In a game, A can give 36 points to B, A can give 42 point to C, B can give 10 points to C. How many points make the game ?

Let the game have $G$ points total.

The rates at which A, B, and C score are constant throughout the game. Let these rates be $a$, $b$, and $c$ points per unit time respectively.

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When A finishes the game (reaches $G$ points):

• A has scored: $G$ points
• B has scored: $G - 36$ points
• C has scored: $G - 42$ points

Time taken: $t_1 = \dfrac{G}{a}$

This gives:

$b \cdot \dfrac{G}{a} = G - 36$

$\dfrac{b}{a} = \dfrac{G - 36}{G}$

Similarly:

$c \cdot \dfrac{G}{a} = G - 42$

$\dfrac{c}{a} = \dfrac{G - 42}{G}$

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When B finishes the game (reaches $G$ points):

• B has scored: $G$ points
• C has scored: $G - 10$ points

Time taken: $t_2 = \dfrac{G}{b}$

This gives:

$c \cdot \dfrac{G}{b} = G - 10$

$\dfrac{c}{b} = \dfrac{G - 10}{G}$

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From the ratios obtained:

$\dfrac{c}{b} = \dfrac{c/a}{b/a}$

$\dfrac{c}{b} = \dfrac{(G-42)/G}{(G-36)/G}$

$\dfrac{c}{b} = \dfrac{G - 42}{G - 36}$

Equating with the other expression for $\dfrac{c}{b}$:

$\dfrac{G - 42}{G - 36} = \dfrac{G - 10}{G}$

$G(G - 42) = (G - 10)(G - 36)$

$G^2 - 42G = G^2 - 46G + 360$

$4G = 360$

$G = 90$

Therefore, the game has 90 points.