To find out how far Johnny is from his house, we need to calculate the total distance he rode in both directions.
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Distance from house to lake: \[ \text{Distance to lake} = \frac{4}{7} \text{ miles} \]
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Distance from lake in the opposite direction: \[ \text{Distance away from lake} = 3 + \frac{1}{8} \text{ miles} = 3.125 \text{ miles} \]
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Total distance Johnny has traveled from his house can be calculated by subtracting the distance to the lake from the distance traveled away from the lake.
First, convert the distance from the lake to a common denominator with the distance to the lake:
Convert \(3.125\) miles to a fraction: \[ 3.125 = 3 + \frac{1}{8} = 3 \cdot \frac{8}{8} + \frac{1}{8} = \frac{24}{8} + \frac{1}{8} = \frac{25}{8} \text{ miles} \]
Now we calculate how far Johnny is from his house: \[ \text{Total distance from house} = \text{Distance from house to lake} + \text{Distance away from lake} = \frac{4}{7} + \frac{25}{8} \]
To add these fractions, we need a common denominator. The least common multiple of \(7\) and \(8\) is \(56\).
Convert \(\frac{4}{7}\) to a denominator of \(56\): \[ \frac{4}{7} = \frac{4 \times 8}{7 \times 8} = \frac{32}{56} \]
Convert \(\frac{25}{8}\) to a denominator of \(56\): \[ \frac{25}{8} = \frac{25 \times 7}{8 \times 7} = \frac{175}{56} \]
Now add the two fractions: \[ \frac{32}{56} + \frac{175}{56} = \frac{207}{56} \]
Now, we convert \(\frac{207}{56}\) to a mixed number: \[ 207 ÷ 56 = 3.6964 \text{ (approximately 3)} \quad \text{with a remainder of } 39. \] So: \[ \frac{207}{56} = 3 \frac{39}{56} \]
Since \(39/56\) is less than \(1\), Johnny is approximately 3 miles away from his house.
Therefore, the answer is: 3 miles.
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