To help the geologist determine how many rocks are in each crate using a balanced scale, we can set up a simple equation to represent the scenario.
Let \( r \) be the number of rocks in each crate and \( c \) be the number of crates.
If the crates weigh \( c \) pounds (since each crate weighs 1 pound) and there are \( r \) rocks in each crate, then the total weight of rocks in the crates is \( c \times r \) pounds.
When the scale is balanced, we can set the total weight of the rocks equal to the weight of the crates plus the number of rocks not in the crates. The equation can be represented as:
\[ \text{Weight of Crates} + \text{Weight of Rocks} = \text{Total Weight on Scale} \]
In terms of our variables, if \( T \) is the total weight read by the scale:
\[ c + c \cdot r = T \]
Now we can solve for \( r \):
\[ c \cdot (r + 1) = T \] \[ r + 1 = \frac{T}{c} \] \[ r = \frac{T}{c} - 1 \]
To determine the number of rocks per crate, the geologist can use this reasoning based on the balanced scales and plug in the specific values for \( T \) and \( c \).
Example Scenarios
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Scenario 1:
- Total Weight \( T = 10 \) pounds
- Number of Crates \( c = 2 \)
Using the equation: \[ r = \frac{10}{2} - 1 = 5 - 1 = 4 \] So, there are 4 rocks in each crate.
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Scenario 2:
- Total Weight \( T = 12 \) pounds
- Number of Crates \( c = 3 \)
Using the equation: \[ r = \frac{12}{3} - 1 = 4 - 1 = 3 \] So, there are 3 rocks in each crate.
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Scenario 3:
- Total Weight \( T = 16 \) pounds
- Number of Crates \( c = 4 \)
Using the equation: \[ r = \frac{16}{4} - 1 = 4 - 1 = 3 \] So, there are 3 rocks in each crate.
With this method, the geologist can easily calculate the number of rocks in each crate across different balance scenarios using mental math.
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