2005 AMC 10B Problem 3

Below is the professionally curated solution for Problem 3 of the 2005 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10B solutions, or check the answer key.

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Concepts:fraction ratio and proportion

Difficulty rating: 870

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

$\dfrac{1}{10}$

$\dfrac{1}{9}$

$\dfrac{1}{3}$

$\dfrac{4}{9}$

$\dfrac{5}{9}$

Solution:

After the first day, $1 - \dfrac13 = \dfrac23$ of the paint remains.

On the second day, $\dfrac13 \cdot \dfrac23 = \dfrac29$ of the original amount is used.

The fraction available on the third day is $\dfrac23 - \dfrac29 = \dfrac49.$

Thus, D is the correct answer.

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