Question

Loki, Moe, Nick y Ott son buenos amigos. Ott no tenía dinero, pero los demás sí. Moe le dio a Ott un quinto de su dinero, Loki le dio a Ott un cuarto de su dinero y Nick le dio a Ott un tercio de su dinero. Cada uno le dio a Ott la misma cantidad de dinero. ¿Qué parte fraccionaria del dinero del grupo tiene Ott ahora?

Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?

$\dfrac{1}{10}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{2}{5}$

$\dfrac{1}{2}$

Solución:

Sin pérdida de generalidad, supón que cada uno le dio a Ott $\$1.$ Esto significa que Moe tenía $\$5,$ Loki tenía $\$4,$ y Nick tenía $\$3$ originalmente.

Ott ahora tiene $\$3,$ y la cantidad total de dinero es $\$5 + \$4 + \$3 = \$12.$

Esto significa que Ott tiene $\dfrac{3}{12} = \dfrac{1}{4}$ del dinero del grupo.

Por lo tanto, B es la respuesta correcta.

WLOG, assume that everyone gave Ott $\$1.$ This means that Moe had $\$5,$ Loki had $\$4,$ and Nick had $\$3$ originally.

Ott now has $\$3,$ and the total amount of money is $\$5 + \$4 + \$3 = \$12.$

This means that Ott has $\dfrac{3}{12} = \dfrac{1}{4}$ of the group's money.

Thus, B is the correct answer.

2002 AMC 8 Problema 25

Solución:

El Problema 25 en otros años