer in the herd is:
(a) 63
(b) 72
(c) 81
(d) 90

3. If there is one attendant for every four grazing deer, how many attendants are there?
(a) 8
(b) 9
(c) 16
(d) 18

4. The ratio between the number of deer grazing, those playing, and those drinking water from the pond is:
(a) [tex]3:2:1[/tex]
(b) [tex]4:3:1[/tex]
(c) [tex]4:3:2[/tex]
(d) [tex]9:3:1[/tex]



Answer :

GinnyAnswer
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.