2022 AMC 8 Problems/Problem 21
Contents
- 1 Problem
- 2 Solution 1 (Inequalities)
- 3 Solution 2 (Answer Choices)
- 4 Solution 3 (Cheap)
- 5 Video Solution by Math-X (First understand the problem!!!)
- 6 Video Solution (🚀58 seconds. Straightforward Explanation.🚀)
- 7 Video Solution
- 8 Video Solution by OmegaLearn
- 9 Video Solution
- 10 Video Solution
- 11 Video Solution
- 12 Video Solution
- 13 Video Solution
- 14 See Also
Problem
Steph scored 
baskets out of 
attempts in the first half of a game, and 
baskets out of attempts in the second half. Candace took 
attempts in the first half and 
attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
![[asy] size(7cm); draw((-8,27)--(72,27)); draw((16,0)--(16,35)); draw((40,0)--(40,35)); label("12", (28,3)); draw((25,6.5)--(25,12)--(31,12)--(31,6.5)--cycle); draw((25,5.5)--(31,5.5)); label("18", (56,3)); draw((53,6.5)--(53,12)--(59,12)--(59,6.5)--cycle); draw((53,5.5)--(59,5.5)); draw((53,5.5)--(59,5.5)); label("20", (28,18)); label("15", (28,24)); draw((25,21)--(31,21)); label("10", (56,18)); label("10", (56,24)); draw((53,21)--(59,21)); label("First Half", (28,31)); label("Second Half", (56,31)); label("Candace", (2.35,6)); label("Steph", (0,21)); [/asy]](http://latex.artofproblemsolving.com/4/2/a/42aefbe42671b2aeb0b90d88e171e5e03a00b73e.png)
Solution 1 (Inequalities)
Let 
be the number of shots that Candace made in the first half, and let 
be the number of shots Candace made in the second half. Since Candace and Steph took the same number of attempts, with an equal percentage of baskets scored, we have 
In addition, we have the following inequalities: ![\[\frac{x}{12}<\frac{15}{20} \implies x<9,\]](http://latex.artofproblemsolving.com/7/1/6/716a42f88961e64e0d35c406547f428f6f675fe6.png)
and ![\[\frac{y}{18}<\frac{10}{10} \implies y<18.\]](http://latex.artofproblemsolving.com/a/3/3/a33cc629e050ccf31e08c13b068e5dd97ddd5290.png)
Pairing this up with 
we see the only possible solution is 
for an answer of 
~wamofan
Solution 2 (Answer Choices)
Clearly, Steph made 
shots in total. Also, due to parity reasons, the difference between the amount of shots Candace made in the first and second half must be odd. Thus, we can just test 7, 9, and 11, and after doing so we find that the answer is 
Solution 3 (Cheap)
Steph made 75 percent of his shots in the first half. He makes all of his shots in the second half. The most baskets Candace could have made in the first half is 8 baskets. The most she could have made in the second half is 17 baskets. Steph makes 25 and misses 5 baskets and the only way for Candace to make 25 shots is to make 8 in the first half and 17 in the second. Thus, 
,
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=gkFZrlmVOdkOwwTT&t=3793
~Math-X
Video Solution (🚀58 seconds. Straightforward Explanation.🚀)
~Education, the Study of Everything
Video Solution
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Video Solution by OmegaLearn
https://youtu.be/DzQZtQvNDwA?t=460
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=IbsSecIq8FE
~Mathematical Dexterity
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=1995
~Interstigation
Video Solution
https://www.youtube.com/watch?v=nXHHM1884Jo
~David
Video Solution
~STEMbreezy
Video Solution
~savannahsolver
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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. 
