18. $P, Q$ and $R$ are the three angles of a triangle, when each has been rounded to the nearest degree.
Which of the following is the complete list of possible values of $P+Q+R$ ?
A $179^{\circ}, 180^{\circ}$ or $181^{\circ}$
B $180^{\circ}, 181^{\circ}$ or $182^{\circ}$
C $178^{\circ}, 179^{\circ}$ or $180^{\circ}$
D $180^{\circ}$
E $178^{\circ}, 179^{\circ}, 180^{\circ}, 181^{\circ}$ or $182^{\circ}$
Solution
A

We suppose that the actual angles of the triangle are $P^{\prime}, Q^{\prime}$ and $R^{\prime}$, which are rounded to $P, Q$ and $R$, respectively.

The sum of the angles of a triangle is $180^{\circ}$ and therefore
$$
P^{\prime}+Q^{\prime}+R^{\prime}=180^{\circ} \text {. }
$$

When an angle is rounded up to the nearest degree, it is increased by at most $0.5^{\circ}$; when it is rounded down to the nearest degree, it is decreased by at most $0.5^{\circ}$. Therefore,
$$
\begin{aligned}
& P^{\prime}-0.5^{\circ} \leq P \leq P^{\prime}+0.5^{\circ}, \\
& Q^{\prime}-0.5^{\circ} \leq Q \leq Q^{\prime}+0.5^{\circ}, \\
\text { and } \quad & R^{\prime}-0.5^{\circ} \leq R \leq R^{\prime}+0.5^{\circ} .
\end{aligned}
$$

Adding these inequalities gives
$$
P^{\prime}+Q^{\prime}+R^{\prime}-1.5^{\circ} \leq P+Q+R \leq P^{\prime}+Q^{\prime}+R^{\prime}+1.5^{\circ} .
$$

Therefore, as $P^{\prime}+Q^{\prime}+R^{\prime}=180^{\circ}$,
$$
178.5^{\circ} \leq P+Q+R \leq 181.5^{\circ} \text {. }
$$

Each of $P, Q$ and $R$ is an integer number of degrees. Hence $P+Q+R$ is also an integer number of degrees. It follows that the only possible values of $P+Q+R$ are $179^{\circ}, 180^{\circ}$ and $181^{\circ}$.

To complete the solution we show that each of these possible values actually occurs for some triangle. This is shown by the following examples.
$P^{\prime}=60.3^{\circ}, Q^{\prime}=60.3^{\circ}, R^{\prime}=59.4^{\circ}$ gives $P=60^{\circ}, Q=60^{\circ}, R=59^{\circ}$. Hence $P+Q+R=179^{\circ}$.
$P^{\prime}=60^{\circ}, Q^{\prime}=60^{\circ}, R^{\prime}=60^{\circ}$ gives $P=60^{\circ}, Q=60^{\circ}, R=60^{\circ}$. Hence $P+Q+R=180^{\circ}$.
$P^{\prime}=59.7^{\circ}, Q^{\prime}=59.7^{\circ}, R^{\prime}=60.6^{\circ}$ gives $P=60^{\circ}, Q=60^{\circ}, R=61^{\circ}$. Hence $P+Q+R=181^{\circ}$.
We deduce that $179^{\circ}, 180^{\circ}, 181^{\circ}$ is a complete list of the possible values of $P+Q+R$.