Difference between revisions of "2024 AMC 8 Problems/Problem 8"
(→Solution 1) |
|||
(39 intermediate revisions by 22 users not shown) | |||
Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
+ | On Monday Taye has <math>\$2</math>. Every day, he either gains <math>\$3</math> or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, <math>3</math> days later? | ||
− | ==Solution 1== | + | <math>\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7</math> |
+ | |||
+ | ==Solution 1 (BRUTE FORCE)== | ||
+ | How many values could be on the first day? Only <math>2</math> dollars. The second day, you can either add <math>3</math> dollars, or double, so you can have <math>5</math> dollars, or <math>4</math>. For each of these values, you have <math>2</math> values for each. For <math>5</math> dollars, you have <math>10</math> dollars or <math>8</math>, and for <math>4</math> dollars, you have <math>8</math> dollars or \$<math>7</math>. Now, you have <math>2</math> values for each of these. For <math>10</math> dollars, you have <math>13</math> dollars or <math>20</math>, for <math>8</math> dollars, you have <math>16</math> dollars or <math>11</math>, for <math>8</math> dollars, you have <math>16</math> dollars or <math>11</math>, and for <math>7</math> dollars, you have <math>14</math> dollars or <math>10</math>. | ||
+ | |||
+ | On the final day, there are 11, 11, 16, and 16 repeating, leaving you with <math>8-2 = \boxed{\textbf{(D)\ 6}}</math> different values. | ||
+ | |||
+ | ~ cxsmi (minor formatting edits) | ||
+ | |||
+ | ==Solution 2 (Slightly less BRUTE FORCE)== | ||
+ | Continue as in Solution 1 to get <math>7</math>, <math>8</math>, or <math>10</math> dollars by the 2nd day. The only way to get the same dollar amount occurring twice by branching (multiply by <math>2</math> or adding <math>3</math>) from here is if <math>7+3=10\cdot 2</math> or <math>7+3=8\cdot 2</math> which both aren't true. Hence our answer is <math>3\cdot2=\boxed{\textbf{(D)\ 6}}</math>. | ||
+ | |||
+ | ~ Sahan Wijetunga | ||
+ | |||
+ | ==Video Solution 1 (easy to digest) by Power Solve== | ||
+ | https://youtu.be/16YYti_pDUg?si=5kw0dc_bZwASNiWm&t=121 | ||
+ | |||
+ | ==Video Solution by Math-X (First fully understand the problem!!!)== | ||
+ | https://youtu.be/BaE00H2SHQM?si=6wjacdxeAgtpc0fW&t=1762 | ||
+ | ~Math-X | ||
+ | |||
+ | ==Video Solution by NiuniuMaths (Easy to understand!)== | ||
+ | https://www.youtube.com/watch?v=V-xN8Njd_Lc | ||
+ | |||
+ | ~NiuniuMaths | ||
+ | |||
+ | ==Video Solution 2 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=L83DxusGkSY | ||
+ | |||
+ | == Video Solution by CosineMethod [🔥Fast and Easy🔥]== | ||
+ | |||
+ | https://www.youtube.com/watch?v=alDY4yhaEEg | ||
+ | |||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/ktzijuZtDas&t=791 | ||
+ | |||
+ | ==Video Solution by Daily Dose of Math (Certified, Simple, and Logical)== | ||
+ | |||
+ | https://youtu.be/8GHuS5HEoWc | ||
+ | |||
+ | ~Thesmartgreekmathdude | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2024|num-b=7|num-a=9}} | ||
+ | {{MAA Notice}} |
Latest revision as of 21:57, 18 August 2024
Contents
- 1 Problem
- 2 Solution 1 (BRUTE FORCE)
- 3 Solution 2 (Slightly less BRUTE FORCE)
- 4 Video Solution 1 (easy to digest) by Power Solve
- 5 Video Solution by Math-X (First fully understand the problem!!!)
- 6 Video Solution by NiuniuMaths (Easy to understand!)
- 7 Video Solution 2 by SpreadTheMathLove
- 8 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 9 Video Solution by Interstigation
- 10 Video Solution by Daily Dose of Math (Certified, Simple, and Logical)
- 11 See Also
Problem
On Monday Taye has . Every day, he either gains
or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday,
days later?
Solution 1 (BRUTE FORCE)
How many values could be on the first day? Only dollars. The second day, you can either add
dollars, or double, so you can have
dollars, or
. For each of these values, you have
values for each. For
dollars, you have
dollars or
, and for
dollars, you have
dollars or $
. Now, you have
values for each of these. For
dollars, you have
dollars or
, for
dollars, you have
dollars or
, for
dollars, you have
dollars or
, and for
dollars, you have
dollars or
.
On the final day, there are 11, 11, 16, and 16 repeating, leaving you with different values.
~ cxsmi (minor formatting edits)
Solution 2 (Slightly less BRUTE FORCE)
Continue as in Solution 1 to get ,
, or
dollars by the 2nd day. The only way to get the same dollar amount occurring twice by branching (multiply by
or adding
) from here is if
or
which both aren't true. Hence our answer is
.
~ Sahan Wijetunga
Video Solution 1 (easy to digest) by Power Solve
https://youtu.be/16YYti_pDUg?si=5kw0dc_bZwASNiWm&t=121
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=6wjacdxeAgtpc0fW&t=1762
~Math-X
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=L83DxusGkSY
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=alDY4yhaEEg
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=791
Video Solution by Daily Dose of Math (Certified, Simple, and Logical)
~Thesmartgreekmathdude
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.