Difference between revisions of "2023 AMC 12B Problems/Problem 1"
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==Solution 1== | ==Solution 1== | ||
+ | The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that wouldn't affect our answer. 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 | ||
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+ | ==Solution 2== | ||
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 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. | ||
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~Technodoggo | ~Technodoggo | ||
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==Video Solution by SpreadTheMathLove== | ==Video Solution by SpreadTheMathLove== |
Revision as of 20:24, 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.
Contents
[hide]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
The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that wouldn't affect our answer. The fourth glass has a glass that is one third. So the total amount of juice will be . If we divide the total amount of juice by 4, we get , which should be the amount of juice in each glass. This means that each of the first three glasses will have to contribute to the fourth glass.
~Sir Ian Seo the Great & lprado
Solution 2
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 the fourth child has more juice on top of their initial .)
Solving, we see that
~Technodoggo
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.