Question
Patrick walked 3/5 miles on the first day, 5/7 miles on the second day, third day he walked half of the first day’s distance and the last day he walked the first two days' distance. How much distance Patrick has walked altogether?
46/35 miles
57/35 miles
1025/350 miles
1050/350 miles
46/35 miles
57/35 miles
1025/350 miles
1050/350 miles
Answers
GPT-4o mini
To find out how much distance Patrick has walked altogether, we will calculate the distance he walked on each day and then sum them up.
1. **First day**: Patrick walked \( \frac{3}{5} \) miles.
2. **Second day**: Patrick walked \( \frac{5}{7} \) miles.
3. **Third day**: Patrick walked half of the first day's distance, which is:
\[
\text{Third day distance} = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10} \text{ miles}.
\]
4. **Fourth day**: Patrick walked the first two days' distance combined:
\[
\text{Fourth day distance} = \frac{3}{5} + \frac{5}{7}.
\]
To add these fractions, we need a common denominator. The least common multiple of 5 and 7 is 35:
- Convert \( \frac{3}{5} \) to a denominator of 35:
\[
\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}.
\]
- Convert \( \frac{5}{7} \) to a denominator of 35:
\[
\frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35}.
\]
- Now add the two fractions:
\[
\frac{3}{5} + \frac{5}{7} = \frac{21}{35} + \frac{25}{35} = \frac{46}{35} \text{ miles}.
\]
Now, we sum the distances from all four days:
\[
\text{Total distance} = \left( \frac{3}{5} + \frac{5}{7} + \frac{3}{10} + \frac{46}{35} \right).
\]
We’ll change all these fractions to a common denominator, which is 70 (the least common multiple of 5, 7, and 10):
1. Convert \( \frac{3}{5} \) to a denominator of 70:
\[
\frac{3}{5} = \frac{3 \times 14}{5 \times 14} = \frac{42}{70}.
\]
2. Convert \( \frac{5}{7} \) to a denominator of 70:
\[
\frac{5}{7} = \frac{5 \times 10}{7 \times 10} = \frac{50}{70}.
\]
3. Convert \( \frac{3}{10} \) to a denominator of 70:
\[
\frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70}.
\]
4. Convert \( \frac{46}{35} \) to a denominator of 70:
\[
\frac{46}{35} = \frac{46 \times 2}{35 \times 2} = \frac{92}{70}.
\]
Now sum these values:
\[
\text{Total distance} = \frac{42}{70} + \frac{50}{70} + \frac{21}{70} + \frac{92}{70} = \frac{42 + 50 + 21 + 92}{70} = \frac{205}{70}.
\]
To simplify \( \frac{205}{70} \), we find the greatest common divisor of 205 and 70, which is 5:
\[
\frac{205 \div 5}{70 \div 5} = \frac{41}{14}.
\]
Finally, we will keep the answer in terms of our calculations instead of expressing it further, but if we convert it back, to find exact customer options given might be there for options \(46/35, 57/35, 1025/350, 1050/350\):
The equivalent total distance can be cross checked to match options stated above.
The third option
- \(1025/350\) = 205/70 as equals to \( \frac{41}{14} \).
So the total distance Patrick has walked altogether is:
**57/35 miles**.
1. **First day**: Patrick walked \( \frac{3}{5} \) miles.
2. **Second day**: Patrick walked \( \frac{5}{7} \) miles.
3. **Third day**: Patrick walked half of the first day's distance, which is:
\[
\text{Third day distance} = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10} \text{ miles}.
\]
4. **Fourth day**: Patrick walked the first two days' distance combined:
\[
\text{Fourth day distance} = \frac{3}{5} + \frac{5}{7}.
\]
To add these fractions, we need a common denominator. The least common multiple of 5 and 7 is 35:
- Convert \( \frac{3}{5} \) to a denominator of 35:
\[
\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}.
\]
- Convert \( \frac{5}{7} \) to a denominator of 35:
\[
\frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35}.
\]
- Now add the two fractions:
\[
\frac{3}{5} + \frac{5}{7} = \frac{21}{35} + \frac{25}{35} = \frac{46}{35} \text{ miles}.
\]
Now, we sum the distances from all four days:
\[
\text{Total distance} = \left( \frac{3}{5} + \frac{5}{7} + \frac{3}{10} + \frac{46}{35} \right).
\]
We’ll change all these fractions to a common denominator, which is 70 (the least common multiple of 5, 7, and 10):
1. Convert \( \frac{3}{5} \) to a denominator of 70:
\[
\frac{3}{5} = \frac{3 \times 14}{5 \times 14} = \frac{42}{70}.
\]
2. Convert \( \frac{5}{7} \) to a denominator of 70:
\[
\frac{5}{7} = \frac{5 \times 10}{7 \times 10} = \frac{50}{70}.
\]
3. Convert \( \frac{3}{10} \) to a denominator of 70:
\[
\frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70}.
\]
4. Convert \( \frac{46}{35} \) to a denominator of 70:
\[
\frac{46}{35} = \frac{46 \times 2}{35 \times 2} = \frac{92}{70}.
\]
Now sum these values:
\[
\text{Total distance} = \frac{42}{70} + \frac{50}{70} + \frac{21}{70} + \frac{92}{70} = \frac{42 + 50 + 21 + 92}{70} = \frac{205}{70}.
\]
To simplify \( \frac{205}{70} \), we find the greatest common divisor of 205 and 70, which is 5:
\[
\frac{205 \div 5}{70 \div 5} = \frac{41}{14}.
\]
Finally, we will keep the answer in terms of our calculations instead of expressing it further, but if we convert it back, to find exact customer options given might be there for options \(46/35, 57/35, 1025/350, 1050/350\):
The equivalent total distance can be cross checked to match options stated above.
The third option
- \(1025/350\) = 205/70 as equals to \( \frac{41}{14} \).
So the total distance Patrick has walked altogether is:
**57/35 miles**.