Question

question 6 of 6
read the problem below and find the solution. draw a diagram on your own
paper to help solve it.
melanie made a circular necklace. she puts beads evenly spaced around the
necklace. if the 4th bead is directly across from the 11th bead, how many
beads are on the necklace?
(do not include units in your answer.)

Explanation:

Step1: Find the number of beads between 4th and 11th

To find the number of beads between the 4th and 11th bead, we calculate \(11 - 4 - 1\) (subtracting 1 because we don't count the 11th bead itself when finding the number of beads in between). So \(11 - 4 - 1 = 6\).

Step2: Determine total beads

Since the 4th bead is directly across from the 11th bead, the number of beads between them on one side is 6, so the number of beads on the other side (from 11th back to 4th going the other way around the circle) is also 6. Then we add the 4th and 11th beads themselves. Wait, actually, in a circle, if two beads are directly across, the number of beads between them on each semicircle is the same. The number of beads between 4th and 11th (excluding 4th and including 11th? Wait no, let's think again. If bead 4 is across from bead 11, then the arc from 4 to 11 has \(11 - 4 = 7\) intervals? Wait no, beads are evenly spaced. Let's consider the positions. Let the number of beads be \(n\). In a circle, the position opposite to bead \(k\) is bead \(k+\frac{n}{2}\) (mod \(n\)). So if bead 4 is opposite bead 11, then \(11=4 + \frac{n}{2}\) (since it's a circle, we can assume \(11>4\) and the distance is \(\frac{n}{2}\) beads apart). So solving for \(n\): \(11 - 4=\frac{n}{2}\), so \(7=\frac{n}{2}\), then \(n = 14\)? Wait no, wait, maybe I made a mistake. Wait, the number of beads between 4th and 11th: from 4 to 11, how many beads? Let's list them: 4,5,6,7,8,9,10,11. Wait, no, the number of beads from 4 to 11 inclusive is \(11 - 4 + 1=8\)? No, wait, no. Wait, if bead 4 is across from bead 11, then the number of beads between them (not including 4 and 11) should be equal on both sides. Wait, let's think of the circle. If you have bead 4 and bead 11 directly across, then the number of beads between 4 and 11 (moving clockwise) is \(11 - 4 - 1 = 6\) (because we start at 4, then 5,6,7,8,9,10, then 11. So between 4 and 11 (excluding 4 and 11) are 5,6,7,8,9,10: 6 beads. Then the number of beads between 11 and 4 (moving the other way) should also be 6. Then total beads: 6 (between 4 and 11) + 6 (between 11 and 4) + 2 (beads 4 and 11) = 14? Wait, no, that's double - counting. Wait, actually, in a circle, if two points are diametrically opposite, the number of elements between them on each semicircle is the same. So the distance between bead 4 and bead 11 is \(\frac{n}{2}\) (since it's a circle, the number of beads between them along the circumference is half the total number, because they are directly across). So the number of beads from 4 to 11 (including 11, excluding 4) is \(11 - 4=7\) beads. But since they are directly across, this 7 beads should be half of the total number of beads? Wait, no, that can't be. Wait, let's use the formula for circular arrangements. If bead \(i\) is opposite bead \(j\), then \(j=i+\frac{n}{2}\) (mod \(n\)). So \(11 = 4+\frac{n}{2}\) (assuming \(n\) is even, which it must be for there to be a bead directly across). So \(11 - 4=\frac{n}{2}\), so \(7=\frac{n}{2}\), then \(n = 14\). Wait, but let's check. If \(n = 14\), then the bead opposite to 4 is \(4 + 7=11\), which matches. So that works.

Answer:

14