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February 1, 2025 16:12
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o3-mini-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> | |
Let \( x \) be the number of octopuses and \( y \) be the number of starfish. Each octopus has 8 arms and each starfish has 5 arms, so the total number of arms is given by: | |
\[ | |
8x + 5y = 41 | |
\] | |
We need to find nonnegative integers \( x \) and \( y \) that satisfy this equation. | |
### Step 1: Solve for \( y \) | |
We can start by testing small values of \( x \): | |
1. **For \( x = 0 \):** | |
\[ | |
8(0) + 5y = 41 \implies 5y = 41 \implies y = \frac{41}{5} = 8.2 \quad \text{(not an integer)} | |
\] | |
2. **For \( x = 1 \):** | |
\[ | |
8(1) + 5y = 41 \implies 8 + 5y = 41 \implies 5y = 33 \implies y = \frac{33}{5} = 6.6 \quad \text{(not an integer)} | |
\] | |
3. **For \( x = 2 \):** | |
\[ | |
8(2) + 5y = 41 \implies 16 + 5y = 41 \implies 5y = 25 \implies y = 5 | |
\] | |
Here, \( y = 5 \) is an integer. | |
### Step 2: Verify the Solution | |
With \( x = 2 \) and \( y = 5 \): | |
\[ | |
8(2) + 5(5) = 16 + 25 = 41 | |
\] | |
The equation holds true. | |
### Conclusion | |
The number of starfish in the aquarium is: | |
\[ | |
\boxed{5} | |
\] |
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