Average: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 3)

Let the trader buy 58 items for 1044 rupees from the first supplier. Let the trader buy 34 items for 748 rupees from the second supplier. Total cost = 1044 + 748 = 1792 rupees. Total items = 58 + 34 = 92. Average price per item = total cost / total items = 1792 / 92 = 448/23 rupees. Check: 448/23 x 92 = 1792, matching the total cost of 1792 rupees.

Let the trader buy 48 items for 1248 rupees from the first supplier. Let the trader buy 21 items for 210 rupees from the second supplier. Total cost = 1248 + 210 = 1458 rupees. Total items = 48 + 21 = 69. Average price per item = total cost / total items = 1458 / 69 = 486/23 rupees. Check: 486/23 x 69 = 1458, matching the total cost of 1458 rupees.

Let the trader buy 49 items for 1127 rupees from the first supplier. Let the trader buy 24 items for 456 rupees from the second supplier. Total cost = 1127 + 456 = 1583 rupees. Total items = 49 + 24 = 73. Average price per item = total cost / total items = 1583 / 73 = 1583/73 rupees. Check: 1583/73 x 73 = 1583, matching the total cost of 1583 rupees.

Let the trader buy 54 items for 1296 rupees from the first supplier. Let the trader buy 62 items for 1860 rupees from the second supplier. Total cost = 1296 + 1860 = 3156 rupees. Total items = 54 + 62 = 116. Average price per item = total cost / total items = 3156 / 116 = 789/29 rupees. Check: 789/29 x 116 = 3156, matching the total cost of 3156 rupees.

Let the trader buy 32 items for 896 rupees from the first supplier. Let the trader buy 26 items for 780 rupees from the second supplier. Total cost = 896 + 780 = 1676 rupees. Total items = 32 + 26 = 58. Average price per item = total cost / total items = 1676 / 58 = 838/29 rupees. Check: 838/29 x 58 = 1676, matching the total cost of 1676 rupees.

Let the trader buy 78 items for 1404 rupees from the first supplier. Let the trader buy 65 items for 1365 rupees from the second supplier. Total cost = 1404 + 1365 = 2769 rupees. Total items = 78 + 65 = 143. Average price per item = total cost / total items = 2769 / 143 = 213/11 rupees. Check: 213/11 x 143 = 2769, matching the total cost of 2769 rupees.

Let k = 7. The first 4 multiples of k are 7, 14, 21, 28. These form an arithmetic sequence: k, 2k, 3k, …, 4k. Sum = k x (1+2+…+4) = 7 x 4 x 5 / 2 = 70. Average = sum / count = 70 / 4 = 35/2. Check: the shortcut average = k(4+1)/2 = 7×5/2 = 35/2, matching 35/2.

Let k = 5. The first 5 multiples of k are 5, 10, 15, 20, 25. These form an arithmetic sequence: k, 2k, 3k, …, 5k. Sum = k x (1+2+…+5) = 5 x 5 x 6 / 2 = 75. Average = sum / count = 75 / 5 = 15. Check: the shortcut average = k(5+1)/2 = 5×6/2 = 15, matching 15.

Let k = 12. The first 10 multiples of k are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120. These form an arithmetic sequence: k, 2k, 3k, …, 10k. Sum = k x (1+2+…+10) = 12 x 10 x 11 / 2 = 660. Average = sum / count = 660 / 10 = 66. Check: the shortcut average = k(10+1)/2 = 12×11/2 = 66, matching 66.

Let k = 9. The first 6 multiples of k are 9, 18, 27, 36, 45, 54. These form an arithmetic sequence: k, 2k, 3k, …, 6k. Sum = k x (1+2+…+6) = 9 x 6 x 7 / 2 = 189. Average = sum / count = 189 / 6 = 63/2. Check: the shortcut average = k(6+1)/2 = 9×7/2 = 63/2, matching 63/2.

Let k = 11. The first 8 multiples of k are 11, 22, 33, 44, 55, 66, 77, 88. These form an arithmetic sequence: k, 2k, 3k, …, 8k. Sum = k x (1+2+…+8) = 11 x 8 x 9 / 2 = 396. Average = sum / count = 396 / 8 = 99/2. Check: the shortcut average = k(8+1)/2 = 11×9/2 = 99/2, matching 99/2.

Let k = 5. The first 6 multiples of k are 5, 10, 15, 20, 25, 30. These form an arithmetic sequence: k, 2k, 3k, …, 6k. Sum = k x (1+2+…+6) = 5 x 6 x 7 / 2 = 105. Average = sum / count = 105 / 6 = 35/2. Check: the shortcut average = k(6+1)/2 = 5×7/2 = 35/2, matching 35/2.

Let k = 12. The first 6 multiples of k are 12, 24, 36, 48, 60, 72. These form an arithmetic sequence: k, 2k, 3k, …, 6k. Sum = k x (1+2+…+6) = 12 x 6 x 7 / 2 = 252. Average = sum / count = 252 / 6 = 42. Check: the shortcut average = k(6+1)/2 = 12×7/2 = 42, matching 42.

Let k = 3. The first 5 multiples of k are 3, 6, 9, 12, 15. These form an arithmetic sequence: k, 2k, 3k, …, 5k. Sum = k x (1+2+…+5) = 3 x 5 x 6 / 2 = 45. Average = sum / count = 45 / 5 = 9. Check: the shortcut average = k(5+1)/2 = 3×6/2 = 9, matching 9.

Let n = 35. The numbers are 1, 2, 3, …, 35 — the first 35 natural numbers. Sum of the first n natural numbers = n(n+1)/2. Sum = 35 x 36 / 2 = 630. Average = sum / count = 630 / 35 = 18. Check: the shortcut average = (n+1)/2 = (35+1)/2 = 18, matching directly.

Let n = 17. The numbers are 2, 4, 6, …, 34 — the first 17 even natural numbers. Sum of the first n even numbers = n(n+1). Sum = 17 x 18 = 306. Average = sum / count = 306 / 17 = 18. Check: the shortcut average = n+1 = 17+1 = 18, matching directly.

Let n = 12. The numbers are 1, 2, 3, …, 12 — the first 12 natural numbers. Sum of the first n natural numbers = n(n+1)/2. Sum = 12 x 13 / 2 = 78. Average = sum / count = 78 / 12 = 13/2. Check: the shortcut average = (n+1)/2 = (12+1)/2 = 13/2, matching directly.

Let n = 38. The numbers are 1, 3, 5, …, 75 — the first 38 odd natural numbers. Sum of the first n odd numbers = n^2. Sum = 38 x 38 = 1444. Average = sum / count = 1444 / 38 = 38. Check: the shortcut average = n = 38, matching directly.

Let n = 21. The numbers are 1, 3, 5, …, 41 — the first 21 odd natural numbers. Sum of the first n odd numbers = n^2. Sum = 21 x 21 = 441. Average = sum / count = 441 / 21 = 21. Check: the shortcut average = n = 21, matching directly.

Let n = 31. The numbers are 2, 4, 6, …, 62 — the first 31 even natural numbers. Sum of the first n even numbers = n(n+1). Sum = 31 x 32 = 992. Average = sum / count = 992 / 31 = 32. Check: the shortcut average = n+1 = 31+1 = 32, matching directly.

Let n = 22. The numbers are 2, 4, 6, …, 44 — the first 22 even natural numbers. Sum of the first n even numbers = n(n+1). Sum = 22 x 23 = 506. Average = sum / count = 506 / 22 = 23. Check: the shortcut average = n+1 = 22+1 = 23, matching directly.

Let n = 38. The numbers are 2, 4, 6, …, 76 — the first 38 even natural numbers. Sum of the first n even numbers = n(n+1). Sum = 38 x 39 = 1482. Average = sum / count = 1482 / 38 = 39. Check: the shortcut average = n+1 = 38+1 = 39, matching directly.

Let n = 34. The numbers are 2, 4, 6, …, 68 — the first 34 even natural numbers. Sum of the first n even numbers = n(n+1). Sum = 34 x 35 = 1190. Average = sum / count = 1190 / 34 = 35. Check: the shortcut average = n+1 = 34+1 = 35, matching directly.

Let n = 26. The numbers are 1, 2, 3, …, 26 — the first 26 natural numbers. Sum of the first n natural numbers = n(n+1)/2. Sum = 26 x 27 / 2 = 351. Average = sum / count = 351 / 26 = 27/2. Check: the shortcut average = (n+1)/2 = (26+1)/2 = 27/2, matching directly.

Let the numbers be 174, 181, 163, 23. There are 4 numbers in the list. Average = sum of numbers / count of numbers. Sum = 541. Average = 541 / 4 = 541/4. Check: 541/4 x 4 = 541, which matches the sum 541.

Let the numbers be 162, 113, 128, 118, 80, 18. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 619. Average = 619 / 6 = 619/6. Check: 619/6 x 6 = 619, which matches the sum 619.

Let the numbers be 65, 74, 59, 116, 100. There are 5 numbers in the list. Average = sum of numbers / count of numbers. Sum = 414. Average = 414 / 5 = 414/5. Check: 414/5 x 5 = 414, which matches the sum 414.

Let the numbers be 174, 50, 59, 87, 179, 83. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 632. Average = 632 / 6 = 316/3. Check: 316/3 x 6 = 632, which matches the sum 632.

Let the numbers be 171, 84, 104, 166, 83, 36. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 644. Average = 644 / 6 = 322/3. Check: 322/3 x 6 = 644, which matches the sum 644.

Let the numbers be 110, 186, 37, 184. There are 4 numbers in the list. Average = sum of numbers / count of numbers. Sum = 517. Average = 517 / 4 = 517/4. Check: 517/4 x 4 = 517, which matches the sum 517.

Let the numbers be 197, 183, 78, 184, 17, 46. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 705. Average = 705 / 6 = 235/2. Check: 235/2 x 6 = 705, which matches the sum 705.

Let the numbers be 181, 166, 49, 178, 122, 193. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 889. Average = 889 / 6 = 889/6. Check: 889/6 x 6 = 889, which matches the sum 889.

Let the numbers be 121, 82, 49, 11. There are 4 numbers in the list. Average = sum of numbers / count of numbers. Sum = 263. Average = 263 / 4 = 263/4. Check: 263/4 x 4 = 263, which matches the sum 263.

Let the numbers be 45, 21, 180, 175. There are 4 numbers in the list. Average = sum of numbers / count of numbers. Sum = 421. Average = 421 / 4 = 421/4. Check: 421/4 x 4 = 421, which matches the sum 421.

Let the numbers be 124, 189, 65, 84, 11. There are 5 numbers in the list. Average = sum of numbers / count of numbers. Sum = 473. Average = 473 / 5 = 473/5. Check: 473/5 x 5 = 473, which matches the sum 473.

Let the numbers be 164, 198, 86, 61, 157, 182. There are 6 numbers in the list. Average = sum of numbers / count of numbers. Sum = 848. Average = 848 / 6 = 424/3. Check: 424/3 x 6 = 848, which matches the sum 848.