Difference between revisions of "2015 AMC 8 Problems/Problem 23"
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The numbers have a sum of <math>6+5+12+4+8=35</math>, which averages to <math>7</math>, which means <math>A, B, C, D,</math> and | The numbers have a sum of <math>6+5+12+4+8=35</math>, which averages to <math>7</math>, which means <math>A, B, C, D,</math> and | ||
| − | <math>E</math> will have the values <math>5, 6, 7, 8</math> and <math>9</math>, respectively. Now it's the process of elimination: Cup <math>A</math> | + | <math>E</math> will have the values <math>5, 6, 7, 8</math> and <math>9</math>, respectively. Now, it's the process of elimination: Cup <math>A</math> |
will have a sum of <math>5</math>, so putting a <math>3.5</math> slip in the cup will leave <math>5-3.5=1.5</math>; However, | will have a sum of <math>5</math>, so putting a <math>3.5</math> slip in the cup will leave <math>5-3.5=1.5</math>; However, | ||
all of our slips are bigger than <math>1.5</math>, so this is impossible. Cup <math>B</math> has a sum of <math>6</math>, but we are told that it already has a <math>3</math> slip, leaving <math>6-3=3</math>, which is too small for the | all of our slips are bigger than <math>1.5</math>, so this is impossible. Cup <math>B</math> has a sum of <math>6</math>, but we are told that it already has a <math>3</math> slip, leaving <math>6-3=3</math>, which is too small for the | ||
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exactly, and the smallest sum of two slips is <math>2+2=4</math>, which is too big, so this case is also impossible. Cup <math>E</math> has a sum of <math>9</math>, but we are told it already has a <math>2</math> slip, so we | exactly, and the smallest sum of two slips is <math>2+2=4</math>, which is too big, so this case is also impossible. Cup <math>E</math> has a sum of <math>9</math>, but we are told it already has a <math>2</math> slip, so we | ||
are left with <math>9-2=7</math>, which is identical to Cup C and thus also impossible. | are left with <math>9-2=7</math>, which is identical to Cup C and thus also impossible. | ||
| − | With all other choices removed, we are left with the answer: Cup <math>\boxed{\textbf{(D)}~D}</math> | + | With all other choices removed, we are left with the answer: Cup <math>\boxed{\textbf{(D)}~D}</math>. |
==Video Solution== | ==Video Solution== | ||
Revision as of 01:24, 8 May 2023
Contents
Problem
Tom has twelve slips of paper which he wants to put into five cups labeled 
, 
, 
, 
, 
. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from to . The numbers on the papers are 
and 
. If a slip with 
goes into cup and a slip with 
goes into cup , then the slip with 
must go into what cup?
Solution
The numbers have a sum of 
, which averages to 
, which means 
and
will have the values 
and 
, respectively. Now, it's the process of elimination: Cup
will have a sum of 
, so putting a slip in the cup will leave 
; However,
all of our slips are bigger than 
, so this is impossible. Cup has a sum of 
, but we are told that it already has a slip, leaving 
, which is too small for the
slip. Cup is a little bit trickier but still manageable. It must have a value of , so adding the slip leaves room for 
. This looks good at first, as we have slips smaller than that, but upon closer inspection, we see that no slip fits
exactly, and the smallest sum of two slips is 
, which is too big, so this case is also impossible. Cup has a sum of , but we are told it already has a slip, so we
are left with 
, which is identical to Cup C and thus also impossible.
With all other choices removed, we are left with the answer: Cup 
.
Video Solution
~savannahsolver
See Also
| 2015 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
