Gavin goes for a run at a constant pace of 9 minutes per mile. Ten minutes later, Lars goes for a run, along the same route, at a constant pace of 7 minutes per mile. How many minutes does it take for Lars to reach Gavin?
Answers
Answer:
35 Minutes
Step-by-step explanation:
First, we want to convert both of these rates to miles per 1 minute. You can do this by taking the reciprocal of the values. This gets us 1/9 mile per a minute and 1/7 mile per a minute.
Second, we want to make the fractions have a common denominator. The least common multiple of 7 and 9 is 63. We have to multiply the numerator and denominator in 1/9 by 7 to get a demoninator of 63. That would get us 7/63. We would also have to multiply the numerator and denominator in 1/7 by 9 to get a denominator of 63. That would get us 9/63.
Third, we want to find the amount of miles Gavin ran in the first 10 minutes when Lars wasn't running. We can do this by multiplying his miles per a minute rate, 7/63 miles per a minute, by 10 since that is the time he is running in minutes. This would get us 70/63.
Forth, we would want to find the amount of miles Lars runs more than Gavin in each minute. We can do this by finding the absolute difference between Lar's rate of running and Gavin's rate of running. This would get us (9/63)-(7/63) which is equal to 2/63. This means that Lars runs 2/63 of a mile more than Gavin in each minute they spend running.
Fifth, we divide the amount of miles that Gavin ran in the first 10 minutes by the rate of miles Lars runs more than Gavin in each minute since we need to find the time it will take Lars to cover the distance that Gavin ran in the head start by using his the speed that he runs that Gavin doesn't. This would get us the expression (70/63)/(2/63) which is equal to 35. This means that Lars will catch up to Gavin in 35 minutes, which is our final answer.