Difference between revisions of "2023 AMC 10B Problems/Problem 1"

Revision as of 14:22, 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

We let $x$ denote how much juice we take from each of the first $3$ children and give to the $4$ th child.

We can write the following equation: $1-x=\dfrac13+3x$ , since each value represents how much juice each child (equally) has in the end. (Each of the first three children now have $1-x$ juice, and hte fourth child has $3x$ more juice on top of their initial $\dfrac13$ .)

Solving, we see that $x=\boxed{\textbf{(C) }\dfrac16}.$

~Technodoggo

Solution 2

Say the capacity of one full glass is 1 ml. Quickly we see the four glasses have 1ml, 1ml, 1ml, and 1/3 ml. Thus, the amount of liquid between all of the glasses is $1+1+1+\frac{1}{3}=\frac{10}{3}$ . In order for all four glasses to have the same amount of orange juice, they have to each have $\frac{\frac{10}{3}}{4}=\frac{5}{6}$ . This means each of the three full glasses must contribute $1-\frac{5}{6} = \frac{1}{6}$ of the amount they have to the fourth glass. $\boxed{\textbf{(C)}\frac{1}{6}}$

~vsinghminhas (tweak - aa)

Vide Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=SUnhwbA5_So

@@ Line 16: / Line 16: @@
 ==Solution 2==
-We begin by assigning a variable <math>x</math> to the capacity of one full glass. Quickly we see the four glasses have <math>x+x+x+\frac{x}{3}=\frac{10x}{3}</math> full glasses filled. In order for all four glasses to have the same amount of orange juice, they have to each have <math>\frac{\frac{10x}{3}}{4}=\frac{5x}{6}</math>. This means each of the three full glasses must contribute <math>1x-\frac{5x}{6} = \frac{1x}{6}</math> of the of the amount they have to the fourth glass. <math>\boxed{\textbf{(C)}\frac{1}{6}}</math>
+Say the capacity of one full glass is 1 ml. Quickly we see the four glasses have 1ml, 1ml, 1ml, and 1/3 ml. Thus, the amount of liquid between all of the glasses is <math>1+1+1+\frac{1}{3}=\frac{10}{3}</math>. In order for all four glasses to have the same amount of orange juice, they have to each have <math>\frac{\frac{10}{3}}{4}=\frac{5}{6}</math>. This means each of the three full glasses must contribute <math>1-\frac{5}{6} = \frac{1}{6}</math> of the amount they have to the fourth glass. <math>\boxed{\textbf{(C)}\frac{1}{6}}</math>
-~vsinghminhas
+~vsinghminhas (tweak - aa)
 ==Vide Solution 1 by SpreadTheMathLove==
 https://www.youtube.com/watch?v=SUnhwbA5_So

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