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Ji-Hun and Kana are running a marathon. Due to the volume of people running, the starts occur in waves. Kana starts at 9:35 a.m. and runs at an average rate of 7 miles per hour (mph). Ji-Hun starts at 10:00 a.m. and runs at an average rate of 8mph. Assuming constant paces and no stops, to the nearest tenth of a mile, in how many miles will Ji-Hun catch up to Kana?

Ji \mathrm{Ji} -Hun and Kana are running a marathon. Due to the volume of people running, the starts occur in waves. Kana starts at 9:35 9: 35 a.m\mathrm{a} . \mathrm{m} . and runs at an average rate of 77 miles per hour (mph). Ji-Hun starts at 10:0010: 00 a.m\mathrm{a} . \mathrm{m} . and runs at an average rate of 8mph 8 \mathrm{mph} . Assuming constant paces and no stops, to the nearest tenth of a mile, in how many miles will Ji-Hun catch up to Kana?

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Q. Ji \mathrm{Ji} -Hun and Kana are running a marathon. Due to the volume of people running, the starts occur in waves. Kana starts at 9:35 9: 35 a.m\mathrm{a} . \mathrm{m} . and runs at an average rate of 77 miles per hour (mph). Ji-Hun starts at 10:0010: 00 a.m\mathrm{a} . \mathrm{m} . and runs at an average rate of 8mph 8 \mathrm{mph} . Assuming constant paces and no stops, to the nearest tenth of a mile, in how many miles will Ji-Hun catch up to Kana?
  1. Calculate Time Difference: Determine the time difference between Kana's and Ji-Hun's start times.\newlineKana starts at 9:359:35 a.m. and Ji-Hun starts at 10:0010:00 a.m., so the time difference is 2525 minutes.\newlineConvert the time difference to hours since the speeds are given in miles per hour.\newline2525 minutes is equal to 2560\frac{25}{60} hours.
  2. Calculate Kana's Distance: Calculate the distance Kana will have run by the time Ji-Hun starts.\newlineUsing the formula Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}, we calculate Kana's distance.\newlineKana's speed is 77 mph, and the time is 2560\frac{25}{60} hours.\newlineDistance=7 mph×(2560) hours.\text{Distance} = 7 \text{ mph} \times \left(\frac{25}{60}\right) \text{ hours}.
  3. Set Up Catch-Up Equation: Perform the calculation from Step 22 to find the distance.\newlineDistance = 7×(2560)7 \times \left(\frac{25}{60}\right)\newlineDistance = 7×0.41677 \times 0.4167 (rounded to four decimal places)\newlineDistance = 2.91672.9167 miles (rounded to four decimal places)\newlineKana will have run approximately 2.91672.9167 miles by the time Ji-Hun starts.
  4. Solve for Distance: Set up an equation to determine when Ji-Hun will catch up to Kana.\newlineLet xx be the number of miles Ji-Hun runs until he catches up to Kana.\newlineSince Kana has a head start of 2.91672.9167 miles, the distance Ji-Hun needs to run to catch up to Kana is xx miles, and the distance Kana runs during the same time is x−2.9167x - 2.9167 miles.\newlineJi-Hun's speed is 88 mph, and Kana's speed is 77 mph.\newlineThe time it takes for both runners to cover these distances will be the same, so we can set up the equation:\newlineTime for Ji-Hun = Time for Kana\newlinex8=x−2.91677\frac{x}{8} = \frac{x - 2.9167}{7}
  5. Round Final Answer: Solve the equation from Step 44 to find the value of xx.x8=x−2.91677\frac{x}{8} = \frac{x - 2.9167}{7}Multiply both sides by 5656 (the least common multiple of 88 and 77) to clear the fractions:7x=8(x−2.9167)7x = 8(x - 2.9167)7x=8x−23.33367x = 8x - 23.3336Now, subtract 8x8x from both sides to solve for xx:7x−8x=−23.33367x - 8x = -23.3336−x=−23.3336-x = -23.3336Multiply both sides by −1-1 to find the positive value of xx:x=23.3336x = 23.3336Round the value of xx to the nearest tenth of a mile.x≈23.3milesx \approx 23.3 \text{miles} \newlineJi-Hun will catch up to Kana after approximately 23.323.3 miles.

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