Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2023 AMC 12B Problems/Problem 1"
(condense with amc 10)
Line 1: Line 1:
 +
{{duplicate|[[2023 AMC 10B Problems/Problem 1|2023 AMC 10B #1]] and [[2023 AMC 12B Problems/Problem 1|2023 AMC 12B #1]]}}
 +
 
==Problem==
 
==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 <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?
 
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?
Line 5: Line 7:
  
 
==Solution 1==
 
==Solution 1==
 +
 +
We let <math>x</math> denote how much juice we take from each of the first <math>3</math> children and give to the <math>4</math>th child.
 +
 +
We can write the following equation: <math>1-x=\dfrac13+3x</math>, since each value represents how much juice each child (equally) has in the end. (Each of the first three children now have <math>1-x</math> juice, and hte fourth child has <math>3x</math> more juice on top of their initial <math>\dfrac13</math>.)
 +
 +
Solving, we see that <math>x=\boxed{\textbf{(C) }\dfrac16}.</math>
 +
 +
~Technodoggo
 +
 +
==Solution 2==
 +
We begin by assigning a variable <math>a</math> to the capacity of one full glass. Quickly we see the four glasses are equivalent to <math>1a+1a+1a+\frac{a}{3}=\frac{10a}{3}</math>. In order for all four glasses to have the same amount of orange juice, they have to each have <math>\frac{\frac{10a}{3}}{4}=\frac{5a}{6}</math>. This means each full glass must contribute <math>1a-\frac{5a}{6} = \frac{1a}{6}</math> where <math>a=1</math>. <math>\boxed{\textbf{(C) }\frac{1}{6}}</math>
 +
 +
~vsinghminhas
 +
 +
==Solution 3==
 
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.
 
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.
  
 
~Sir Ian Seo the Great & lprado
 
~Sir Ian Seo the Great & lprado
 +
 +
==Video Solution by SpreadTheMathLove==
 +
 +
https://www.youtube.com/watch?v=SUnhwbA5_So
 +
 +
==See also==
 +
{{AMC10 box|year=2023|ab=B|before=First Problem|num-a=2}}
 +
{{AMC12 box|year=2023|ab=B|before=First Problem|num-a=2}}
 +
 +
[[Category:Rate Problems]]
 +
{{MAA Notice}}

Revision as of 18:48, 15 November 2023

The following problem is from both the 2023 AMC 10B #1 and 2023 AMC 12B #1, so both problems redirect to this page.

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

We begin by assigning a variable $a$ to the capacity of one full glass. Quickly we see the four glasses are equivalent to $1a+1a+1a+\frac{a}{3}=\frac{10a}{3}$. In order for all four glasses to have the same amount of orange juice, they have to each have $\frac{\frac{10a}{3}}{4}=\frac{5a}{6}$. This means each full glass must contribute $1a-\frac{5a}{6} = \frac{1a}{6}$ where $a=1$. $\boxed{\textbf{(C) }\frac{1}{6}}$

~vsinghminhas

Solution 3

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

Video Solution by SpreadTheMathLove

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

See also

2023 AMC 10B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_12B_Problems/Problem_1&oldid=203467"