Let's use variables to represent the number of small dogs and large dogs that Ryan walked.
Let \( x \) be the number of small dogs Ryan walked, and \( y \) be the number of large dogs he walked.
We know two things based on the information provided:
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The total money earned from walking the dogs: \[ 5x + 8y = 55 \]
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The total number of dogs walked is not specified, but we can express \( y \) in terms of \( x \) with any additional information or constraints.
From the first equation, we can solve for \( y \): \[ 8y = 55 - 5x \] \[ y = \frac{55 - 5x}{8} \]
Now \( y \) must be a non-negative integer, which means \( 55 - 5x \) must be non-negative and divisible by 8. Therefore: \[ 55 - 5x \geq 0 \implies x \leq 11 \] Next, we check values of \( x \) to find valid integer solutions for \( y \):
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If \( x = 0 \): \[ y = \frac{55 - 5(0)}{8} = \frac{55}{8} \text{ (not an integer)} \]
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If \( x = 1 \): \[ y = \frac{55 - 5(1)}{8} = \frac{50}{8} \text{ (not an integer)} \]
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If \( x = 2 \): \[ y = \frac{55 - 5(2)}{8} = \frac{45}{8} \text{ (not an integer)} \]
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If \( x = 3 \): \[ y = \frac{55 - 5(3)}{8} = \frac{40}{8} = 5 \text{ (integer)} \]
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If \( x = 4 \): \[ y = \frac{55 - 5(4)}{8} = \frac{35}{8} \text{ (not an integer)} \]
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If \( x = 5 \): \[ y = \frac{55 - 5(5)}{8} = \frac{30}{8} \text{ (not an integer)} \]
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If \( x = 6 \): \[ y = \frac{55 - 5(6)}{8} = \frac{25}{8} \text{ (not an integer)} \]
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If \( x = 7 \): \[ y = \frac{55 - 5(7)}{8} = \frac{20}{8} = 2.5 \text{ (not an integer)} \]
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If \( x = 8 \): \[ y = \frac{55 - 5(8)}{8} = \frac{15}{8} \text{ (not an integer)} \]
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If \( x = 9 \): \[ y = \frac{55 - 5(9)}{8} = \frac{10}{8} \text{ (not an integer)} \]
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If \( x = 10 \): \[ y = \frac{55 - 5(10)}{8} = \frac{5}{8} \text{ (not an integer)} \]
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If \( x = 11 \): \[ y = \frac{55 - 5(11)}{8} = \frac{0}{8} = 0 \text{ (is an integer)} \]
The valid pairs \((x, y)\) are:
- \( (3, 5) \): Ryan walked 3 small dogs and 5 large dogs.
- \( (11, 0) \): Ryan walked 11 small dogs and no large dogs.
Therefore, possible numbers of small dogs he walked are \( 3 \) or \( 11 \). Based on the task requirements, we can choose one of these values; if only a single answer is needed, we can use:
\[ \text{Ryan walked } 3 \text{ small dogs.} \]
If more context is needed, feel free to provide additional constraints regarding the number of dogs, and I can further refine the answer!