# 1990 AJHSME Problems/Problem 24

## Problem

Three $\Delta$'s and a $\diamondsuit$ will balance nine $\bullet$'s. One $\Delta$ will balance a $\diamondsuit$ and a $\bullet$.

$[asy] unitsize(5.5); fill((0,0)--(-4,-2)--(4,-2)--cycle,black); draw((-12,2)--(-12,0)--(12,0)--(12,2)); draw(ellipse((-12,5),8,3)); draw(ellipse((12,5),8,3)); label("\Delta \hspace{2 mm}\Delta \hspace{2 mm}\Delta \hspace{2 mm}\diamondsuit ",(-12,6.5),S); label("\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm} \bullet ",(12,5.2),N); label("\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet ",(12,5.2),S); fill((44,0)--(40,-2)--(48,-2)--cycle,black); draw((34,2)--(34,0)--(54,0)--(54,2)); draw(ellipse((34,5),6,3)); draw(ellipse((54,5),6,3)); label("\Delta ",(34,6.5),S); label("\bullet \hspace{2 mm}\diamondsuit ",(54,6.5),S); [/asy]$

How many $\bullet$'s will balance the two $\diamondsuit$'s in this balance?

$[asy] unitsize(5.5); fill((0,0)--(-4,-2)--(4,-2)--cycle,black); draw((-12,4)--(-12,2)--(12,-2)--(12,0)); draw(ellipse((-12,7),6.5,3)); draw(ellipse((12,3),6.5,3)); label("?",(-12,8.5),S); label("\diamondsuit \hspace{2 mm}\diamondsuit ",(12,4.5),S); [/asy]$

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

## Solution

For simplicity, suppose $\Delta = a$, $\diamondsuit = b$ and $\bullet = c$. Then, $$3a+b=9c$$ $$a=b+c$$ and we want to know what $2b$ is in terms of $c$. Substituting the second equation into the first, we have $$4b=6c\Rightarrow 2b=3c$$

Thus, we need $3$ $\bullet$'s $\rightarrow \boxed{\text{C}}$.