Competition problem — Algebra — Maths Olympiad Prep

Library / /101 of 173

Algebra Difficulty 2.8 Junior Find the answer

In the sum shown, $P, Q$ and $R$ represent three different single digits. What is the value of $P+Q+R$ ?

$\text{P 7 R + 39 R R Q 0}$

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

$\begin{array}{r} P 75 \\ +\quad 395 \\ \hline 5 Q 0 \end{array}$

Since $1+7+9=17$ , we get $Q=7$ and then $1+P+3=5$ and so $P=1$ , giving the final sum

$\begin{array}{r} 11 \\ 175 \\ +\quad 395 \\ \hline 570 \end{array}$

Therefore, $P+Q+R=1+7+5=13$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.