AMC 8 Challenge Lab
Lesson 4: Algebraic Expressions & Equations
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
Algebraic Expressions & Equations
• 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. If 6x+5=71, what is x?��A. 16 B. 10 C. 11 D. 12 E. 15
2. If x²−13x+42=0, what is the sum of its roots?��A. 13 B. 6 C. 7 D. 42 E. 1
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. The average of 12, 20, 18 and x is 16. What is x?��A. 16 B. 14 C. 10 D. 18 E. 19
4. The arithmetic sequence starts 3, 6, 9, ... . What is the 15th term?��A. 50 B. 48 C. 42 D. 49 E. 45
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. What is 26×11−26?��A. 265 B. 312 C. 260 D. 249 E. 286
6. If 5x+2=32, what is x?��A. 11 B. 10 C. 6 D. 7 E. 5
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. If x²−13x+42=0, what is the sum of its roots?��A. 42 B. 7 C. 13 D. 6 E. 1
8. The average of 10, 21, 23 and x is 16. What is x?��A. 10 B. 18 C. 14 D. 6 E. 16
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. The arithmetic sequence starts 3, 5, 7, ... . What is the 12th term?��A. 23 B. 30 C. 27 D. 25 E. 24
10. What is 13×21−13?��A. 260 B. 286 C. 239 D. 265 E. 273
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. If 4x+7=51, what is x?��A. 11 B. 16 C. 12 D. 10 E. 15
12. If x²−13x+42=0, what is the sum of its roots?��A. 13 B. 7 C. 42 D. 6 E. 1
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. The average of 15, 22, 22 and x is 16. What is x?��A. 9 B. 16 C. 1 D. 19 E. 5
14. The arithmetic sequence starts 3, 10, 17, ... . What is the 9th term?��A. 66 B. 64 C. 52 D. 63 E. 59
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. What is 18×16−18?��A. 275 B. 254 C. 288 D. 306 E. 270
16. If 3x+4=22, what is x?��A. 11 B. 6 C. 5 D. 10 E. 7
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. If x²−13x+42=0, what is the sum of its roots?��A. 7 B. 6 C. 13 D. 1 E. 42
18. The average of 13, 14, 21 and x is 16. What is x?��A. 20 B. 12 C. 21 D. 20 E. 16
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. The arithmetic sequence starts 3, 9, 15, ... . What is the 14th term?��A. 81 B. 84 C. 87 D. 75 E. 86
20. What is 23×11−23?��A. 219 B. 230 C. 235 D. 253 E. 276
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. If 2x+1=23, what is x?��A. 12 B. 10 C. 11 D. 16 E. 3
22. If x²−13x+42=0, what is the sum of its roots?��A. 6 B. 1 C. 42 D. 13 E. 7
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. The average of 11, 15, 20 and x is 16. What is x?��A. 14 B. 15 C. 18 D. 22 E. 16
24. The arithmetic sequence starts 3, 8, 13, ... . What is the 11th term?��A. 55 B. 53 C. 48 D. 58 E. 58
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. What is 28×21−28?��A. 588 B. 616 C. 560 D. 539 E. 565
26. If 7x+6=48, what is x?��A. 7 B. 13 C. 6 D. 11 E. 5
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. If x²−13x+42=0, what is the sum of its roots?��A. 6 B. 42 C. 1 D. 7 E. 13
28. The average of 16, 16, 19 and x is 16. What is x?��A. 18 B. 9 C. 16 D. 17 E. 13
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. The arithmetic sequence starts 3, 7, 11, ... . What is the 8th term?��A. 31 B. 35 C. 36 D. 27 E. 32
30. What is 15×16−15?��A. 255 B. 230 C. 209 D. 225 E. 240
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. If 6x+3=69, what is x?��A. 9 B. 11 C. 10 D. 16 E. 12
32. If x²−13x+42=0, what is the sum of its roots?��A. 1 B. 7 C. 42 D. 6 E. 13
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. The average of 14, 17, 18 and x is 16. What is x?��A. 11 B. 15 C. 19 D. 16 E. 20
34. The arithmetic sequence starts 3, 6, 9, ... . What is the 13th term?��A. 42 B. 39 C. 43 D. 36 E. 44
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. What is 20×11−20?��A. 240 B. 189 C. 205 D. 200 E. 220
36. If 5x+8=38, what is x?��A. 5 B. 7 C. 6 D. 13 E. 11
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. If x²−13x+42=0, what is the sum of its roots?��A. 6 B. 7 C. 42 D. 1 E. 13
38. The average of 12, 18, 23 and x is 16. What is x?��A. 15 B. 16 C. 17 D. 7 E. 11
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. The arithmetic sequence starts 3, 5, 7, ... . What is the 10th term?��A. 26 B. 20 C. 23 D. 19 E. 21
40. What is 25×21−25?��A. 550 B. 500 C. 505 D. 525 E. 479
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. If 4x+5=49, what is x?��A. 16 B. 9 C. 10 D. 12 E. 11
42. If x²−13x+42=0, what is the sum of its roots?��A. 7 B. 13 C. 6 D. 1 E. 42
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. The average of 10, 19, 22 and x is 16. What is x?��A. 13 B. 9 C. 16 D. 18 E. 17
44. The arithmetic sequence starts 3, 10, 17, ... . What is the 15th term?��A. 101 B. 106 C. 94 D. 108 E. 105
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. What is 12×16−12?��A. 192 B. 185 C. 204 D. 180 E. 164
46. If 3x+2=20, what is x?��A. 6 B. 10 C. 11 D. 7 E. 5
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. If x²−13x+42=0, what is the sum of its roots?��A. 1 B. 6 C. 13 D. 7 E. 42
48. The average of 15, 20, 21 and x is 16. What is x?��A. 16 B. 12 C. 18 D. 4 E. 8
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. The arithmetic sequence starts 3, 9, 15, ... . What is the 12th term?��A. 74 B. 72 C. 75 D. 69 E. 63
50. What is 17×11−17?��A. 159 B. 175 C. 204 D. 170 E. 187
Answer Key
Questions 1-50
1. C�2. A�3. B�4. E�5. C�6. C�7. C�8. A�9. D�10. A
11. A�12. A�13. E�14. E�15. E�16. B�17. C�18. E�19. A�20. B
21. C�22. D�23. C�24. B�25. C�26. C�27. E�28. E�29. A�30. D
31. B�32. E�33. B�34. B�35. D�36. C�37. E�38. E�39. E�40. B
41. E�42. B�43. A�44. A�45. D�46. A�47. C�48. E�49. D�50. D
Solutions 1-2
Check the method, not just the letter.
1. Answer: C. 11��Subtract 5, then divide by 6: x=(71−5)/6=11.
2. Answer: A. 13��For x²−Sx+P=0, the sum of roots is S=13.
Solutions 3-4
Check the method, not just the letter.
3. Answer: B. 14��The total must be 4×16=64. Subtract the known sum 50 to get x=14.
4. Answer: E. 45��a_n=a_1+(n−1)d=3+(15−1)(3)=45.
Solutions 5-6
Check the method, not just the letter.
5. Answer: C. 260��Factor 26: 26(11−1)=26×10=260.
6. Answer: C. 6��Subtract 2, then divide by 5: x=(32−2)/5=6.
Solutions 7-8
Check the method, not just the letter.
7. Answer: C. 13��For x²−Sx+P=0, the sum of roots is S=13.
8. Answer: A. 10��The total must be 4×16=64. Subtract the known sum 54 to get x=10.
Solutions 9-10
Check the method, not just the letter.
9. Answer: D. 25��a_n=a_1+(n−1)d=3+(12−1)(2)=25.
10. Answer: A. 260��Factor 13: 13(21−1)=13×20=260.
Solutions 11-12
Check the method, not just the letter.
11. Answer: A. 11��Subtract 7, then divide by 4: x=(51−7)/4=11.
12. Answer: A. 13��For x²−Sx+P=0, the sum of roots is S=13.
Solutions 13-14
Check the method, not just the letter.
13. Answer: E. 5��The total must be 4×16=64. Subtract the known sum 59 to get x=5.
14. Answer: E. 59��a_n=a_1+(n−1)d=3+(9−1)(7)=59.
Solutions 15-16
Check the method, not just the letter.
15. Answer: E. 270��Factor 18: 18(16−1)=18×15=270.
16. Answer: B. 6��Subtract 4, then divide by 3: x=(22−4)/3=6.
Solutions 17-18
Check the method, not just the letter.
17. Answer: C. 13��For x²−Sx+P=0, the sum of roots is S=13.
18. Answer: E. 16��The total must be 4×16=64. Subtract the known sum 48 to get x=16.
Solutions 19-20
Check the method, not just the letter.
19. Answer: A. 81��a_n=a_1+(n−1)d=3+(14−1)(6)=81.
20. Answer: B. 230��Factor 23: 23(11−1)=23×10=230.
Solutions 21-22
Check the method, not just the letter.
21. Answer: C. 11��Subtract 1, then divide by 2: x=(23−1)/2=11.
22. Answer: D. 13��For x²−Sx+P=0, the sum of roots is S=13.
Solutions 23-24
Check the method, not just the letter.
23. Answer: C. 18��The total must be 4×16=64. Subtract the known sum 46 to get x=18.
24. Answer: B. 53��a_n=a_1+(n−1)d=3+(11−1)(5)=53.
Solutions 25-26
Check the method, not just the letter.
25. Answer: C. 560��Factor 28: 28(21−1)=28×20=560.
26. Answer: C. 6��Subtract 6, then divide by 7: x=(48−6)/7=6.
Solutions 27-28
Check the method, not just the letter.
27. Answer: E. 13��For x²−Sx+P=0, the sum of roots is S=13.
28. Answer: E. 13��The total must be 4×16=64. Subtract the known sum 51 to get x=13.
Solutions 29-30
Check the method, not just the letter.
29. Answer: A. 31��a_n=a_1+(n−1)d=3+(8−1)(4)=31.
30. Answer: D. 225��Factor 15: 15(16−1)=15×15=225.
Solutions 31-32
Check the method, not just the letter.
31. Answer: B. 11��Subtract 3, then divide by 6: x=(69−3)/6=11.
32. Answer: E. 13��For x²−Sx+P=0, the sum of roots is S=13.
Solutions 33-34
Check the method, not just the letter.
33. Answer: B. 15��The total must be 4×16=64. Subtract the known sum 49 to get x=15.
34. Answer: B. 39��a_n=a_1+(n−1)d=3+(13−1)(3)=39.
Solutions 35-36
Check the method, not just the letter.
35. Answer: D. 200��Factor 20: 20(11−1)=20×10=200.
36. Answer: C. 6��Subtract 8, then divide by 5: x=(38−8)/5=6.
Solutions 37-38
Check the method, not just the letter.
37. Answer: E. 13��For x²−Sx+P=0, the sum of roots is S=13.
38. Answer: E. 11��The total must be 4×16=64. Subtract the known sum 53 to get x=11.
Solutions 39-40
Check the method, not just the letter.
39. Answer: E. 21��a_n=a_1+(n−1)d=3+(10−1)(2)=21.
40. Answer: B. 500��Factor 25: 25(21−1)=25×20=500.
Solutions 41-42
Check the method, not just the letter.
41. Answer: E. 11��Subtract 5, then divide by 4: x=(49−5)/4=11.
42. Answer: B. 13��For x²−Sx+P=0, the sum of roots is S=13.
Solutions 43-44
Check the method, not just the letter.
43. Answer: A. 13��The total must be 4×16=64. Subtract the known sum 51 to get x=13.
44. Answer: A. 101��a_n=a_1+(n−1)d=3+(15−1)(7)=101.
Solutions 45-46
Check the method, not just the letter.
45. Answer: D. 180��Factor 12: 12(16−1)=12×15=180.
46. Answer: A. 6��Subtract 2, then divide by 3: x=(20−2)/3=6.
Solutions 47-48
Check the method, not just the letter.
47. Answer: C. 13��For x²−Sx+P=0, the sum of roots is S=13.
48. Answer: E. 8��The total must be 4×16=64. Subtract the known sum 56 to get x=8.
Solutions 49-50
Check the method, not just the letter.
49. Answer: D. 69��a_n=a_1+(n−1)d=3+(12−1)(6)=69.
50. Answer: D. 170��Factor 17: 17(11−1)=17×10=170.