Question

A student was given a piece of rope and told to cut it into two equal pieces, keep one piece, and pass the other piece to the next student. Each student was to repeat this process until every student in the class had exactly one piece of rope. Which of the following could be the fraction of the original rope that one of the students had?

• A. $\frac{1}{14}$
• B. $\frac{1}{15}$
• C. $\frac{1}{16}$
• D. $\frac{1}{17}$
• E. $\frac{1}{18}$

Answer 1

Answer: (C)

$\frac{1}{16}$

The first student gets $\frac{1}{2}$ of the rope.

The second student gets $\frac{1}{2}$ of $\frac{1}{2}$ of the rope, which is $\frac{1}{4}$.

The third student gets $\frac{1}{2}$ of $\frac{1}{4}$ of the rope, which is $\frac{1}{8}$.

The fourth student gets $\frac{1}{2}$ of $\frac{1}{8}$, which is $\frac{1}{16}$.

That’s an answer choice!

Listing things out also helps reveal the pattern.

After the second student, you might recognize that the portion of rope each student receives is a power of $\frac{1}{2}$!

The only answer choice that’s a power of $\frac{1}{2}$ is ${\left(\frac{1}{2}\right)}^4=\frac{1}{16}$.