Average: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 2)

Let the one-way distance be d = 36 x 48 = 1728 km, chosen so both speeds divide it evenly. Time for the outward trip = d / 36 = 1728/36 = 48 hours. Time for the return trip = d / 48 = 1728/48 = 36 hours. Total distance = 2 x 1728 = 3456 km. Total time = 48 + 36 = 84 hours. Average speed = total distance / total time = 3456 / 84 = 288/7 km/h. This matches the harmonic-mean shortcut, 2 x 36 x 48 / (36+48). Check: it is less than the simple average (36+48)/2 = 42, since harmonic mean is always at most the arithmetic mean.

Let there be 9 students with an average age of 12 years. Total age of all 9 students = 9 x 12 = 108 years. One student aged 14 years is excluded. Total age of the remaining 8 students = 108 – 14 = 94 years. Average age of the remaining students = 94 / 8 = 47/4 years. Check: 47/4 x 8 = 94, which matches 94.

Let there be 13 students with an average age of 16 years. Total age of all 13 students = 13 x 16 = 208 years. One student aged 17 years is excluded. Total age of the remaining 12 students = 208 – 17 = 191 years. Average age of the remaining students = 191 / 12 = 191/12 years. Check: 191/12 x 12 = 191, which matches 191.

Let there be 8 students with an average age of 12 years. Total age of all 8 students = 8 x 12 = 96 years. One student aged 11 years is excluded. Total age of the remaining 7 students = 96 – 11 = 85 years. Average age of the remaining students = 85 / 7 = 85/7 years. Check: 85/7 x 7 = 85, which matches 85.

Let there be 8 students with an average age of 12 years. Total age of all 8 students = 8 x 12 = 96 years. One student aged 13 years is excluded. Total age of the remaining 7 students = 96 – 13 = 83 years. Average age of the remaining students = 83 / 7 = 83/7 years. Check: 83/7 x 7 = 83, which matches 83.

Let there be 11 students with an average age of 14 years. Total age of all 11 students = 11 x 14 = 154 years. One student aged 18 years is excluded. Total age of the remaining 10 students = 154 – 18 = 136 years. Average age of the remaining students = 136 / 10 = 68/5 years. Check: 68/5 x 10 = 136, which matches 136.

Let there be 11 students with an average age of 16 years. Total age of all 11 students = 11 x 16 = 176 years. One student aged 11 years is excluded. Total age of the remaining 10 students = 176 – 11 = 165 years. Average age of the remaining students = 165 / 10 = 33/2 years. Check: 33/2 x 10 = 165, which matches 165.

Let there be 15 students with an average age of 17 years. Total age of all 15 students = 15 x 17 = 255 years. One student aged 18 years is excluded. Total age of the remaining 14 students = 255 – 18 = 237 years. Average age of the remaining students = 237 / 14 = 237/14 years. Check: 237/14 x 14 = 237, which matches 237.

Let there be 13 students with an average age of 16 years. Total age of all 13 students = 13 x 16 = 208 years. One student aged 16 years is excluded. Total age of the remaining 12 students = 208 – 16 = 192 years. Average age of the remaining students = 192 / 12 = 16 years. Check: 16 x 12 = 192, which matches 192.

Let there be 7 students with an average age of 13 years. Total age of all 7 students = 7 x 13 = 91 years. One student aged 14 years is excluded. Total age of the remaining 6 students = 91 – 14 = 77 years. Average age of the remaining students = 77 / 6 = 77/6 years. Check: 77/6 x 6 = 77, which matches 77.

Let there be 12 students with an average age of 12 years. Total age of all 12 students = 12 x 12 = 144 years. One student aged 16 years is excluded. Total age of the remaining 11 students = 144 – 16 = 128 years. Average age of the remaining students = 128 / 11 = 128/11 years. Check: 128/11 x 11 = 128, which matches 128.

Let there be 6 persons whose average weight increases by 7 kg after a swap. The person who left weighed 34 kg. Because the average of all 6 persons rises by 7 kg, the total weight rises by 6 x 7 = 42 kg. New person’s weight = old member’s weight + total increase = 34 + 42 = 76 kg. Check: replacing 34 kg with 76 kg adds 42 kg to the group; 42 / 6 = 7 kg per person, matching the given increase.

Let there be 10 persons whose average weight increases by 1 kg after a swap. The person who left weighed 12 kg. Because the average of all 10 persons rises by 1 kg, the total weight rises by 10 x 1 = 10 kg. New person’s weight = old member’s weight + total increase = 12 + 10 = 22 kg. Check: replacing 12 kg with 22 kg adds 10 kg to the group; 10 / 10 = 1 kg per person, matching the given increase.

Let there be 9 persons whose average weight increases by 6 kg after a swap. The person who left weighed 70 kg. Because the average of all 9 persons rises by 6 kg, the total weight rises by 9 x 6 = 54 kg. New person’s weight = old member’s weight + total increase = 70 + 54 = 124 kg. Check: replacing 70 kg with 124 kg adds 54 kg to the group; 54 / 9 = 6 kg per person, matching the given increase.

Let there be 14 persons whose average weight increases by 5 kg after a swap. The person who left weighed 76 kg. Because the average of all 14 persons rises by 5 kg, the total weight rises by 14 x 5 = 70 kg. New person’s weight = old member’s weight + total increase = 76 + 70 = 146 kg. Check: replacing 76 kg with 146 kg adds 70 kg to the group; 70 / 14 = 5 kg per person, matching the given increase.

Let there be 5 persons whose average weight increases by 7 kg after a swap. The person who left weighed 56 kg. Because the average of all 5 persons rises by 7 kg, the total weight rises by 5 x 7 = 35 kg. New person’s weight = old member’s weight + total increase = 56 + 35 = 91 kg. Check: replacing 56 kg with 91 kg adds 35 kg to the group; 35 / 5 = 7 kg per person, matching the given increase.

Let there be 5 persons whose average weight increases by 5 kg after a swap. The person who left weighed 57 kg. Because the average of all 5 persons rises by 5 kg, the total weight rises by 5 x 5 = 25 kg. New person’s weight = old member’s weight + total increase = 57 + 25 = 82 kg. Check: replacing 57 kg with 82 kg adds 25 kg to the group; 25 / 5 = 5 kg per person, matching the given increase.

Let there be 11 persons whose average weight increases by 4 kg after a swap. The person who left weighed 41 kg. Because the average of all 11 persons rises by 4 kg, the total weight rises by 11 x 4 = 44 kg. New person’s weight = old member’s weight + total increase = 41 + 44 = 85 kg. Check: replacing 41 kg with 85 kg adds 44 kg to the group; 44 / 11 = 4 kg per person, matching the given increase.

Let there be 10 persons whose average weight increases by 2 kg after a swap. The person who left weighed 54 kg. Because the average of all 10 persons rises by 2 kg, the total weight rises by 10 x 2 = 20 kg. New person’s weight = old member’s weight + total increase = 54 + 20 = 74 kg. Check: replacing 54 kg with 74 kg adds 20 kg to the group; 20 / 10 = 2 kg per person, matching the given increase.

Let there be 14 persons whose average weight increases by 5 kg after a swap. The person who left weighed 60 kg. Because the average of all 14 persons rises by 5 kg, the total weight rises by 14 x 5 = 70 kg. New person’s weight = old member’s weight + total increase = 60 + 70 = 130 kg. Check: replacing 60 kg with 130 kg adds 70 kg to the group; 70 / 14 = 5 kg per person, matching the given increase.

Let there be 5 persons whose average weight increases by 6 kg after a swap. The person who left weighed 13 kg. Because the average of all 5 persons rises by 6 kg, the total weight rises by 5 x 6 = 30 kg. New person’s weight = old member’s weight + total increase = 13 + 30 = 43 kg. Check: replacing 13 kg with 43 kg adds 30 kg to the group; 30 / 5 = 6 kg per person, matching the given increase.

Let the trader buy 40 items for 440 rupees from the first supplier. Let the trader buy 68 items for 1972 rupees from the second supplier. Total cost = 440 + 1972 = 2412 rupees. Total items = 40 + 68 = 108. Average price per item = total cost / total items = 2412 / 108 = 67/3 rupees. Check: 67/3 x 108 = 2412, matching the total cost of 2412 rupees.

Let the trader buy 34 items for 952 rupees from the first supplier. Let the trader buy 37 items for 407 rupees from the second supplier. Total cost = 952 + 407 = 1359 rupees. Total items = 34 + 37 = 71. Average price per item = total cost / total items = 1359 / 71 = 1359/71 rupees. Check: 1359/71 x 71 = 1359, matching the total cost of 1359 rupees.

Let the trader buy 77 items for 2233 rupees from the first supplier. Let the trader buy 21 items for 357 rupees from the second supplier. Total cost = 2233 + 357 = 2590 rupees. Total items = 77 + 21 = 98. Average price per item = total cost / total items = 2590 / 98 = 185/7 rupees. Check: 185/7 x 98 = 2590, matching the total cost of 2590 rupees.

Let the trader buy 54 items for 1404 rupees from the first supplier. Let the trader buy 27 items for 378 rupees from the second supplier. Total cost = 1404 + 378 = 1782 rupees. Total items = 54 + 27 = 81. Average price per item = total cost / total items = 1782 / 81 = 22 rupees. Check: 22 x 81 = 1782, matching the total cost of 1782 rupees.