AMC 8 Challenge Lab
Lesson 5: Rates, Proportions & Word Problems
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
Rates, Proportions & Word Problems
• 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. A cyclist travels 55 miles per hour for 5 hours. How many miles does the cyclist travel?��A. 275 B. 280 C. 220 D. 330 E. 60
2. What is 20% of 160?��A. 42 B. 32 C. 8 D. 27 E. 140
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. If 3x+6=36, what is x?��A. 9 B. 12 C. 10 D. 11 E. 15
4. The average of 13, 18, 20 and x is 19. What is x?��A. 17 B. 25 C. 19 D. 21 E. 29
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. There are 7 shirt choices, 7 pants choices and 2 hat choices. How many outfits are possible?��A. 98 B. 49 C. 96 D. 16 E. 100
6. A cyclist travels 30 miles per hour for 2 hours. How many miles does the cyclist travel?��A. 60 B. 30 C. 32 D. 62 E. 90
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. What is 20% of 100?��A. 20 B. 5 C. 15 D. 80 E. 30
8. If 2x+3=13, what is x?��A. 4 B. 9 C. 5 D. 10 E. 6
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. The average of 11, 19, 19 and x is 19. What is x?��A. 27 B. 16 C. 19 D. 31 E. 23
10. There are 7 shirt choices, 4 pants choices and 4 hat choices. How many outfits are possible?��A. 108 B. 116 C. 112 D. 15 E. 28
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. A cyclist travels 55 miles per hour for 3 hours. How many miles does the cyclist travel?��A. 168 B. 58 C. 165 D. 110 E. 220
12. What is 20% of 200?��A. 10 B. 180 C. 50 D. 35 E. 40
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. If 7x+8=78, what is x?��A. 11 B. 10 C. 9 D. 15 E. 15
14. The average of 16, 20, 18 and x is 19. What is x?��A. 19 B. 18 C. 27 D. 26 E. 22
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. There are 7 shirt choices, 5 pants choices and 3 hat choices. How many outfits are possible?��A. 108 B. 15 C. 102 D. 35 E. 105
16. A cyclist travels 30 miles per hour for 4 hours. How many miles does the cyclist travel?��A. 90 B. 34 C. 120 D. 150 E. 124
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. What is 20% of 140?��A. 38 B. 23 C. 28 D. 7 E. 120
18. If 6x+5=35, what is x?��A. 5 B. 4 C. 6 D. 11 E. 10
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. The average of 14, 21, 23 and x is 19. What is x?��A. 14 B. 19 C. 22 D. 23 E. 18
20. There are 7 shirt choices, 6 pants choices and 2 hat choices. How many outfits are possible?��A. 82 B. 42 C. 84 D. 15 E. 86
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. A cyclist travels 55 miles per hour for 5 hours. How many miles does the cyclist travel?��A. 220 B. 60 C. 330 D. 275 E. 280
22. What is 20% of 80?��A. 60 B. 16 C. 4 D. 26 E. 11
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. If 5x+2=52, what is x?��A. 10 B. 7 C. 15 D. 9 E. 11
24. The average of 12, 22, 22 and x is 19. What is x?��A. 19 B. 16 C. 18 D. 20 E. 24
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. There are 7 shirt choices, 7 pants choices and 4 hat choices. How many outfits are possible?��A. 49 B. 196 C. 200 D. 192 E. 18
26. A cyclist travels 30 miles per hour for 2 hours. How many miles does the cyclist travel?��A. 32 B. 90 C. 62 D. 30 E. 60
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. What is 20% of 180?��A. 36 B. 160 C. 46 D. 31 E. 9
28. If 4x+7=27, what is x?��A. 6 B. 11 C. 4 D. 10 E. 5
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. The average of 10, 14, 21 and x is 19. What is x?��A. 31 B. 27 C. 19 D. 35 E. 15
30. There are 7 shirt choices, 4 pants choices and 3 hat choices. How many outfits are possible?��A. 28 B. 87 C. 14 D. 81 E. 84
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. A cyclist travels 55 miles per hour for 3 hours. How many miles does the cyclist travel?��A. 58 B. 220 C. 110 D. 165 E. 168
32. What is 20% of 120?��A. 24 B. 6 C. 19 D. 34 E. 100
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. If 3x+4=34, what is x?��A. 10 B. 11 C. 7 D. 15 E. 9
34. The average of 15, 15, 20 and x is 19. What is x?��A. 26 B. 30 C. 19 D. 16 E. 22
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. There are 7 shirt choices, 5 pants choices and 2 hat choices. How many outfits are possible?��A. 35 B. 68 C. 70 D. 72 E. 14
36. A cyclist travels 30 miles per hour for 4 hours. How many miles does the cyclist travel?��A. 120 B. 34 C. 150 D. 124 E. 90
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. What is 20% of 220?��A. 44 B. 39 C. 11 D. 200 E. 54
38. If 2x+1=11, what is x?��A. 4 B. 6 C. 3 D. 5 E. 10
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. The average of 13, 16, 19 and x is 19. What is x?��A. 19 B. 16 C. 28 D. 24 E. 32
40. There are 7 shirt choices, 6 pants choices and 4 hat choices. How many outfits are possible?��A. 168 B. 172 C. 42 D. 17 E. 164
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. A cyclist travels 55 miles per hour for 5 hours. How many miles does the cyclist travel?��A. 220 B. 275 C. 60 D. 330 E. 280
42. What is 20% of 160?��A. 42 B. 8 C. 32 D. 140 E. 27
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. If 7x+6=76, what is x?��A. 15 B. 11 C. 10 D. 9 E. 13
44. The average of 11, 17, 18 and x is 19. What is x?��A. 15 B. 30 C. 26 D. 34 E. 19
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. There are 7 shirt choices, 7 pants choices and 3 hat choices. How many outfits are possible?��A. 17 B. 49 C. 150 D. 144 E. 147
46. A cyclist travels 30 miles per hour for 2 hours. How many miles does the cyclist travel?��A. 90 B. 62 C. 32 D. 30 E. 60
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. What is 20% of 100?��A. 15 B. 20 C. 80 D. 30 E. 5
48. If 6x+3=33, what is x?��A. 10 B. 9 C. 6 D. 4 E. 5
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. The average of 16, 18, 23 and x is 19. What is x?��A. 23 B. 15 C. 24 D. 19 E. 23
50. There are 7 shirt choices, 4 pants choices and 2 hat choices. How many outfits are possible?��A. 28 B. 56 C. 54 D. 13 E. 58
Answer Key
Questions 1-50
1. A�2. B�3. C�4. B�5. A�6. A�7. A�8. C�9. A�10. C
11. C�12. E�13. B�14. E�15. E�16. C�17. C�18. A�19. E�20. C
21. D�22. B�23. A�24. D�25. B�26. E�27. A�28. E�29. A�30. E
31. D�32. A�33. A�34. A�35. C�36. A�37. A�38. D�39. C�40. A
41. B�42. C�43. C�44. B�45. E�46. E�47. B�48. E�49. D�50. B
Solutions 1-2
Check the method, not just the letter.
1. Answer: A. 275��Distance = rate × time = 55×5=275.
2. Answer: B. 32��20% of 160 = 20/100 × 160 = 32.
Solutions 3-4
Check the method, not just the letter.
3. Answer: C. 10��Subtract 6, then divide by 3: x=(36−6)/3=10.
4. Answer: B. 25��The total must be 4×19=76. Subtract the known sum 51 to get x=25.
Solutions 5-6
Check the method, not just the letter.
5. Answer: A. 98��Multiplication principle: 7×7×2=98.
6. Answer: A. 60��Distance = rate × time = 30×2=60.
Solutions 7-8
Check the method, not just the letter.
7. Answer: A. 20��20% of 100 = 20/100 × 100 = 20.
8. Answer: C. 5��Subtract 3, then divide by 2: x=(13−3)/2=5.
Solutions 9-10
Check the method, not just the letter.
9. Answer: A. 27��The total must be 4×19=76. Subtract the known sum 49 to get x=27.
10. Answer: C. 112��Multiplication principle: 7×4×4=112.
Solutions 11-12
Check the method, not just the letter.
11. Answer: C. 165��Distance = rate × time = 55×3=165.
12. Answer: E. 40��20% of 200 = 20/100 × 200 = 40.
Solutions 13-14
Check the method, not just the letter.
13. Answer: B. 10��Subtract 8, then divide by 7: x=(78−8)/7=10.
14. Answer: E. 22��The total must be 4×19=76. Subtract the known sum 54 to get x=22.
Solutions 15-16
Check the method, not just the letter.
15. Answer: E. 105��Multiplication principle: 7×5×3=105.
16. Answer: C. 120��Distance = rate × time = 30×4=120.
Solutions 17-18
Check the method, not just the letter.
17. Answer: C. 28��20% of 140 = 20/100 × 140 = 28.
18. Answer: A. 5��Subtract 5, then divide by 6: x=(35−5)/6=5.
Solutions 19-20
Check the method, not just the letter.
19. Answer: E. 18��The total must be 4×19=76. Subtract the known sum 58 to get x=18.
20. Answer: C. 84��Multiplication principle: 7×6×2=84.
Solutions 21-22
Check the method, not just the letter.
21. Answer: D. 275��Distance = rate × time = 55×5=275.
22. Answer: B. 16��20% of 80 = 20/100 × 80 = 16.
Solutions 23-24
Check the method, not just the letter.
23. Answer: A. 10��Subtract 2, then divide by 5: x=(52−2)/5=10.
24. Answer: D. 20��The total must be 4×19=76. Subtract the known sum 56 to get x=20.
Solutions 25-26
Check the method, not just the letter.
25. Answer: B. 196��Multiplication principle: 7×7×4=196.
26. Answer: E. 60��Distance = rate × time = 30×2=60.
Solutions 27-28
Check the method, not just the letter.
27. Answer: A. 36��20% of 180 = 20/100 × 180 = 36.
28. Answer: E. 5��Subtract 7, then divide by 4: x=(27−7)/4=5.
Solutions 29-30
Check the method, not just the letter.
29. Answer: A. 31��The total must be 4×19=76. Subtract the known sum 45 to get x=31.
30. Answer: E. 84��Multiplication principle: 7×4×3=84.
Solutions 31-32
Check the method, not just the letter.
31. Answer: D. 165��Distance = rate × time = 55×3=165.
32. Answer: A. 24��20% of 120 = 20/100 × 120 = 24.
Solutions 33-34
Check the method, not just the letter.
33. Answer: A. 10��Subtract 4, then divide by 3: x=(34−4)/3=10.
34. Answer: A. 26��The total must be 4×19=76. Subtract the known sum 50 to get x=26.
Solutions 35-36
Check the method, not just the letter.
35. Answer: C. 70��Multiplication principle: 7×5×2=70.
36. Answer: A. 120��Distance = rate × time = 30×4=120.
Solutions 37-38
Check the method, not just the letter.
37. Answer: A. 44��20% of 220 = 20/100 × 220 = 44.
38. Answer: D. 5��Subtract 1, then divide by 2: x=(11−1)/2=5.
Solutions 39-40
Check the method, not just the letter.
39. Answer: C. 28��The total must be 4×19=76. Subtract the known sum 48 to get x=28.
40. Answer: A. 168��Multiplication principle: 7×6×4=168.
Solutions 41-42
Check the method, not just the letter.
41. Answer: B. 275��Distance = rate × time = 55×5=275.
42. Answer: C. 32��20% of 160 = 20/100 × 160 = 32.
Solutions 43-44
Check the method, not just the letter.
43. Answer: C. 10��Subtract 6, then divide by 7: x=(76−6)/7=10.
44. Answer: B. 30��The total must be 4×19=76. Subtract the known sum 46 to get x=30.
Solutions 45-46
Check the method, not just the letter.
45. Answer: E. 147��Multiplication principle: 7×7×3=147.
46. Answer: E. 60��Distance = rate × time = 30×2=60.
Solutions 47-48
Check the method, not just the letter.
47. Answer: B. 20��20% of 100 = 20/100 × 100 = 20.
48. Answer: E. 5��Subtract 3, then divide by 6: x=(33−3)/6=5.
Solutions 49-50
Check the method, not just the letter.
49. Answer: D. 19��The total must be 4×19=76. Subtract the known sum 57 to get x=19.
50. Answer: B. 56��Multiplication principle: 7×4×2=56.