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February 1, 2025 16:35
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r1-aquarium
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<USER> | |
### 2. Aquarium | |
In the aquarium, there are octopuses with eight arms and starfish with five arms. | |
**How many starfish are there in the aquarium if the total number of arms of all the animals living in it is 41?** | |
<ASSISTANT> | |
To determine the number of starfish in the aquarium, let's break down the problem step by step. | |
### Step 1: Define Variables | |
- Let \( s \) represent the number of starfish. | |
- Let \( o \) represent the number of octopuses. | |
### Step 2: Understand the Arm Counts | |
- Each starfish has **5 arms**. | |
- Each octopus has **8 arms**. | |
### Step 3: Set Up the Equation | |
The total number of arms from all animals is **41**. Therefore: | |
\[ | |
5s + 8o = 41 | |
\] | |
### Step 4: Solve for Possible Integer Solutions | |
We need to find non-negative integers \( s \) and \( o \) that satisfy the equation. | |
Let's solve for \( s \): | |
\[ | |
5s = 41 - 8o \\ | |
s = \frac{41 - 8o}{5} | |
\] | |
For \( s \) to be an integer, \( 41 - 8o \) must be divisible by **5**. | |
### Step 5: Find Valid Values of \( o \) | |
Let's test possible values of \( o \): | |
1. **If \( o = 2 \):** | |
\[ | |
s = \frac{41 - 8(2)}{5} = \frac{41 - 16}{5} = \frac{25}{5} = 5 | |
\] | |
This gives \( s = 5 \), which is a valid integer. | |
2. **If \( o = 7 \):** | |
\[ | |
s = \frac{41 - 8(7)}{5} = \frac{41 - 56}{5} = \frac{-15}{5} = -3 | |
\] | |
Negative number of starfish is not possible. | |
3. **If \( o = 0 \):** | |
\[ | |
s = \frac{41 - 8(0)}{5} = \frac{41}{5} = 8.2 | |
\] | |
Not an integer. | |
4. **If \( o = 1 \):** | |
\[ | |
s = \frac{41 - 8(1)}{5} = \frac{33}{5} = 6.6 | |
\] | |
Not an integer. | |
5. **If \( o = 3 \):** | |
\[ | |
s = \frac{41 - 8(3)}{5} = \frac{41 - 24}{5} = \frac{17}{5} = 3.4 | |
\] | |
Not an integer. | |
6. **If \( o = 4 \):** | |
\[ | |
s = \frac{41 - 8(4)}{5} = \frac{41 - 32}{5} = \frac{9}{5} = 1.8 | |
\] | |
Not an integer. | |
7. **If \( o = 5 \):** | |
\[ | |
s = \frac{41 - 8(5)}{5} = \frac{41 - 40}{5} = \frac{1}{5} = 0.2 | |
\] | |
Not an integer. | |
8. **If \( o = 6 \):** | |
\[ | |
s = \frac{41 - 8(6)}{5} = \frac{41 - 48}{5} = \frac{-7}{5} = -1.4 | |
\] | |
Negative number of starfish is not possible. | |
### Step 6: Conclusion | |
The only valid solution is when \( o = 2 \) and \( s = 5 \). | |
\[ | |
\boxed{5} | |
\] |
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