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(A) Suppose $x$ points make the game.When $A$ scores $x$ points,$B$ scores $(x - 20)$ points and $C$ scores $(x - 32)$ points.This implies the ratio o

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$100$

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(A) Suppose $x$ points make the game.When $A$ scores $x$ points,$B$ scores $(x - 20)$ points and $C$ scores $(x - 32)$ points.This implies the ratio of the speeds of $A$ and $B$ is $x : (x - 20)$ and the ratio of the speeds of $A$ and $C$ is $x : (x - 32)$.When $B$ scores $x$ points,$C$ scores $(x - 15)$ points,so the ratio of the speeds of $B$ and $C$ is $x : (x - 15)$.We know that $\frac{	ext{Speed of } A}{	ext{Speed of } C} = \frac{	ext{Speed of } A}{	ext{Speed of } B} 	imes \frac{	ext{Speed of } B}{	ext{Speed of } C}$.Substituting the ratios: $\frac{x}{x - 32} = \frac{x}{x - 20} 	imes \frac{x}{x - 15}$.Canceling $x$ from both sides: $\frac{1}{x - 32} = \frac{x}{(x - 20)(x - 15)}$.$(x - 20)(x - 15) = x(x - 32)$.$x^2 - 35x + 300 = x^2 - 32x$.$300 = 35x - 32x$.$300 = 3x$.$x = 100$.Thus,$100$ points make the game.

In a game,$A$ can give $B$ $20$ points,$A$ can give $C$ $32$ points,and $B$ can give $C$ $15$ points. How many points make the game?

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