Question
Sara 4/5 laps, Jamie 5/8 laps, Keenan 5/9 laps, totaling 20 laps; how many laps did each run?
Answers
GPT-4o mini
To determine how many laps each person ran—Sara, Jamie, and Keenan—we can set up equations based on the fractions provided and the total number of laps they completed together.
Let's denote the total number of laps run by Sara, Jamie, and Keenan as \( x \). According to the problem:
- Sara ran \( \frac{4}{5} x \) laps
- Jamie ran \( \frac{5}{8} x \) laps
- Keenan ran \( \frac{5}{9} x \) laps
The total number of laps they ran is given as follows:
\[
\frac{4}{5} x + \frac{5}{8} x + \frac{5}{9} x = 20
\]
To solve this equation, we first need a common denominator for the fractions involved. The denominators are 5, 8, and 9. The least common multiple (LCM) of 5, 8, and 9 is 360.
Now we will express each fraction with a denominator of 360:
- For Sara:
\[
\frac{4}{5} x = \frac{4 \times 72}{5 \times 72} x = \frac{288}{360} x
\]
- For Jamie:
\[
\frac{5}{8} x = \frac{5 \times 45}{8 \times 45} x = \frac{225}{360} x
\]
- For Keenan:
\[
\frac{5}{9} x = \frac{5 \times 40}{9 \times 40} x = \frac{200}{360} x
\]
Now substituting these into the equation:
\[
\frac{288}{360} x + \frac{225}{360} x + \frac{200}{360} x = 20
\]
Combining the terms on the left-hand side:
\[
\frac{288 + 225 + 200}{360} x = 20
\]
Calculating the sum in the numerator:
\[
288 + 225 + 200 = 713
\]
So we have:
\[
\frac{713}{360} x = 20
\]
To isolate \( x \), we multiply both sides by the reciprocal of the fraction:
\[
x = 20 \times \frac{360}{713}
\]
Calculating this, we find:
\[
x = \frac{7200}{713}
\]
Now we will calculate each individual's laps:
1. **Sara's laps:**
\[
\frac{4}{5} x = \frac{4}{5} \times \frac{7200}{713} = \frac{28800}{3565} \text{ laps}
\]
2. **Jamie's laps:**
\[
\frac{5}{8} x = \frac{5}{8} \times \frac{7200}{713} = \frac{45000}{5704} \text{ laps}
\]
3. **Keenan's laps:**
\[
\frac{5}{9} x = \frac{5}{9} \times \frac{7200}{713} = \frac{40000}{6417} \text{ laps}
\]
Finally, the number of laps each person ran can be calculated with approximation or simplification, however, they total to 20 laps. These values are the respresentation of total laps each ran, based on the fractions given.
Let's denote the total number of laps run by Sara, Jamie, and Keenan as \( x \). According to the problem:
- Sara ran \( \frac{4}{5} x \) laps
- Jamie ran \( \frac{5}{8} x \) laps
- Keenan ran \( \frac{5}{9} x \) laps
The total number of laps they ran is given as follows:
\[
\frac{4}{5} x + \frac{5}{8} x + \frac{5}{9} x = 20
\]
To solve this equation, we first need a common denominator for the fractions involved. The denominators are 5, 8, and 9. The least common multiple (LCM) of 5, 8, and 9 is 360.
Now we will express each fraction with a denominator of 360:
- For Sara:
\[
\frac{4}{5} x = \frac{4 \times 72}{5 \times 72} x = \frac{288}{360} x
\]
- For Jamie:
\[
\frac{5}{8} x = \frac{5 \times 45}{8 \times 45} x = \frac{225}{360} x
\]
- For Keenan:
\[
\frac{5}{9} x = \frac{5 \times 40}{9 \times 40} x = \frac{200}{360} x
\]
Now substituting these into the equation:
\[
\frac{288}{360} x + \frac{225}{360} x + \frac{200}{360} x = 20
\]
Combining the terms on the left-hand side:
\[
\frac{288 + 225 + 200}{360} x = 20
\]
Calculating the sum in the numerator:
\[
288 + 225 + 200 = 713
\]
So we have:
\[
\frac{713}{360} x = 20
\]
To isolate \( x \), we multiply both sides by the reciprocal of the fraction:
\[
x = 20 \times \frac{360}{713}
\]
Calculating this, we find:
\[
x = \frac{7200}{713}
\]
Now we will calculate each individual's laps:
1. **Sara's laps:**
\[
\frac{4}{5} x = \frac{4}{5} \times \frac{7200}{713} = \frac{28800}{3565} \text{ laps}
\]
2. **Jamie's laps:**
\[
\frac{5}{8} x = \frac{5}{8} \times \frac{7200}{713} = \frac{45000}{5704} \text{ laps}
\]
3. **Keenan's laps:**
\[
\frac{5}{9} x = \frac{5}{9} \times \frac{7200}{713} = \frac{40000}{6417} \text{ laps}
\]
Finally, the number of laps each person ran can be calculated with approximation or simplification, however, they total to 20 laps. These values are the respresentation of total laps each ran, based on the fractions given.