Question: log of practice session
ran $5 \frac{1}{4}$ laps
walked $1 \frac{3}{4}$ laps
ran $4 \frac{2}{3}$ laps
walked $1 \frac{1}{3}$ laps
ran $3 \frac{4}{5}$ laps
13. John trains by running on the school track. When he gets tired, he
walks until he is able to run again. His log for one training day is shown
to the left.
a) Determine how many laps around the track John ran.
b) Determine how many laps he walked around the track.
c) Determine how many more laps John ran than he walked.
d) If one lap around the school track is 400 m, determine the total
distance John travelled.
1. **State the problem:**
John's training log shows laps he ran and walked. We need to find:
a) Total laps ran
b) Total laps walked
c) Difference between laps ran and walked
d) Total distance traveled if one lap = 400 m
2. **Convert mixed numbers to improper fractions for easier addition:**
- Ran laps:
$5 \frac{1}{4} = \frac{21}{4}$
$4 \frac{2}{3} = \frac{14}{3}$
$3 \frac{4}{5} = \frac{19}{5}$
- Walked laps:
$1 \frac{3}{4} = \frac{7}{4}$
$1 \frac{1}{3} = \frac{4}{3}$
3. **Find common denominators and add ran laps:**
- Common denominator for 4, 3, 5 is 60.
- Convert each:
$\frac{21}{4} = \frac{21 \times 15}{60} = \frac{315}{60}$
$\frac{14}{3} = \frac{14 \times 20}{60} = \frac{280}{60}$
$\frac{19}{5} = \frac{19 \times 12}{60} = \frac{228}{60}$
- Sum ran laps:
$$\frac{315}{60} + \frac{280}{60} + \frac{228}{60} = \frac{315 + 280 + 228}{60} = \frac{823}{60}$$
4. **Add walked laps:**
- Common denominator for 4 and 3 is 12.
- Convert each:
$\frac{7}{4} = \frac{7 \times 3}{12} = \frac{21}{12}$
$\frac{4}{3} = \frac{4 \times 4}{12} = \frac{16}{12}$
- Sum walked laps:
$$\frac{21}{12} + \frac{16}{12} = \frac{37}{12}$$
5. **Calculate how many more laps John ran than walked:**
- Find common denominator for $\frac{823}{60}$ and $\frac{37}{12}$ which is 60.
- Convert walked laps:
$$\frac{37}{12} = \frac{37 \times 5}{60} = \frac{185}{60}$$
- Subtract:
$$\frac{823}{60} - \frac{185}{60} = \frac{823 - 185}{60} = \frac{638}{60}$$
- Simplify fraction by dividing numerator and denominator by 2:
$$\frac{\cancel{638}^{319}}{\cancel{60}^{30}} = \frac{319}{30}$$
- Convert to mixed number:
$$319 \div 30 = 10 \text{ remainder } 19 \Rightarrow 10 \frac{19}{30}$$
6. **Calculate total distance traveled:**
- Total laps = ran + walked = $\frac{823}{60} + \frac{37}{12}$
- Convert $\frac{37}{12}$ to denominator 60:
$$\frac{37}{12} = \frac{185}{60}$$
- Sum:
$$\frac{823}{60} + \frac{185}{60} = \frac{1008}{60}$$
- Simplify by dividing numerator and denominator by 12:
$$\frac{\cancel{1008}^{84}}{\cancel{60}^{5}} = \frac{84}{5}$$
- Convert to mixed number:
$$84 \div 5 = 16 \text{ remainder } 4 \Rightarrow 16 \frac{4}{5}$$
- Distance in meters:
$$16 \frac{4}{5} \text{ laps} \times 400 \text{ m/lap} = \frac{84}{5} \times 400 = \frac{84 \times 400}{5} = \frac{33600}{5} = 6720 \text{ m}$$
**Final answers:**
- a) John ran $\frac{823}{60} = 13 \frac{43}{60}$ laps.
- b) John walked $\frac{37}{12} = 3 \frac{1}{12}$ laps.
- c) John ran $10 \frac{19}{30}$ laps more than he walked.
- d) Total distance traveled is 6720 meters.