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

1990 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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1990 AJHSME Problems/Problem 24

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