Pengi Editor's Note
The Pengi editorial team selected this Think Academy 2022 AMC 10B analysis. The problem-level breakdowns with common mistake alerts help students learn not just the correct method but also where test-takers typically go wrong.
Source: Think Academy Blog
2022 AMC 10B Real Questions and Analysis
In this article, you’ll find:
- Representative real questions from each module with detailed solutions.
- The complete 2022 AMC 10B Answer Key.
- The best resources to prepare effectively for the AMC 10.
- A concise topic distribution chart showing which areas appeared most in the 2022 AMC 10B.
- A module-to-question mapping table highlighting the core concepts tested in each module for the 2022 AMC 10B.
Real Question and Solutions Explained
Algebra Example – Problem 1
Question:
Define \(x\diamond y=|x-y|\) for all real numbers \(x,y\). What is the value of \((1\diamond(2\diamond3))-((1\diamond2)\diamond3)\)?
(A) -2 (B) -1 (C) 0 (D) 1 (E) 2
Solution:
Compute \(2\diamond3=|2-3|=1.\)
Then \(1\diamond(2\diamond3)=1\diamond1=|1-1|=0.\)
Compute \(1\diamond2=|1-2|=1.\)
Then \((1\diamond2)\diamond3=1\diamond3=|1-3|=2.\)
Finally, \((1\diamond(2\diamond3))-((1\diamond2)\diamond3)=0-2=-2.\)
Answer:(A)
Common Mistakes:
- Assuming \(\diamond\) is associative (e.g., \(1\diamond(2\diamond3)=(1\diamond2)\diamond3\)).
- Dropping absolute values or signs.
Number Theory Example – Problem 8
Question:
Consider the 100 sets \(\{1,2,\ldots,10\},\{11,12,\ldots,20\},\ldots,\{991,992,\ldots,1000\}\). How many of these sets contain exactly two multiples of 7?
(A) 40 (B) 42 (C) 43 (D) 49 (E) 50
Solution:
Each set is \([10k+1,10k+10]\).
The number of multiples of 7 in each block repeats every 7 sets (periodicity 70).
Pattern across one cycle of 7 sets: \(1,1,2,1,2,1,2\) — three of them have two multiples.
There are 100 sets = 14 complete cycles (98 sets) + 2 remaining sets (both with 1 multiple).
Total = \(14\times3=42.\)
Answer:(B)
Common Mistakes:
- Ignoring the periodic repetition by 70.
- Counting endpoints incorrectly when both are included.
Geometry Example – Problem 2
Question:
In rhombus 𝐴𝐵𝐶𝐷, point 𝑃 lies on segment \(\overline{AD}\) so that \(\overline{BP}\perp\overline{AD}\), \(AP=3\), and \(PD=2\). What is the area of 𝐴𝐵𝐶𝐷?
(A) 3.5 (B) 10 (C) 6.5 (D) 20 (E) 25
Solution:
Place \(A(0,0)\) and \(D(5,0)\Rightarrow AD=5.\)
Point \(P\) is at \(x=3\); let \(B=(3,h)\) since \(BP\perp AD.\)
In a rhombus, \(|AB|=|AD|\Rightarrow\sqrt{(3-0)^2+h^2}=5\Rightarrow9+h^2=25\Rightarrow h=4.\)
Height \(BP=4\); base \(AD=5.\)
Area \(=AD\times BP=5\times4=20.\)
Answer: (D)
Common Mistakes:
- Treating the rhombus as a rectangle or using diagonal formulas unnecessarily.
- Forgetting that area = base × perpendicular height.
Combinatorics Example – Problem 12
Question:
A pair of fair 6-sided dice is rolled \(n\) times. What is the least \(n\) such that the probability the sum equals 7 at least once is greater than \(\frac{1}{2}\)?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
Solution:
For one roll, \(P(\text{sum}=7)=\frac{1}{6}.\)
Probability of no 7 in \(n\) rolls is \((\frac{5}{6})^n.\)
We need \(1-(\frac{5}{6})^n>\frac{1}{2}\iff(\frac{5}{6})^n<\frac{1}{2}.\) Step 4: Take logarithms: \(n>\frac{\ln(\frac{1}{2})}{\ln(\frac{5}{6})}\approx3.80.\)
Smallest integer \(n=4.\)
Answer:(C)
Common Mistakes:
- Using \((\frac{1}{6})^n>\frac{1}{2}\) instead of complement.
- Rounding \(3.80\) down to 3 instead of up to 4.
2022 AMC 10B Answer Key
| Question | Answer |
|---|---|
| 1 | A |
| 2 | D |
| 3 | D |
| 4 | A |
| 5 | B |
| 6 | A |
| 7 | B |
| 8 | B |
| 9 | D |
| 10 | D |
| 11 | B |
| 12 | C |
| 13 | E |
| 14 | B |
| 15 | D |
| 16 | D |
| 17 | C |
| 18 | B |
| 19 | C |
| 20 | D |
| 21 | E |
| 22 | E |
| 23 | C |
| 24 | B |
| 25 | A |
Last 10 Years AMC 10 Real Questions and Analysis
Think Academy provides in-depth breakdowns of the past decade of AMC 10 exams. Click below to explore:
- Year-by-year topic trend insights and concept distributions
- Real AMC 10 exams from the last 10 years
- Official answer keys
- Representative questions, detailed solutions, and common mistakes
| AMC 10A | AMC 10B |
|---|---|
| 2024 AMC 10A | 2024 AMC10B |
| 2023 AMC 10A | 2023 AMC10B |
| 2022 AMC 10A | 2022 AMC10B |
| 2021 AMC 10A | 2021 AMC10B |
| 2020 AMC 10A | 2020 AMC10B |
| 2019 AMC 10A | 2019 AMC10B |
| 2018 AMC 10A | 2018 AMC10B |
| 2017 AMC 10A | 2017 AMC10B |
| 2016 AMC 10A | 2016 AMC10B |
2022 AMC 10B Topic Distribution
The 2022 AMC 10B featured 25 questions to be completed in 75 minutes, emphasizing advanced problem-solving and proof-based reasoning skills.
Learn more about AMC 10 Format and Scoring Here: AMC 10 FAQ and Resources: Your Ultimate Guide
Detailed Module Analysis
| Module | Question Numbers | What It Tests (Brief) |
|---|---|---|
| Algebra (+ Arithmetic Reasoning) | 1, 4, 5, 7, 9, 10, 13, 15, 18, 21, 24 | Quadratic equations, linear and nonlinear equations, inequalities, rational expressions, and function manipulation |
| Number Theory | 6, 8, 17, 25 | Integer properties, prime factorization, divisibility, and modular arithmetic |
| Geometry | 2, 16, 20, 22 | Plane and coordinate geometry, emphasizing similarity, circle geometry, and quadrilateral properties (especially rhombi and parallelograms) |
| Counting & Probability / Combinatorics | 3, 11, 12, 14, 19, 23 | Counting principles, logical reasoning, and probability-based problem solving. |
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