Difference between revisions of "2023 AMC 10B Problems/Problem 1"
Technodoggo (talk | contribs) |
Vsinghminhas (talk | contribs) |
||
| Line 1: | Line 1: | ||
| + | ==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? | ||
| + | |||
| + | <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> | ||
| + | |||
==Solution 1== | ==Solution 1== | ||
| Line 8: | Line 14: | ||
~Technodoggo | ~Technodoggo | ||
| + | |||
| + | ==Solution 2== | ||
| + | We begin by assigning a variable <math>x</math> to the capacity of one full glass. Quickly we see the 4 glasses have <math>x+x+x+\frac{x}{3}=\frac{10a}{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{10a}{3}}{4}=\frac{5a}{6}</math>. This means each of the three full glasses must contribute <math>1a-\frac{5a}{6} = \frac{1a}{6}</math> of the of the amount they have to the fourth glass. <math>\boxed{\textbf{(C)}\frac{1}{6}}</math> | ||
| + | |||
| + | ~vsinghminhas | ||
Revision as of 14:06, 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 
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?
Solution 1
We let 
denote how much juice we take from each of the first 
children and give to the 
th child.
We can write the following equation: 
, since each value represents how much juice each child (equally) has in the end. (Each of the first three children now have 
juice, and hte fourth child has 
more juice on top of their initial 
.)
Solving, we see that 
~Technodoggo
Solution 2
We begin by assigning a variable to the capacity of one full glass. Quickly we see the 4 glasses have 
full glasses filled. In order for all four glasses to have the same amount of orange juice, they have to each have 
. This means each of the three full glasses must contribute 
of the of the amount they have to the fourth glass. 
~vsinghminhas
