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Number theory Difficulty 2.4 Junior Find the answer Canada

In the sum shown, $P$ , $Q$ and $R$ represent three different single digits.

$\begin{array}{cccc} & P&7&R\\ +&3&9&R\\ \hline &R&Q&0\\ \end{array}$

The value of $P+Q+R$ is

$13$
$12$
$14$
$3$
$4$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since the second number being added is greater than 300 and the sum has hundreds digit $R$ , then $R$ cannot be 0.

From the ones column, we see that the ones digit of $R+R$ is 0. Since $R \neq 0$ , then $R=5$ .

This makes the sum $\begin{array}{cccc} & P&\overset{1}{7}&5\\ +&3&9&5\\ \hline &5&Q&0\\ \end{array}$ Since $1+7+9=17$ , we get $Q = 7$ and then $1+P+3=5$ and so $P=1$ , giving the final sum $\begin{array}{cccc} & \overset{1}{1}&\overset{1}{7}&5\\ +&3&9&5\\ \hline &5&7&0\\ \end{array}$ Therefore, $P+Q+R=1+7+5=13$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.