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Difference between revisions of "2023 AMC 12B Problems/Problem 1"

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==Problem==
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Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only <math>\frac{1}{3}</math> full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
  
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<math>\textbf{(A) } \frac{1}{12} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{6} \qquad\textbf{(D) } \frac{1}{8} \qquad\textbf{(E) } \frac{2}{9}</math>
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==Solution 1==
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The first three glasses each have a full glass. The fourth glass has a glass that is one third. So the total amount of juice will be <math>1+1+1+\frac{1}{3} = \frac{10}{3}</math>. If we divide the total amount of juice by 4, we get <math>\frac{5}{6}</math>, which should be the amount of juice in each glass. This means that each of the first three glasses will have to contribute <math>1 - \frac{5}{6} = \boxed{\frac{1}{6}}</math> to the fourth glass.
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~Sir Ian Seo the Great & lprado

Revision as of 17:23, 15 November 2023

Problem

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

$\textbf{(A) } \frac{1}{12} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{6} \qquad\textbf{(D) } \frac{1}{8} \qquad\textbf{(E) } \frac{2}{9}$

Solution 1

The first three glasses each have a full glass. The fourth glass has a glass that is one third. So the total amount of juice will be $1+1+1+\frac{1}{3} = \frac{10}{3}$. If we divide the total amount of juice by 4, we get $\frac{5}{6}$, which should be the amount of juice in each glass. This means that each of the first three glasses will have to contribute $1 - \frac{5}{6} = \boxed{\frac{1}{6}}$ to the fourth glass.

~Sir Ian Seo the Great & lprado

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