AMC 8 Challenge Lab
Lesson 14: Multi-Step AMC 8 Problem Solving
Recognize • Model • Solve • Verify
50 original AMC 8-style multiple-choice problems with complete English solutions.
AMC 8 Strategy
Build speed without sacrificing accuracy.
1. Read for the exact target and hidden restrictions.
2. Translate words into a diagram, equation, table, or smaller case.
3. Estimate before calculating; eliminate impossible choices.
4. Look for shortcuts: symmetry, factoring, parity, ratios, or patterns.
5. Verify the result and make sure it answers the question asked.
Lesson Focus
Multi-Step AMC 8 Problem Solving
• Concept recognition
• Efficient contest strategy
• Mental-math checks
• Multiple representations
• Answer-choice verification
Practice 1-2
Choose the best answer. Solutions begin after Question 50.
1. What is 20×8−20?��A. 160 B. 132 C. 140 D. 145 E. 180
2. What is 40% of 140?��A. 100 B. 56 C. 66 D. 3 E. 51
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. If 6x+5=23, what is x?��A. 4 B. 2 C. 3 D. 8 E. 11
4. What is the sum of the interior angles of a 10-gon?��A. 1260 B. 1620 C. 1440 D. 1800 E. 1445
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. A rectangle is 10 units by 9 units. What is its area?��A. 100 B. 81 C. 90 D. 38 E. 19
6. There are 6 shirt choices, 7 pants choices and 3 hat choices. How many outfits are possible?��A. 16 B. 42 C. 129 D. 126 E. 123
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. A bag has 4 red and 8 blue marbles. What is the probability of choosing a red marble?��A. 8/12 B. 4/13 C. 4/12 D. 4/8 E. 1/12
8. The average of 10, 17, 21 and x is 16. What is x?��A. 12 B. 21 C. 16 D. 20 E. 20
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. The arithmetic sequence starts 3, 9, 15, ... . What is the 10th term?��A. 60 B. 62 C. 57 D. 51 E. 63
10. What is the remainder when 18×15 is divided by 7?��A. 6 B. 4 C. 9 D. 3 E. 5
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. A right triangle has legs 3 and 4. What is its hypotenuse?��A. 7 B. 1 C. 5 D. 6 E. 10
12. What is 13×15−13?��A. 167 B. 187 C. 182 D. 195 E. 208
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. What is 15% of 200?��A. 25 B. 40 C. 185 D. 13 E. 30
14. If 5x+8=28, what is x?��A. 13 B. 4 C. 3 D. 9 E. 5
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. What is the sum of the interior angles of a 9-gon?��A. 1260 B. 1265 C. 1620 D. 1080 E. 1440
16. A rectangle is 11 units by 7 units. What is its area?��A. 70 B. 36 C. 77 D. 88 E. 18
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. There are 7 shirt choices, 6 pants choices and 2 hat choices. How many outfits are possible?��A. 86 B. 15 C. 42 D. 84 E. 82
18. A bag has 3 red and 9 blue marbles. What is the probability of choosing a red marble?��A. 9/12 B. 1/12 C. 3/12 D. 3/13 E. 3/9
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. The average of 14, 21, 20 and x is 17. What is x?��A. 18 B. 13 C. 17 D. 18 E. 9
20. The arithmetic sequence starts 4, 9, 14, ... . What is the 13th term?��A. 65 B. 59 C. 69 D. 69 E. 64
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. What is the remainder when 17×21 is divided by 11?��A. 9 B. 5 C. 10 D. 6 E. 7
22. A right triangle has legs 7 and 24. What is its hypotenuse?��A. 26 B. 31 C. 25 D. 17 E. 84
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. What is 24×7−24?��A. 192 B. 144 C. 149 D. 168 E. 137
24. What is 20% of 100?��A. 15 B. 80 C. 30 D. 5 E. 20
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. If 4x+3=23, what is x?��A. 10 B. 6 C. 5 D. 4 E. 7
26. What is the sum of the interior angles of a 8-gon?��A. 1080 B. 1260 C. 1085 D. 1440 E. 900
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. A rectangle is 12 units by 5 units. What is its area?��A. 34 B. 17 C. 72 D. 55 E. 60
28. There are 3 shirt choices, 5 pants choices and 4 hat choices. How many outfits are possible?��A. 64 B. 56 C. 12 D. 60 E. 15
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. A bag has 2 red and 3 blue marbles. What is the probability of choosing a red marble?��A. 1/5 B. 2/5 C. 2/6 D. 3/5 E. 2/3
30. The average of 11, 16, 19 and x is 18. What is x?��A. 18 B. 30 C. 15 D. 22 E. 26
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. The arithmetic sequence starts 5, 9, 13, ... . What is the 8th term?��A. 32 B. 29 C. 33 D. 37 E. 38
32. What is the remainder when 16×11 is divided by 8?��A. 2 B. 8 C. 1 D. 0 E. 5
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. A right triangle has legs 8 and 15. What is its hypotenuse?��A. 60 B. 18 C. 23 D. 17 E. 7
34. What is 17×14−17?��A. 255 B. 207 C. 226 D. 238 E. 221
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. What is 25% of 160?��A. 50 B. 40 C. 35 D. 135 E. 6
36. If 3x+6=24, what is x?��A. 9 B. 5 C. 7 D. 6 E. 8
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. What is the sum of the interior angles of a 7-gon?��A. 900 B. 905 C. 720 D. 1080 E. 1260
38. A rectangle is 13 units by 11 units. What is its area?��A. 156 B. 24 C. 132 D. 143 E. 48
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. There are 4 shirt choices, 4 pants choices and 3 hat choices. How many outfits are possible?��A. 45 B. 16 C. 11 D. 48 E. 51
40. A bag has 7 red and 4 blue marbles. What is the probability of choosing a red marble?��A. 7/12 B. 4/11 C. 7/11 D. 7/4 E. 1/11
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. The average of 15, 20, 18 and x is 19. What is x?��A. 17 B. 19 C. 27 D. 23 E. 28
42. The arithmetic sequence starts 6, 9, 12, ... . What is the 11th term?��A. 36 B. 41 C. 39 D. 33 E. 40
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. What is the remainder when 15×17 is divided by 5?��A. 1 B. 5 C. 2 D. 5 E. 0
44. A right triangle has legs 5 and 12. What is its hypotenuse?��A. 13 B. 30 C. 14 D. 17 E. 7
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. What is 28×21−28?��A. 565 B. 616 C. 588 D. 560 E. 539
46. What is 30% of 220?��A. 190 B. 66 C. 76 D. 7 E. 61
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. If 2x+1=15, what is x?��A. 3 B. 8 C. 12 D. 6 E. 7
48. What is the sum of the interior angles of a 6-gon?��A. 720 B. 900 C. 540 D. 725 E. 1080
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. A rectangle is 14 units by 9 units. What is its area?��A. 140 B. 23 C. 46 D. 126 E. 117
50. There are 5 shirt choices, 7 pants choices and 2 hat choices. How many outfits are possible?��A. 70 B. 72 C. 68 D. 35 E. 14
Answer Key
Questions 1-50
1. C�2. B�3. C�4. C�5. C�6. D�7. C�8. C�9. C�10. B
11. C�12. C�13. E�14. B�15. A�16. C�17. D�18. C�19. B�20. E
21. B�22. C�23. B�24. E�25. C�26. A�27. E�28. D�29. B�30. E
31. C�32. D�33. D�34. E�35. B�36. D�37. A�38. D�39. D�40. C
41. D�42. A�43. E�44. A�45. D�46. B�47. E�48. A�49. D�50. A
Solutions 1-2
Check the method, not just the letter.
1. Answer: C. 140��Factor 20: 20(8−1)=20×7=140.
2. Answer: B. 56��40% of 140 = 40/100 × 140 = 56.
Solutions 3-4
Check the method, not just the letter.
3. Answer: C. 3��Subtract 5, then divide by 6: x=(23−5)/6=3.
4. Answer: C. 1440��Use (n−2)180°: (10−2)180°=1440°.
Solutions 5-6
Check the method, not just the letter.
5. Answer: C. 90��Area = 10×9=90 square units.
6. Answer: D. 126��Multiplication principle: 6×7×3=126.
Solutions 7-8
Check the method, not just the letter.
7. Answer: C. 4/12��There are 12 total marbles and 4 red, so the probability is 4/12.
8. Answer: C. 16��The total must be 4×16=64. Subtract the known sum 48 to get x=16.
Solutions 9-10
Check the method, not just the letter.
9. Answer: C. 57��a_n=a_1+(n−1)d=3+(10−1)(6)=57.
10. Answer: B. 4��Reduce modulo 7: 18%7=4, 15%7=1; product remainder = 4.
Solutions 11-12
Check the method, not just the letter.
11. Answer: C. 5��By the Pythagorean theorem, c=√(3²+4²)=5.
12. Answer: C. 182��Factor 13: 13(15−1)=13×14=182.
Solutions 13-14
Check the method, not just the letter.
13. Answer: E. 30��15% of 200 = 15/100 × 200 = 30.
14. Answer: B. 4��Subtract 8, then divide by 5: x=(28−8)/5=4.
Solutions 15-16
Check the method, not just the letter.
15. Answer: A. 1260��Use (n−2)180°: (9−2)180°=1260°.
16. Answer: C. 77��Area = 11×7=77 square units.
Solutions 17-18
Check the method, not just the letter.
17. Answer: D. 84��Multiplication principle: 7×6×2=84.
18. Answer: C. 3/12��There are 12 total marbles and 3 red, so the probability is 3/12.
Solutions 19-20
Check the method, not just the letter.
19. Answer: B. 13��The total must be 4×17=68. Subtract the known sum 55 to get x=13.
20. Answer: E. 64��a_n=a_1+(n−1)d=4+(13−1)(5)=64.
Solutions 21-22
Check the method, not just the letter.
21. Answer: B. 5��Reduce modulo 11: 17%11=6, 21%11=10; product remainder = 5.
22. Answer: C. 25��By the Pythagorean theorem, c=√(7²+24²)=25.
Solutions 23-24
Check the method, not just the letter.
23. Answer: B. 144��Factor 24: 24(7−1)=24×6=144.
24. Answer: E. 20��20% of 100 = 20/100 × 100 = 20.
Solutions 25-26
Check the method, not just the letter.
25. Answer: C. 5��Subtract 3, then divide by 4: x=(23−3)/4=5.
26. Answer: A. 1080��Use (n−2)180°: (8−2)180°=1080°.
Solutions 27-28
Check the method, not just the letter.
27. Answer: E. 60��Area = 12×5=60 square units.
28. Answer: D. 60��Multiplication principle: 3×5×4=60.
Solutions 29-30
Check the method, not just the letter.
29. Answer: B. 2/5��There are 5 total marbles and 2 red, so the probability is 2/5.
30. Answer: E. 26��The total must be 4×18=72. Subtract the known sum 46 to get x=26.
Solutions 31-32
Check the method, not just the letter.
31. Answer: C. 33��a_n=a_1+(n−1)d=5+(8−1)(4)=33.
32. Answer: D. 0��Reduce modulo 8: 16%8=0, 11%8=3; product remainder = 0.
Solutions 33-34
Check the method, not just the letter.
33. Answer: D. 17��By the Pythagorean theorem, c=√(8²+15²)=17.
34. Answer: E. 221��Factor 17: 17(14−1)=17×13=221.
Solutions 35-36
Check the method, not just the letter.
35. Answer: B. 40��25% of 160 = 25/100 × 160 = 40.
36. Answer: D. 6��Subtract 6, then divide by 3: x=(24−6)/3=6.
Solutions 37-38
Check the method, not just the letter.
37. Answer: A. 900��Use (n−2)180°: (7−2)180°=900°.
38. Answer: D. 143��Area = 13×11=143 square units.
Solutions 39-40
Check the method, not just the letter.
39. Answer: D. 48��Multiplication principle: 4×4×3=48.
40. Answer: C. 7/11��There are 11 total marbles and 7 red, so the probability is 7/11.
Solutions 41-42
Check the method, not just the letter.
41. Answer: D. 23��The total must be 4×19=76. Subtract the known sum 53 to get x=23.
42. Answer: A. 36��a_n=a_1+(n−1)d=6+(11−1)(3)=36.
Solutions 43-44
Check the method, not just the letter.
43. Answer: E. 0��Reduce modulo 5: 15%5=0, 17%5=2; product remainder = 0.
44. Answer: A. 13��By the Pythagorean theorem, c=√(5²+12²)=13.
Solutions 45-46
Check the method, not just the letter.
45. Answer: D. 560��Factor 28: 28(21−1)=28×20=560.
46. Answer: B. 66��30% of 220 = 30/100 × 220 = 66.
Solutions 47-48
Check the method, not just the letter.
47. Answer: E. 7��Subtract 1, then divide by 2: x=(15−1)/2=7.
48. Answer: A. 720��Use (n−2)180°: (6−2)180°=720°.
Solutions 49-50
Check the method, not just the letter.
49. Answer: D. 126��Area = 14×9=126 square units.
50. Answer: A. 70��Multiplication principle: 5×7×2=70.