On a number line, X is at 5 and Y is at 17. Which point is two thirds of the way from X to Y?

Step-by-step solution

1. 1

   Find the length of the segment. Distance on a number line is the difference of the coordinates:

   $XY=17-5=12$

2. 2

   Take the required fraction of that length.

   $\frac23(12)=8$

   This $8$ is a distance travelled, not a coordinate — the next step is where most errors happen.

3. 3

   Measure from the starting point, in the stated direction. The phrase "from $X$ to $Y$" means starting at $X=5$ and moving toward $Y$, so add:

   $5+8=13$

   $\boxed{13}$

4. 4

   See why 17 − 8 = 9 is wrong. Subtracting from $Y$ answers a different question — it finds the point two thirds of the way from $Y$ to $X$, which is one third of the way from $X$. Since $9$ is only $4$ units past $X$ out of $12$, it is the one-third point, not the two-thirds point.

5. 5

   Check the answer with the two sub-lengths. From $X$ to $13$ is $8$ units and from $13$ to $Y$ is $4$ units, so the point divides $XY$ in the ratio $8:4=2:1$ — exactly what "two thirds of the way" means. The general formula is

   $P=X+t(Y-X)=5+\tfrac23(12)=13$