On the number line, $S$ is three-quarters of the way from $P$ to $Q$. Also, $T$ is one-third of the way from $P$ to $Q$. What is the distance along the number line from $T$ to $S$ ?

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-093.jpg?height=122&width=437&top_left_y=264&top_left_x=1294)
(A) 20
(B) 15
(C) 6
(D) 25
(E) 31

## Part B: Each correct answer is worth 6.
Solution 1

Since $Q$ is at 46 and $P$ is at -14 , then the distance along the number line from $P$ is $46-(-14)=60$.

Since $S$ is three-quarters of the way from $P$ to $Q$, then $S$ is at $-14+\frac{3}{4}(60)=-14+45=31$.

Since $T$ is one-third of the way from $P$ to $Q$, then $T$ is at $-14+\frac{1}{3}(60)=-14+20=6$.

Thus, the distance along the number line from $T$ to $S$ is $31-6=25$.

## Solution 2

Since $Q$ is at 46 and $P$ is at -14 , then the distance along the number line from $P$ to $Q$ is $46-(-14)=60$.

Since $S$ is three-quarters of the way from $P$ to $Q$ and $T$ is one-third of the way from $P$ to $Q$, then the distance from $T$ to $S$ is $60\left(\frac{3}{4}-\frac{1}{3}\right)=60\left(\frac{9}{12}-\frac{4}{12}\right)=60\left(\frac{5}{12}\right)=25$.

ANSWER: (D)