PR is 3 1/3 metres and PQ is 5 5/6 metres, with P, Q, R on one line. Find QR.

Step-by-step solution

1. 1

   Decide the order of the points before doing any arithmetic. A length can never be negative, so the longest of the named segments must be the whole. Since

   $PQ=5\tfrac56>PR=3\tfrac13$

   the point $R$ must lie between $P$ and $Q$, giving $PR+RQ=PQ$. Assuming instead that $Q$ lies between $P$ and $R$ leads to $QR=PR-PQ<0$, which is impossible for a distance — a clear signal that the assumed order was wrong.

2. 2

   Convert both mixed numbers to improper fractions.

   $3\tfrac13=\frac{10}{3},\qquad 5\tfrac56=\frac{35}{6}$

3. 3

   Put them over a common denominator. The least common denominator of $3$ and $6$ is $6$:

   $\frac{10}{3}=\frac{20}{6}$

4. 4

   Subtract to find $QR$.

   $QR=PQ-PR=\frac{35}{6}-\frac{20}{6}=\frac{15}{6}=\frac{5}{2}=2\tfrac12\ \text{m}$

5. 5

   Check the segment addition. Adding the parts back: $3\tfrac13+2\tfrac12=\tfrac{10}{3}+\tfrac52=\tfrac{20+15}{6}=\tfrac{35}{6}=5\tfrac56\;\checkmark$, which is exactly $PQ$. All three lengths are positive and the largest is the whole, so the configuration is consistent.