Pengi Editor's Note
This article covers the 2016 AMC 10B exam with detailed worked solutions for representative problems, the complete answer key, and a breakdown of which topics appeared where. Pengi's editorial team selected this piece because studying the 10B alongside the 10A gives students a more complete picture of the range of difficulty and problem types that appear in any given year. Recommended for students in grades 7–10 working through AMC 10 past exams systematically.
Source: Think Academy Blog
2016 AMC 10B Real Questions and Analysis
In this article, you'll find:
- Representative real questions from each module with detailed solutions.
- The complete 2016 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 2016 AMC 10B.
- A module-to-question mapping table highlighting the core concepts tested in each module for the 2016 AMC 10B.
Real Question and Solutions Explained
Algebra Example – Problem 2
Question:
If ( n \heartsuit m = n^3 m^2 ), what is [ \frac{2 \heartsuit 4}{4 \heartsuit 2}? ]
(A) ( \frac{1}{4} ) (B) ( \frac{1}{2} ) (C) 1 (D) 2 (E) 4
Solution:
Compute each term:
[ 2 \heartsuit 4 = 2^3 \times 4^2 = 8 \times 16 = 128, ] [ 4 \heartsuit 2 = 4^3 \times 2^2 = 64 \times 4 = 256. ]
Hence,
[ \frac{2 \heartsuit 4}{4 \heartsuit 2} = \frac{128}{256} = \frac{1}{2}. ]
Answer: (B)
Common Mistakes:
- Mixing exponents (writing ( n^2 m^3 ) instead of ( n^3 m^2 )).
- Computing ( (2/4)^3 (2/4)^2 ) instead of dividing.
- Forgetting order in the definition of the operation.
Number Theory Example – Problem 8
Question:
What is the tens digit of [ 2015^{2016} - 2017 \times 2015^{2016 - 2017}? ]
(A) 0 (B) 1 (C) 3 (D) 5 (E) 8
Solution:
Only the last two digits of ( 2015^{2016} ) matter (mod 100).
Since ( 2015 \equiv 15 \pmod{100} ), find the last two digits of ( 15^{2016} ).
Observe: [ 15^1 \equiv 15, \quad 15^2 \equiv 25, \quad 15^3 \equiv 75, \quad 15^4 \equiv 25 \pmod{100}. ] Thus for ( n \ge 2 ), ( 15^n \equiv 25 \pmod{100}. )
Hence,
[ 2015^{2016} \equiv 25 \pmod{100}. ]
Subtracting 2017 gives
[ 25 - 17 = 8 \pmod{100}. ]
So the tens digit is 0.
Answer: (A)
Common Mistakes:
- Forgetting to reduce modulo 100.
- Assuming repetition every 4 terms without checking the 15 pattern.
- Subtracting 17 incorrectly as mod 10 instead of mod 100.
Geometry Example – Problem 10
Question:
A thin piece of wood in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second, similar triangle has side length 5 inches. What is the weight of the second piece?
(A) 14.0 (B) 16.0 (C) 20.0 (D) 33.3 (E) 55.6
Solution:
For similar figures of uniform thickness and density,
[ \text{weight} \propto (\text{side length})^2. ]
Thus,
[ \frac{W_2}{W_1} = \left( \frac{5}{3} \right)^2 = \frac{25}{9}. ]
Therefore,
[ W_2 = 12 \times \frac{25}{9} = 33.\overline{3}. ]
Answer: (D)
Common Mistakes:
- Scaling by side ratio instead of area ratio.
- Cubing the ratio (confusing with 3D scaling).
- Rounding too early before comparison.
Combinatorics Example – Problem 12
Question:
Two different numbers are selected at random from {1, 2, 3, 4, 5} and multiplied. What is the probability that the product is even?
(A) 0.2 (B) 0.4 (C) 0.5 (D) 0.7 (E) 0.8
Solution:
Total pairs (unordered):
[ C(5,2) = \frac{5!}{2!,(5 - 2)!} = 10. ]
For the product to be odd, both numbers must be odd. There are 3 odd numbers {1, 3, 5}, so
[ C(3,2) = \frac{3!}{2!,(3 - 2)!} = 3. ]
Hence, even cases = 10 − 3 = 7.
Probability:
[ P = \frac{7}{10} = 0.7. ]
Answer: (D)
Common Mistakes:
- Counting ordered pairs instead of combinations.
- Forgetting the phrase "two different numbers."
- Assuming half are even by symmetry without checking.
2016 AMC 10B Answer Key
| Question | Answer |
|---|---|
| 1 | D |
| 2 | B |
| 3 | D |
| 4 | B |
| 5 | D |
| 6 | B |
| 7 | C |
| 8 | A |
| 9 | C |
| 10 | D |
| 11 | B |
| 12 | D |
| 13 | D |
| 14 | D |
| 15 | C |
| 16 | E |
| 17 | D |
| 18 | E |
| 19 | D |
| 20 | C |
| 21 | B |
| 22 | A |
| 23 | C |
| 24 | D |
| 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 AMC 10B |
| 2023 AMC 10A | 2023 AMC 10B |
| 2022 AMC 10A | 2022 AMC 10B |
| 2021 AMC 10A | 2021 AMC 10B |
| 2020 AMC 10A | 2020 AMC 10B |
| 2019 AMC 10A | 2019 AMC 10B |
| 2018 AMC 10A | 2018 AMC 10B |
| 2017 AMC 10A | 2017 AMC 10B |
| 2016 AMC 10A | 2016 AMC 10B |
2016 AMC 10B Topic Distribution
The 2016 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: AMC 10 FAQ and Resources: Your Ultimate Guide
Detailed Module Analysis
| Module | Question Numbers | What It Tests (Brief) |
|---|---|---|
| Algebra | 1–5, 7, 8, 10–13, 17, 24 | Equations, ratios, coordinate geometry |
| Number Theory | 14, 20, 23, 25 | Congruence, divisibility |
| Geometry | 6, 15, 19, 21, 22 | Triangles, solids |
| Combinatorics / Probability | 9, 16, 17, 18 | Counting and probability |
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