1990 AJHSME Problems/Problem 24

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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 $\rightarrow \boxed{\text{C}}$.

See Also

1990 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AJHSME/AMC 8 Problems and Solutions
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