for questions #1-3, use the agrarian barter system as shown below. show your thinking and be sure to include units throughout your process.
the agrarian barter system
4 pigs = 9 ducks
1 goat = 3 ducks
2 pigs = 5 chickens
1. how many pigs would you get for 50 chickens? show, or explain, how you arrived at your answer.
2. how many chickens would you get for 6 goats? show, or explain, how you arrived at your answer.
3. using the exchange rates below, what could you get in exchange for 36 chickens? (find as many options as you can)
4 pigs = 9 ducks 9 goats = 2 horses
1 goat = 3 ducks 5 ducks = 13 geese
2 pigs = 5 chickens 1 cow = 12 geese
20 rabbits = 1 pig 5 horses = 3 bulls
3 goats = 2 sheep 1 bull = 32 pigeons

Question

for questions #1-3, use the agrarian barter system as shown below. show your thinking and be sure to include units throughout your process.
the agrarian barter system
4 pigs = 9 ducks
1 goat = 3 ducks
2 pigs = 5 chickens
1. how many pigs would you get for 50 chickens? show, or explain, how you arrived at your answer.
2. how many chickens would you get for 6 goats? show, or explain, how you arrived at your answer.
3. using the exchange rates below, what could you get in exchange for 36 chickens? (find as many options as you can)
4 pigs = 9 ducks	9 goats = 2 horses
1 goat = 3 ducks	5 ducks = 13 geese
2 pigs = 5 chickens	1 cow = 12 geese
20 rabbits = 1 pig	5 horses = 3 bulls
3 goats = 2 sheep	1 bull = 32 pigeons

Accepted Answer

### Question 1
# Explanation:
## Step1: Relate pigs and chickens
We know that $2$ pigs $ = 5$ chickens. Let $x$ be the number of pigs for $50$ chickens. We can set up a proportion: $\frac{2\ \text{pigs}}{5\ \text{chickens}}=\frac{x\ \text{pigs}}{50\ \text{chickens}}$
## Step2: Solve the proportion
Cross - multiply: $5\times x=2\times50$
$5x = 100$
Divide both sides by $5$: $x=\frac{100}{5}=20$
# Answer:
$20$ pigs

### Question 2
# Explanation:
## Step1: Relate goats to ducks, then ducks to pigs, then pigs to chickens
First, from $1$ goat $ = 3$ ducks, for $6$ goats, the number of ducks is $6\times3 = 18$ ducks.
From $4$ pigs $ = 9$ ducks, we can find out how many pigs are equivalent to $18$ ducks. Let $y$ be the number of pigs for $18$ ducks. $\frac{4\ \text{pigs}}{9\ \text{ducks}}=\frac{y\ \text{pigs}}{18\ \text{ducks}}$, cross - multiply: $9y=4\times18$, $9y = 72$, $y = 8$ pigs.
From $2$ pigs $ = 5$ chickens, let $z$ be the number of chickens for $8$ pigs. $\frac{2\ \text{pigs}}{5\ \text{chickens}}=\frac{8\ \text{pigs}}{z\ \text{chickens}}$, cross - multiply: $2z=5\times8$, $2z = 40$, $z = 20$ chickens. (Alternative way: We can also first find the relation between goats and chickens. Since $2$ pigs $ = 5$ chickens and $4$ pigs $ = 9$ ducks, $2$ pigs $=\frac{9}{2}$ ducks. And $1$ goat $ = 3$ ducks, so $1$ goat $=\frac{2}{9}\times2$ pigs $=\frac{4}{9}$ pigs. Then from $2$ pigs $ = 5$ chickens, $1$ pig $=\frac{5}{2}$ chickens. So $1$ goat $=\frac{4}{9}\times\frac{5}{2}=\frac{10}{9}$ chickens. Then for $6$ goats, the number of chickens is $6\times\frac{10}{9}=\frac{20}{3}$? Wait, no, the first method is correct. Wait, there is a mistake in the alternative way. Let's correct the first method. Wait, actually, we can also use the relation between pigs and chickens directly. Wait, we know that $2$ pigs $ = 5$ chickens and $4$ pigs $ = 9$ ducks and $1$ goat $ = 3$ ducks. Let's do it step by step:

1. Convert goats to ducks: $6$ goats $\times3$ ducks/goat $ = 18$ ducks.
2. Convert ducks to pigs: Since $4$ pigs $ = 9$ ducks, the number of pigs for $18$ ducks: $\frac{4\ \text{pigs}}{9\ \text{ducks}}\times18\ \text{ducks}=8$ pigs.
3. Convert pigs to chickens: Since $2$ pigs $ = 5$ chickens, the number of chickens for $8$ pigs: $\frac{5\ \text{chickens}}{2\ \text{pigs}}\times8\ \text{pigs}=20$ chickens.
# Answer:
$20$ chickens

### Question 3
We know that $2$ pigs $ = 5$ chickens. Let's find the number of pigs for $36$ chickens first. $\frac{2\ \text{pigs}}{5\ \text{chickens}}=\frac{a\ \text{pigs}}{36\ \text{chickens}}$, $5a=2\times36$, $a=\frac{72}{5} = 14.4$ pigs (not a whole number, but we can use other conversions).

**Option 1: Chickens to pigs to rabbits**
From $2$ pigs $ = 5$ chickens, for $36$ chickens, number of pigs $p$: $\frac{2}{5}=\frac{p}{36}$, $p=\frac{72}{5}=14.4$ pigs. From $20$ rabbits $ = 1$ pig, number of rabbits $r = 14.4\times20 = 288$ rabbits.

**Option 2: Chickens to pigs to ducks**
From $2$ pigs $ = 5$ chickens, $p=\frac{2\times36}{5}=14.4$ pigs. From $4$ pigs $ = 9$ ducks, number of ducks $d$: $\frac{4}{9}=\frac{14.4}{d}$, $4d=9\times14.4$, $4d = 129.6$, $d = 32.4$ ducks.

**Option 3: Chickens to pigs to goats (via ducks)**
From $2$ pigs $ = 5$ chickens, $p = 14.4$ pigs. From $4$ pigs $ = 9$ ducks, $d=\frac{9\times14.4}{4}=32.4$ ducks. From $1$ goat $ = 3$ ducks, number of goats $g=\frac{32.4}{3}=10.8$ goats.

**Option 4: Chickens to pigs to sheep (via goats)**
From above, $g = 10.8$ goats. From $3$ goats $ = 2$ sheep, number of sheep $s$: $\frac{3}{2}=\frac{10.8}{s}$, $3s=2\times10.8$, $3s = 21.6$, $s = 7.2$ sheep.

**Option 5: Chickens to pigs to geese (via ducks)**
From $p = 14.4$ pigs, $d = 32.4$ ducks. From $5$ ducks $ = 13$ geese, number of geese $e$: $\frac{5}{13}=\frac{32.4}{e}$, $5e=13\times32.4$, $5e = 421.2$, $e = 84.24$ geese.

**Option 6: Chickens to pigs to cows (via geese)**
From $e = 84.24$ geese, from $1$ cow $ = 12$ geese, number of cows $c=\frac{84.24}{12}=7.02$ cows.

**Option 7: Chickens to pigs to horses (via goats)**
From $g = 10.8$ goats, from $9$ goats $ = 2$ horses, number of horses $h$: $\frac{9}{2}=\frac{10.8}{h}$, $9h=2\times10.8$, $9h = 21.6$, $h = 2.4$ horses.

**Option 8: Chickens to pigs to bulls (via horses)**
From $h = 2.4$ horses, from $5$ horses $ = 3$ bulls, number of bulls $b$: $\frac{5}{3}=\frac{2.4}{b}$, $5b=3\times2.4$, $5b = 7.2$, $b = 1.44$ bulls.

**Option 9: Chickens to pigs to pigeons (via bulls)**
From $b = 1.44$ bulls, from $1$ bull $ = 32$ pigeons, number of pigeons $q=1.44\times32 = 46.08$ pigeons.

We can also start with other conversion chains. For example, using the relation between goats and ducks and then to other animals. But the most straightforward is to use the chicken - pig relation and then branch out.

Some whole - number based options:

Since $2$ pigs $ = 5$ chickens, $10$ chickens $ = 4$ pigs, $20$ chickens $ = 8$ pigs, $30$ chickens $ = 12$ pigs. For $36$ chickens, we can write $36=30 + 6$. $30$ chickens $ = 12$ pigs and $6$ chickens:

From $2$ pigs $ = 5$ chickens, $6$ chickens $=\frac{2\times6}{5}=2.4$ pigs. So total pigs for $36$ chickens $=12 + 2.4=14.4$ pigs (as before).

If we consider the relation between chickens and ducks via pigs:

From $2$ pigs $ = 5$ chickens and $4$ pigs $ = 9$ ducks, we can say that $5$ chickens $=\frac{9}{2}$ ducks, so $1$ chicken $=\frac{9}{10}$ ducks. Then $36$ chickens $=36\times\frac{9}{10}=32.4$ ducks.

From $1$ goat $ = 3$ ducks, $32.4$ ducks $=\frac{32.4}{3}=10.8$ goats.

From $3$ goats $ = 2$ sheep, $10.8$ goats $=\frac{2\times10.8}{3}=7.2$ sheep.

And so on.

The possible exchange options (non - exhaustive) are:
- $14.4$ pigs
- $288$ rabbits (from $14.4$ pigs and $20$ rabbits per pig)
- $32.4$ ducks (from $14.4$ pigs and $4$ pigs $ = 9$ ducks)
- $10.8$ goats (from $32.4$ ducks and $1$ goat $ = 3$ ducks)
- $7.2$ sheep (from $10.8$ goats and $3$ goats $ = 2$ sheep)
- $84.24$ geese (from $32.4$ ducks and $5$ ducks $ = 13$ geese)
- $7.02$ cows (from $84.24$ geese and $1$ cow $ = 12$ geese)
- $2.4$ horses (from $10.8$ goats and $9$ goats $ = 2$ horses)
- $1.44$ bulls (from $2.4$ horses and $5$ horses $ = 3$ bulls)
- $46.08$ pigeons (from $1.44$ bulls and $1$ bull $ = 32$ pigeons)

If we consider combinations of whole - number animals by using the basic exchange rates:

For example, using the rate $2$ pigs $ = 5$ chickens:

- If we take $10$ chickens, we can get $4$ pigs (since $5$ chickens $ = 2$ pigs, $10$ chickens $ = 4$ pigs).
- If we take $20$ chickens, we can get $8$ pigs.
- If we take $30$ chickens, we can get $12$ pigs.

For the remaining $6$ chickens:

From $2$ pigs $ = 5$ chickens, $6$ chickens $=\frac{2\times6}{5}=2.4$ pigs.

We can also use the relation between chickens and goats:

From $1$ goat $ = 3$ ducks, $4$ pigs $ = 9$ ducks, $2$ pigs $ = 5$ chickens.

We can write a chain:

Chickens $\to$ Pigs $\to$ Ducks $\to$ Goats $\to$ Sheep $\to$ Horses $\to$ Bulls $\to$ Pigeons or Chickens $\to$ Pigs $\to$ Rabbits or Chickens $\to$ Pigs $\to$ Ducks $\to$ Geese $\to$ Cows etc.

A more practical way is to use the given exchange rates to find equivalent values:

1. **Pigs**:
Using $\frac{2\ \text{pigs}}{5\ \text{chickens}}=\frac{x\ \text{pigs}}{36\ \text{chickens}}$, we get $x=\frac{2\times36}{5}=14.4$ pigs.

2. **Rabbits**:
Since $20$ rabbits $ = 1$ pig, for $14.4$ pigs, the number of rabbits $r = 14.4\times20 = 288$ rabbits.

3. **Ducks**:
From $4$ pigs $ = 9$ ducks, for $14.4$ pigs, $\frac{4\ \text{pigs}}{9\ \text{ducks}}=\frac{14.4\ \text{pigs}}{y\ \text{ducks}}$, $4y=9\times14.4$, $y = 32.4$ ducks.

4. **Goats**:
From $1$ goat $ = 3$ ducks, for $32.4$ ducks, $g=\frac{32.4}{3}=10.8$ goats.

5. **Sheep**:
From $3$ goats $ = 2$ sheep, for $10.8$ goats, $s=\frac{2\times10.8}{3}=7.2$ sheep.

6. **Horses**:
From $9$ goats $ = 2$ horses, for $10.8$ goats, $\frac{9\ \text{goats}}{2\ \text{horses}}=\frac{10.8\ \text{goats}}{h\ \text{horses}}$, $9h = 2\times10.8$, $h = 2.4$ horses.

7. **Bulls**:
From $5$ horses $ = 3$ bulls, for $2.4$ horses, $\frac{5\ \text{horses}}{3\ \text{bulls}}=\frac{2.4\ \text{horses}}{b\ \text{bulls}}$, $5b=3\times2.4$, $b = 1.44$ bulls.

8. **Pigeons**:
From $1$ bull $ = 32$ pigeons, for $1.44$ bulls, $p = 1.44\times32=46.08$ pigeons.

9. **Geese**:
From $5$ ducks $ = 13$ geese, for $32.4$ ducks, $\frac{5\ \text{ducks}}{13\ \text{geese}}=\frac{32.4\ \text{ducks}}{e\ \text{geese}}$, $5e=13\times32.4$, $e = 84.24$ geese.

10. **Cows**:
From $1$ cow $ = 12$ geese, for $84.24$ geese, $c=\frac{84.24}{12}=7.02$ cows.

So, in exchange for $36$ chickens, you could get (among other things):
- $14.4$ pigs
- $288$ rabbits
- $32.4$ ducks
- $10.8$ goats
- $7.2$ sheep
- $2.4$ horses
- $1.44$ bulls
- $46.08$ pigeons
- $84.24$ geese
- $7.02$ cows

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Relate pigs and chickens

Step2: Solve the proportion

Step1: Relate goats to ducks, then ducks to pigs, then pigs to chickens

Answer:

Question 2