Question
Saeed walked ¾ of a mile away from school in a straight line. Justin left the same point at school and ⅚ of a mile along the same straight line as Saeed. If Sally walks to a point exactly halfway between Saeed and Justin, how far from the school will she be?
Answers
GPT-4o mini
To find the distance that Sally will be from the school, we first need to determine the positions of Saeed and Justin relative to the school.
- Saeed walked \( \frac{3}{4} \) miles away from the school.
- Justin walked \( \frac{5}{6} \) miles away from the school.
Next, we will find the halfway point between Saeed and Justin. To do this, we can calculate the distances from the school that Saeed and Justin are at, then find the average of these two distances.
1. **Convert the fractions to have a common denominator.**
The least common multiple of 4 and 6 is 12. We can convert the fractions:
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\]
\[
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\]
2. **Find the halfway point.**
The halfway distance between Saeed and Justin can be calculated as follows:
\[
\text{Halfway point} = \frac{\frac{9}{12} + \frac{10}{12}}{2} = \frac{\frac{19}{12}}{2} = \frac{19}{12} \cdot \frac{1}{2} = \frac{19}{24}
\]
3. **Calculate how far Sally will be from the school.**
The distance from the school to the point halfway between Saeed and Justin is \( \frac{19}{24} \) miles.
Thus, Sally will be \( \frac{19}{24} \) miles from the school.
- Saeed walked \( \frac{3}{4} \) miles away from the school.
- Justin walked \( \frac{5}{6} \) miles away from the school.
Next, we will find the halfway point between Saeed and Justin. To do this, we can calculate the distances from the school that Saeed and Justin are at, then find the average of these two distances.
1. **Convert the fractions to have a common denominator.**
The least common multiple of 4 and 6 is 12. We can convert the fractions:
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\]
\[
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\]
2. **Find the halfway point.**
The halfway distance between Saeed and Justin can be calculated as follows:
\[
\text{Halfway point} = \frac{\frac{9}{12} + \frac{10}{12}}{2} = \frac{\frac{19}{12}}{2} = \frac{19}{12} \cdot \frac{1}{2} = \frac{19}{24}
\]
3. **Calculate how far Sally will be from the school.**
The distance from the school to the point halfway between Saeed and Justin is \( \frac{19}{24} \) miles.
Thus, Sally will be \( \frac{19}{24} \) miles from the school.