Consider that we run 0.5 mile every 3 minutes and our friend runs 2 miles every 14 minutes on treadmills. We want to find the least possible whole numbers of miles we can run if we both start and stop running at the same time. To do so, we will determine how much time we need to complete 1 mile by multiplying our speed by 2.
0.5 mi/3 min
0.5* 2 mi/3* 2 min
1 mi/6 min
We can run 1 mile in 6 minutes. Now we will find how much time our friend needs to complete 1 mile. This time we will divide their speed by 2.
2 mi/14 min
2 mi/ 2/14 min/ 2
1 mi/7 min
Our friends travels 1 mile in 7 minutes. Let's make a list that represents how long it takes us to run each consecutive mile.
Our time &= 0,6,12,18,24,30,36,42,...
Friend's time &= 0,7,14,21,28,35,42,...
To find the least amount of time we can start and stop running at the same time, let's calculate the (LCM) of 6 and 7. Recall that the LCM of two numbers is the smallest whole number that is a of both numbers. To find the LCM, we can follow two steps.
- Express each number as a product of its factors.
- Multiply the highest power of each prime .
Let's start by finding the prime factors with the of each number.
The red are the prime factors of the given number. Remember that 1 is neither prime nor . We can use these factors to write the of each number.
6 &= 2* 3
7 &= 7
Now we will identify the highest power of each prime factor.
6 &= 3* 2
7 &= 7
Let's multiply these powers.
This means that to start and stop running at the same time while running a whole number of miles, we need to run at least 42 minutes. Now let's calculate how many miles we can run in 42 minutes. We run 1 mile every 6 minutes during the 42 times minutes.
1 mile * 42 minutes/6 minutes
Let's simplify this expression to find the number of miles we can run in 42 minutes. For simplicity we will ignore the units for now.
We can run 7 miles in 42 minutes. Now let's see how far our friend can run in 42 minutes if every mile takes them 7 minutes.
1 mile * 42 minutes/7 minutes
To simplify our calculations, we will ignore the units for now.
Our friend can run 6 miles in 42 minutes.
