Question

Johnny rode his bike StartFraction 4 over 7 EndFraction of a mile from his house to the lake on a straight path. Then, he turned around and rode his bike 3 and StartFraction 1 over 8 EndFraction miles in the opposite direction. About how far is Johnny from his house?
1 and one-half miles
2 miles
2 and one-half miles
3 miles

Answer 1

To find out how far Johnny is from his house, we need to calculate the total distance he rode in both directions.

1. **Distance from house to lake**:
\[
\text{Distance to lake} = \frac{4}{7} \text{ miles}
\]

2. **Distance from lake in the opposite direction**:
\[
\text{Distance away from lake} = 3 + \frac{1}{8} \text{ miles} = 3.125 \text{ miles}
\]

3. **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**.