A relay with n laps was run by 4 people. The first runner ran half the laps. The second runner ran half the amount of laps the first runner ran. The third runner ran half the amount of laps the second runner ran. The remaining laps were completed by the fourth runner. How many laps did the fourth runner complete?
a. n/16
b. n/8
c. n/4
d. n/2
e. 7n/8
please answer and explain
a. n/16
To find out how many laps the fourth runner completed, we need to understand the pattern of laps run by each runner.
Let's break it down step by step:
1. The first runner ran half the laps. This means they ran n/2 laps.
2. The second runner ran half the amount of laps the first runner ran. This means they ran (1/2) * (n/2) = n/4 laps.
3. The third runner ran half the amount of laps the second runner ran. This means they ran (1/2) * (n/4) = n/8 laps.
So far we have:
- First runner: n/2 laps
- Second runner: n/4 laps
- Third runner: n/8 laps
Now, we need to find out how many laps the fourth runner completed. We know that the total number of laps is n, and the laps ran by the first three runners are n/2, n/4, and n/8 respectively.
To find the laps run by the fourth runner, we subtract the laps run by the first three runners from the total number of laps:
Total laps - (laps run by first runner + laps run by second runner + laps run by third runner) = Laps run by fourth runner
n - (n/2 + n/4 + n/8) = Laps run by fourth runner
Simplifying the equation:
n - (4n/8 + 2n/8 + n/8)
= n - (7n/8)
= n - 7n/8
= (8n - 7n)/8
= n/8
Therefore, the fourth runner completed n/8 laps.
So, the correct answer is:
b. n/8
To find out how many laps the fourth runner completed, we need to work our way backwards by finding the number of laps each runner ran.
Let's start with the information given:
- The first runner ran half the laps: n/2 laps (where n is the total number of laps in the relay).
- The second runner ran half the amount of laps the first runner ran: (n/2)/2 = n/4 laps.
- The third runner ran half the amount of laps the second runner ran: (n/4)/2 = n/8 laps.
Now, let's calculate the remaining laps for the fourth runner:
The sum of the laps run by the first, second, and third runners is:
n/2 + n/4 + n/8 = (4n + 2n + n)/8 = 7n/8
Therefore, the remaining laps that the fourth runner completed can be calculated by subtracting the sum of the laps run by the other three runners (7n/8) from the total number of laps (n):
n - 7n/8 = (8n - 7n)/8 = n/8
So, the fourth runner completed n/8 laps.
Therefore, the answer is (b) n/8.